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Solving Problems that Include Fractions and Decimals

Introduction.

There are four important operations that you will encounter when solving problems in mathematics. The figures below indicate some of the actions in a problem that lead to different operations.

Addition and subtraction are related operations. Addition typically means to combine two or more numbers, and subtraction involves the difference , or removal, of one number from another.

Combining things, Accumulations, and Amounts of increase all pointing to Addition

Multiplication and division are also related operations. Both operations involve grouping and rates.

Combining several groups with the same size, Scaling a quantity, and Calculating Area all pointing at Multiplication

You have explored how to tell when to use which operation. Now, you will focus on identifying the operation from a word problem, and then use procedures to actually perform the operation and determine a solution to the problem.

Working with Signed Numbers

Signed numbers include integers and other rational numbers that have either a positive or a negative sign.

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Source: Badwater Elevation Sign, Complex01 and Elevation Benchmark, Jeff Kramer, Wikimedia Commons

Use the diagram below to review standard procedures for adding, subtracting, multiplying, and dividing integers.

A description of how to perform Subtraction, Addition, Division, and Multiplication

Adding and Subtracting Decimals

You have applied the rules of integers to solve word problems. Now, you will review ways to add and subtract decimals, and then use what you learn to solve problems relating to addition and subtraction of positive and negative decimals.

Click on the image below to open a base-ten model interactive in a new web browser tab or window. The interactive represents the two addends in an addition problem, or the minuend  and subtrahend in a subtraction problem. Use the manipulative to work through at least 3 problems.

  • Click on a block and drag it on top of its opposite block to remove zero pairs.
  • Click on a block and drag it to the next column to regroup.
  • Click “Next Problem” to move to the next problem when you are ready.

fractions and decimals problem solving

Need additional help for addition?

Need additional help for subtraction?

Use what you noticed in the interactive to answer the following questions.

In the original problem, 4.3 – 1.5, when you dragged a ones rod into the tenths column, it split into 10 tenths. How does that relate to the regrouping that was recorded symbolically in the image shown below?

fractions and decimals problem solving

In an addition problem, such as 6.4 + 4.8, when you regroup 10 tenths into 1 one and drag the ones rod into the ones place, how did that action appear in the regrouping that was recorded symbolically such as the regrouping shown in the image below?

fractions and decimals problem solving

Pause and Reflect

1. Why is it important to line up the decimal point when adding or subtracting decimal numbers?

2. When regrouping 1 one and 3 tenths into 13 tenths, why do you cross out the original 3 in the tenths place and replace it with 13? 

Adding and Subtracting Fractions

You have used models and algorithms to add and subtract decimals, paying special attention to the regrouping that was necessary to perform the computations. Now, you will extend the idea of regrouping to models and procedures used to add and subtract fractions, including mixed numbers.

Consider the following problem.

apples

The example below shows how Marley used fraction strips to solve this problem.

Click the image below to view additional examples, including a video with a worked-out example for you to follow.

fractions and decimals problem solving

1. How is regrouping when subtracting mixed numbers similar to regrouping when subtracting decimals?

2. When adding decimals, you regroup when the sum of the two digits in a place value that is greater than 10. When would you need to regroup as you add mixed numbers?

Multiplying and Dividing Decimals

Now that you’ve investigated addition and subtraction with decimals and fractions, let’s take a closer look at multiplication and division. You will start in this section with decimals, and then use a similar model to multiply and divide fractions and mixed numbers in the next section.

fractions and decimals problem solving

  • Write an expression that you can use to determine the amount of oil that Rachel started with.
  • How would you represent 2.2 and 2.5 as improper fractions with denominators of 10?

The interactive below uses blocks to multiply decimals. When the blocks are combined, they will form a rectangle; the area of the rectangle is the product of the two decimals or the answer to Rachel’s problem.

fractions and decimals problem solving

  • In the first activity, the first decimal is the length of the rectangle, and the second decimal is the width. Represent each decimal by dragging the appropriate blocks and moving them to the area for each decimal.
  • In the second activity, use the information from the decimals and drag the blocks to the open area to create a rectangle. You will use the green blocks to fill in the missing pieces of the rectangle.
  • Is the answer the same as what we found earlier in Anu's solution?
  • Adjust the numerators to create and represent two more multiplication problems. Record those problems on a piece of paper.

Based on what you saw in the interactive, why do you think that the product has the same number of digits to the right of the decimal as the total number of digits to the right of the decimal in the two factors ?

Multiplying and Dividing Fractions

In this section, you will look at models to represent multiplying and dividing fractions.

Multiplying Fractions

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Use the interactive below to represent the problem and graphically illustrate the product. Use the Numerator and Denominator sliders to create each fraction or mixed number. You may also need to use the Zoom in/out sliders to see the entire model.

fractions and decimals problem solving

Need additional directions?

Use the interactive to answer the following questions:

  • What are the dimensions of the shaded rectangle in the solution? Check Your Answer
  • The solid lines represent the boundaries of a rectangle with an area of 1 square unit. The dashed lines represent the boundaries of a number of equal-sized regions within this area. What fraction of 1 does each smaller rectangle represent? Check Your Answer

fractions and decimals problem solving

  • What mixed number does this rearranged figure represent? How does this compare with the product of 3 4 and 6 1 2 ? Check Your Answer

Dividing Fractions

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To solve this problem, Barbara used a fraction strip generator, which gave her the following diagram.

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  • Barbara knew this was a division problem, not a multiplication problem. How did she know that? Check Your Answer
  • Use the diagram to explain why the quotient of 6 1 2 ÷ 1 2 is 13. Check Your Answer

Use the same fraction strip generator that Barbara used to solve the problem below.

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Click the image below to open the fraction strip generator in a new web browser tab or window. Enter the key information from the problem, including the dividend and the divisor , and then use the results to answer the questions that follow.

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In the fraction diagrams, both 5 3 4 and 3 8 are marked off into eighths. Why do you think that is the case? Check Your Answer

To divide 5 3 4 by 3 8 , the number sentence beneath the diagrams shows multiplication of 5 3 4 by 8 3 , which is the reciprocal of 3 8 . Multiplying by 8 3 is the same as multiplying by 8 , and then dividing by 3 . Why do you need to multiply 5 3 4 by 8 , which is the numerator of the reciprocal? Check Your Answer

The next step in the number sentence divides the product of 5 3 4 and 8 by 3 (multiplies 5 3 4 by the fraction 8 3 ) . Why do you need to divide by 3 at this point? Check Your Answer

See the completed fraction diagram for Patrice's ornament problem.

Completed fraction diagram

1. How does the multiplication algorithm connect to the area model that you used in the first interactive?

2. How does the division algorithm connect to the fraction strip model that you used in the interactive?

You studied models that represent operations on rational numbers (fractions and decimals). You also connected those models to the standard algorithms for performing the operations.

The graphic below summarizes procedures to add, subtract, multiply, and divide decimals.

fractions and decimals problem solving

The graphic below summarizes procedures to add, subtract, multiply, or divide fractions, including mixed numbers.

fractions and decimals problem solving

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Relationship Between Fractions and Decimals

The relationship between fractions and decimals is very important to understand to develop a strong base in arithmetic. When a number is represented in form of p/q, where p and q belong to whole numbers and q is not equal to 0, it is termed as a fraction and can be converted into a decimal form by either converting the denominator to a power of 10 or by long division method.

What is the Relationship Between Fractions and Decimals?

Both  fractions  and  decimals  are just two ways to represent numbers. Fractions are written in the form of p/q, where q≠0, while in decimals, the whole number part and fractional part are connected through a decimal point, for example, 0.5. Fractions and decimals represent the relationship of part by whole. In both fractions and decimals, we represent the whole by 1. Let us look at some examples to understand the relationship between fractions and decimals. Consider a full-thin crust pizza with 6 slices. Your mother gave half of it, i.e., 3 slices then in fractional form, we write it as 1/2, and in the decimal form, we write it as 0.5.

Let us consider another example. Emma divides her garden into 12 equal parts. She grows flowers of different colors in each part of the garden. Amongst the 12 slots, she reserved 8 equal portions for red flowers, 2 portions for yellow color flowers, and 2 for blue color flowers. Let us write the portion given to flowers of each color in fraction and in decimal.

  • Red flowers are grown in the 8/12 or 0.666 part of the garden.
  • Yellow flowers are grown in the 2/12 or 0.1666 part of the garden.
  • Blue flowers are also grown in the 2/12 or 0.1666 part of the garden.

Let us look at the fraction and decimal representation given in the chart below to have more clarity about the fractions and decimals relationship.

fraction and decimal chart

Converting Fraction into Decimal

We can convert a fraction to its decimal form by the following two methods.

  • Long Division Method
  • Convert the denominator of the fraction to multiples of 10 like 10, 100, 1000, etc.

Converting Fraction to Decimal by Long Division Method

When a number is present in a fraction form i.e., p/q, to convert it into the decimal form we use the long division method. The steps for converting fractions into decimals are given below. Let us understand these steps of the long division with the help of an example.

Convert 3/8 into decimals.

Step 1: Treat the numerator digit of the fraction as a dividend and the denominator as the divisor . In this case, the numerator is lesser than the denominator. Step 2: Make the dividend greater than the numerator by placing 0 next to the digit and to the quotient . Now we have 30 as a new dividend. (30>8). Step 3: In the quotient, place decimal after 0 and start the division. Step 4: Multiply 8 with a number so that the product is less than equal to 30. 8 times 3 is 24. Here now the digit in the quotient is 3, the remainder is 6. After introducing decimal in the quotient we can attach one 0 at each step of division. Step 5: Now the new dividend is 60. Multiply 8 with a number so that the product is less than equal to 60. 8 times 7 is 56. Here now quotient is 7, the remainder is 4. Step 6: Now the new dividend is 40. Multiply 8 with a number so that the product is less than equal to 40. 8 times 5 is 40. Step 7: The final remainder is 0 and the quotient is 0.375. 3/8 = 0.375.

 fraction to decimal conversion by long division

Convert the Denominator

Another method to convert the fraction to a decimal is by converting the denominator of the fraction to powers of 10 like 10, 100, 1000, etc. Let us understand this with the steps given below. We will take an example to proceed with the given steps.

Convert 3/4 to decimals.

Step1: Think of a number by which we can easily multiply the denominator and numerator to get the power of 10. Step2: Here denominator is 4. 4 times 25 is 100. Step 3: Multiply the numerator also with the same number Step 4: By multiplying the numerator  of the fraction by 25 we get (3 × 25) = 75 Step 5: Now we have a denominator in terms of powers of 10. Step 6: 75/100 = 0.75. The decimal place of the final answer depends upon the number of trailing zeros present in the digit of the denominator.

Converting Decimal into Fraction

Every decimal number can be expressed in the form of a fraction. Steps to convert a decimal number to the fractional form are stated below:

  • Rewrite the number by ignoring the decimal point.
  • Divide the number by the power of 10 such that the number of zeros in that should be equal to the number of decimal places in the given number.
  • Simplify the fraction.

Look at this example for a deeper understanding.

Decimal form = 6.5 = 65/10 = 13/2 (Fraction form of 6.5)

Important Notes:

  • Fractions represent a  ratio  between two numbers, so they show finite value. For example, 1/3, or we can say 1 out of 3 parts.
  • Decimals can also represent infinite values along with finite values. For example, if we convert the above fraction to decimal, we get 0.33333333 and it goes on up to infinity.

Related Articles

Check out the links of interesting articles related to the relationship between fractions and decimals.

  • Long Division

Solved Examples

Example 1. For the following figure, what is the decimal representation of the orange shaded portion of the blue square?

Relationship between fractions and decimal example

It is given that, 1 out of 4 blue-colored squares is shaded orange. So, the fraction of the orange shaded portion to the blue square is 1/4. To convert it into a decimal, we need to multiply both numerator and denominator by 25, so that we will have a power of 10 in the denominator.

1/4×25/25 =25/100 =0.25

Therefore, 0.25 or 1/4 portion of the blue square is shaded orange.

Example 2. Jack bought 1000 oranges from a nearby fruit vendor but later found out that 25 of them were rotten. Can you tell the fraction and the decimal value of the rotten oranges to the total oranges bought by Jack?

Here, we have 25 rotten oranges out of 1000. So our fraction becomes 25/1000. How do we write it as a decimal? Such problems are solved by dividing the numerator by the denominator. Here, we need to divide 25 by 1000. To divide 25 by 1000, we will simply shift it by 3 decimal places on the right. Here in the numerator, we have only 2 digits (25) so here we will introduce one zero before 025. The decimal place of the final answer depends upon the number of trailing zeros present in the digit of the denominator. Thus, after division, we get: 025/1000 = 0.025 ∴ The ratio of rotten oranges to fresh oranges in decimal form is 0.025

Example 3: There are 36 fruits in a basket. 9 are mangoes and the remaining are apples. What fraction of the fruits are apples? Write your answer in decimal too.

Total number of fruits in the basket = 36 Number of mangoes = 9 Number of apples = 36 − 9 = 27 apples ∴ The fraction of apples in the basket is 27/36 and in simplest form, 3/4 Now, we have to convert 3/4 into decimal. To convert 3/4 into decimal, we multiply both numerator and denominator by 25 to get a power of 10 in the denominator.  =3/4×25/25 =75/100 =0.75 ∴ 0.75 or 3/4 fruits in the basket are apples.

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FAQs on   Relationship Between Fractions and Decimals

What is the importance of fraction and decimal.

Fractions and decimals are required when there is the precision required in the value because whole numbers can only be used for counting. For measuring, decimals and fractions are used. Decimal gives a more precise value as compared to fractions as we can even represent the infinite numbers using decimals, which is not possible with fractions.

What is the Difference Between the Fraction and Decimals?

The main difference between the fraction and decimals is that fraction represents a ratio between two, while decimals can also be used for writing infinite values and for more precision. Fractions are of the form p/q, where p and q are whole numbers and q is not equal to 0, while decimals are written with the help of a decimal point in between to separate the whole numbers part and the fractional part of a number, for example, 2.89 is a decimal number.

How are Decimal and Fractions Represented?

Fractions are written in the form of p/q, where q≠0, while in decimals, the whole number part and fractional part is connected through a decimal point, for example, 0.5.

How to Convert Decimals into Fractions?

To convert a decimal to a fraction, we follow three basic steps mentioned below:

  • Rewrite the number by ignoring the decimal point
  • Divide the number by the place value of the last digit in the fractional part of the number
  • Simplify the fraction

What is 22/7 in Decimals?

22/7 is an irrational number, which is expressed as 3.14 in decimals up to two decimal places. It is a non-terminating and non-repeating decimal number that goes up to infinity.

What are the Ways to Convert Fraction to Decimal?

There are two ways to convert a fraction into a decimal, which are given below:

  • Long division method
  • Conversion of the denominator to powers of 10

How to Convert a Mixed Fraction to Decimal?

To convert a mixed fraction to decimal, we first need to convert it into an improper fraction. Then we can divide the numerator by denominator to convert it into decimal.

Decimals, Fractions and Percentages

Decimals, Fractions and Percentages are just different ways of showing the same value:

Here, have a play with it yourself:

Example Values

Here is a table of commonly used values shown in Percent, Decimal and Fraction form:

Conversions!

From percent to decimal.

To convert from percent to decimal divide by 100 and remove the % sign.

An easy way to divide by 100 is to move the decimal point 2 places to the left :

Don't forget to remove the % sign!

From Decimal to Percent

To convert from decimal to percent multiply by 100%

An easy way to multiply by 100 is to move the decimal point 2 places to the right :

 Don't forget to add the % sign!

From Fraction to Decimal

To convert a fraction to a decimal divide the top number by the bottom number:

Example: Convert 2 5 to a decimal

Divide 2 by 5: 2 ÷ 5 = 0.4

Answer: 2 5 = 0.4

From Decimal to Fraction

To convert a decimal to a fraction needs a little more work.

Example: To convert 0.75 to a fraction

From fraction to percentage.

To convert a fraction to a percentage divide the top number by the bottom number, then multiply the result by 100%

Example: Convert 3 8 to a percentage

First divide 3 by 8: 3 ÷ 8 = 0.375

Then multiply by 100%: 0.375 × 100% = 37.5%

Answer: 3 8 = 37.5%

From Percentage to Fraction

To convert a percentage to a fraction , first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above).

Example: To convert 80% to a fraction

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Fractions, Decimals & Percentages

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Decimal Fractions Questions

Decimal fractions questions are offered here to help students comprehend the concept and achieve high results on their final exams. As each question comes with a detailed explanation, students can use these questions to improve their grades. We have also offered practice problems for pupils to help them develop their problem-solving skills. Click here for more information on decimal fractions .

Decimal Fractions Questions with Solutions

1. Find the decimal fractions among the given options:

7/6, 20/40, 144/100, 26/10, 25/30.

A fraction whose denominator is the power of 10, such as 10 1 , 10 2 , 10 3 , etc is called the decimal fraction.

Among the given options, 7/6, 20/40, and 25/30 are not decimal fractions, because the denominator of these fractions is not the powers of 10.

Whereas, 144/100 and 26/10 are decimal fractions, whose denominator is the power of 10.

2. Add the following decimal fractions: (35/10) + (12/100).

Solution: 362/100.

Given decimal fractions: (35/10) +(12/100).

As the denominators of the decimal fractions are different, take the LCM of 10 and 100.

Thus, the LCM of 10 and 100 is 100.

To make the denominator of the decimal fractions same, multiply the numerator and denominator of the decimal fraction by 10.

(35/10) + (12/100) = [(35×10) /(10×10)] + (12/100)

So, (35/10) + (12/100) = (350 + 12)/100

(35/10) + (12/100) = 362/100

Therefore, the sum of the decimal fractions (35/10) and (12/100) is 362/100.

3. How to convert the fraction 1/8 into a decimal fraction.

To convert the fraction 1/8 into a decimal fraction, follow the steps given below:

Step 1: Check the denominator of the given fraction. If it can be written in the prime factorization of either 2 or 5, it can be easily converted into a decimal fraction. So, in the given fraction, 8 is the denominator, whose prime factorization is 2 × 2 × 2.

Step 2: As 10 and 100 are not multiples of 8, we will check with the next power of 10, which is 1000.

Step 3: Now, check whether 1000 is a multiple of 8. And we found that 1000 is the multiple of 8.

Step 4: Now, multiply the numerator and denominator of 1/8 by 125, and we get the decimal fraction.

I.e. 1/8 = (1×125) / (8×125) = 125/ 1000.

Thus, the decimal fraction of 1/8 is 125/1000.

4. Which among the following is a decimal fraction: 5/15, 2/10, 7/20, 65/100.

We know that decimal fractions are fractions, such that the denominator of the fraction should be the powers of 10, such as 10, 100, 1000, and so on.

In the fractions 5/15 and 7/20, the denominator is not a power of 10. Hence, 5/15 and 7/20 are not decimal fractions.

Whereas in the fractions 2/10 and 65/100, the denominators are the powers of 10, such as 10 1 and 10 2 , respectively.

Hence, among the given fractions 2/10 and 65/100 are the decimal fractions.

5. What is the sum of the decimal fractions 25/100 and 30/100?

Given decimal fractions: 25/100 and 30/100.

As the denominators are the same in both decimal fractions, we can directly add the numerators.

(25/100) + (30/100) = (25 + 30)/100

(25/100) + (30/100) = 55/100.

Hence, the sum of the decimal fractions 25/100 and 30/100 is 55/100.

6. Subtract the decimal fraction 21/100 from 50/100.

To find: (50/100) – (21/100)

As the denominator of both decimal fractions is the same, we can directly subtract the numerator values.

(50/100) – (21/100) = (50 – 21)/100

(50/100) – (21/100) = 29/100.

Therefore, subtraction of 21/100 from 50/100 gives 29/100.

7. Find the product of the decimal fractions 15/10 and 45/100.

Like multiplying fractions, we can also perform multiplying decimal fractions.

To find the product of decimal fractions, multiply the numerator of the first decimal fraction with the numerator of the second decimal fraction, and multiply the denominator of the first decimal fraction with the denominator of the second decimal fraction.

(15/10) × (45/100) = (15 × 45) / (10 × 100)

(15/10) × (45/100) = 675 / 1000

Hence, the product of 15/10 and 45/100 is 675/1000.

8. Subtract: 51/100 – 23/10.

Given: 51/100 – 23/10

Since the denominators of the decimal fractions are different, take the LCM of 100 and 10 is 100.

Thus, the expression 51/100 – 23/10 can also be written as:

(51/100) – (23/10) = (51/100) – [(23×10) / (10×10)]

(51/100) – (23/10) = (51/100) – (230/100)

Now, the denominators of both the decimal fractions are the same, subtract the numerators.

(51/100) – (23/10) = (51 – 230) / 100

(51/100) – (23/10) = -179/100

Hence, (51/100) – (23/10) is -179/100.

9. Divide the decimal fraction 25/10 by 15/100.

Given: (25/10) ÷ (15/100)

To divide these decimal fractions, follow the below steps:

Step 1: Take the reciprocal of the second decimal fraction.

I.e., 15/100 becomes 100/15.

Step 2: Now, multiply the first decimal fraction with the reciprocal of the second decimal fraction, to get the answer.

I.e., (25/10) ÷ (15/100) = (25/10) × (100/15)

(25/10) ÷ (15/100) = (25 × 100) ÷ (10 × 15)

(25/10) ÷ (15/100) = 250/15

(25/10) ÷ (15/100) = 50/3

Therefore, the division of 25/10 by 15/100 gives 50/3.

10. Write the decimal fractions for the decimal number 0.6.

Given: Decimal number: 0.6.

To convert the given decimal number into the decimal fraction, multiply and divide the decimal number by the powers of 10.

For example, (0.6 × 10)/10 = 6/10

Similarly, (0.6 × 100)/100 = 60/100

(0.6 × 1000)/1000 = 600/1000, and so on.

Therefore, the decimal fractions of 0.6 are 6/10, 60/100, 600/1000, etc.

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Practice Questions

Answer the following question:

  • Give the equivalent decimal fractions for the decimal number 0.05.
  • Find the sum of the decimal fractions 27/10 and 49/100.
  • Divide the decimal fractions: (56/100) ÷ (24/10).

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Solving problems

14.5 solving problems, worked example 14.9: working with money.

Two friends reach into their pockets and pull out the change they have. Ahmed has \(\text{R}\,\text{2,25}\) and Sujatha has \(\text{R}\,\text{2,55}\). Who has more money? Which amount is larger?

Use your knowledge to find the answer.

We have two numbers, and we must decide which of them is bigger: \(\text{R}\,\text{2,25}\) or \(\text{R}\,\text{2,55}\). Your experience with money probably helps you understand which number is bigger, and who has more money!

Use a number line to check your answer.

It is also helpful to mark the numbers on the number line. The number \(\text{2,25}\) sits exactly in the middle of \(\text{2,2}\) and \(\text{2,3}\). Remember that \(\text{2,2} = \text{2,20}\) and \(\text{2,3} = \text{2,30}\) and between them are these ten numbers: \(\text{2,21}\) and \(\text{2,22}\) and \(\text{2,23}\) and so on, up to \(\text{2,29}\) and then \(\text{2,30}\).

\(\text{2,25}\) is in the middle of these ten divisions. You can see this on the number line below.

The number \(\text{2,55}\) sits in the middle of \(\text{2,50}\) and \(\text{2,60}\).

fractions and decimals problem solving

For two numbers on the number line, the one on the right is always greater than the one on the left. Therefore, \(\text{2,55}\) must be the bigger number. This makes sense because \(\text{R}\,\text{2,55}\) is more money than \(\text{R}\,\text{2,25}\).

Remember that the inequality symbol always opens to the larger number, like a creature eating the larger snack! It is like this:

Therefore, the correct answer is: Sujatha has more money because \(\text{R}\,\text{2,25} < \text{R}\,\text{2,55}\). (In words, the inequality means, “\(\text{R}\,\text{2,25}\) is less than \(\text{R}\,\text{2,55}\)”.)

You will see decimal numbers in calculations with different measurements. For example, when calculating the perimeter or area of a shape, the side measurements could be given in decimal notation.

fractions and decimals problem solving

The side of this regular heptagon is \(\text{9,1} \text{ mm}\). To find its perimeter, you need to add all the sides or multiply the \(\text{9,1} \text{ mm}\) side length by \(7\):

If we walk all the way around the shape we walk \(\text{9,1} \text{ mm}\) seven times, so

Worked Example 14.10 Calculating area with decimals

Consider the following rectangle.

fractions and decimals problem solving

  • Determine the perimeter of the rectangle and select the correct unit.
  • Determine the area of the rectangle and select the correct unit.

Determine the perimeter by adding all the sides.

To find the perimeter of a polygon, we add up all of the sides as if we are walking around the edge of the shape. Perimeter is a distance, so it is measured in units of length (such as centimetres, metres or kilometres).

fractions and decimals problem solving

If we start at Point J and walk around the shape, we get

The opposite sides of a rectangle are the same length. If we let one side be \(l\) and the other side be \(b\), then the perimeter is \[\begin{align} P &= l + b + l + b \\ &= 2l + 2b \end{align}\]

Determine the area by calculating the number of square centimetres taken up by the shape.

To calculate the area of the rectangle, we must work out how many square centimetres (\(\text{cm}^{2}\)) will fit in the space.

We use the formula for area of a rectangle:

Examples logo

Decimals are a fundamental concept in mathematics, representing numbers between whole numbers. Understanding decimals is crucial for students to excel in math, especially in topics like measurement, money, and fractions. This guide breaks down decimals into simple, easy-to-grasp concepts with practical examples, helping teachers convey these ideas effectively to students. Whether it’s adding, subtracting, multiplying, or dividing, decimals play a key role in everyday math applications, making this guide essential for classroom success.

What are Decimals – Definition

Decimals are numbers that contain a decimal point to represent a fraction of a whole. Unlike whole numbers, decimals allow for greater precision by dividing a number into parts smaller than one. Think of them as an extension of the place value system, where each position to the right of the decimal point represents tenths, hundredths, thousandths, and so on. For example, the decimal 0.5 signifies half of a whole, and 1.75 represents one whole plus three-quarters of another. Understanding decimals is vital for accurate measurement, financial calculations, and scientific data representation.

decimal examples

Types of Decimal Numbers

Decimal numbers are a crucial part of mathematics, representing values between whole numbers. They consist of a whole number part, a decimal point, and a fractional part.

Types of decimal numbers include:

  • Terminating Decimals: These decimals have a finite number of digits after the decimal point. For example, 0.75 has two digits after the decimal point and does not continue indefinitely.
  • Repeating Decimals: These decimals have one or more repeating digits after the decimal point. An example is 0.333…, where the 3 repeats indefinitely.
  • Non-repeating, Non-terminating Decimals: These decimals do not have a repeating pattern and continue indefinitely without terminating. An example is the decimal representation of π (pi), which is approximately 3.14159…

Each type plays a specific role in mathematics and real-world applications, helping students understand the nuances of numerical values.

Place Value in Decimals

The place value system in decimals extends the concept of whole numbers to include positions to the right of the decimal point. Each place represents a fraction of ten. For instance, in the decimal number 3.142, the 1 signifies one-tenth (0.1), the 4 represents four one-hundredths (0.04), and the 2 stands for two one-thousandths (0.002). Understanding place value is essential for performing arithmetic operations with decimals and helps students grasp the concept of magnitude and precision in numerical data.

Properties of Decimals

Decimals exhibit unique properties that make them fundamental in mathematics. Key properties include:

  • Commutative Property of Addition and Multiplication: The order of adding or multiplying decimals does not change the result.
  • For example, 0.5 + 0.3 = 0.3 + 0.5, and 0.2 * 0.4 = 0.4 * 0.2.
  • Associative Property of Addition and Multiplication: When adding or multiplying decimals, the grouping does not affect the outcome. For example, (0.5 + 0.2) + 0.3 = 0.5 + (0.2 + 0.3).
  • Distributive Property: Multiplication over addition or subtraction holds true for decimals. For instance, 0.3 * (0.5 + 0.2) = (0.3 * 0.5) + (0.3 * 0.2).

These properties ensure consistency and predictability in calculations, aiding in the teaching and understanding of mathematical concepts involving decimals.

Arithmetic Operations on Decimals

Performing arithmetic operations on decimals is fundamental in mathematics, enabling precise calculations. These operations include addition, subtraction, multiplication, and division, each requiring attention to decimal place alignment for accurate results.

  • Addition: 0.75 + 0.25 = 1.00 Adding two decimals, align the decimal points to ensure each place value is correctly summed, resulting in a whole number.
  • Subtraction: 2.5 – 0.5 = 2.00 Subtracting decimals involves aligning the decimal points and subtracting each digit in its respective place value.
  • Multiplication: 0.5 * 2 = 1.0 When multiplying, the total number of decimal places in the product equals the sum of the decimal places in the factors.
  • Division: 1.5 ÷ 0.5 = 3 Dividing decimals requires converting the divisor to a whole number by adjusting the dividend accordingly, facilitating straightforward division.
  • Mixed Operations: (0.75 + 1.25) * 2 = 4.00 Combining operations, follow the order of operations (PEMDAS/BODMAS) to achieve accurate results.
  • Subtracting Larger from Smaller: 0.5 – 1.0 = -0.5 When the minuend is smaller, the result is a negative decimal, indicating the difference.
  • Multiplying by Powers of 10: 0.25 * 100 = 25 Multiplying decimals by powers of 10 shifts the decimal point to the right, increasing the value.

Decimal to Fraction Conversion

Converting decimals to fractions involves understanding the place value of the decimal and simplifying the resulting fraction to its lowest terms.

  • Terminating Decimal to Fraction: 0.5 = 1/2 Recognizing 0.5 represents half, it directly converts to the fraction 1/2.
  • Repeating Decimal to Fraction: 0.\overline{3} = 1/3 A repeating decimal like 0.\overline{3} signifies an infinite series of 3s, equivalent to the fraction 1/3.
  • Non-Repeating Decimal: 0.25 = 1/4 The decimal 0.25, representing twenty-five hundredths, simplifies to the fraction 1/4.
  • Complex Repeating Decimal: 0.\overline{66} = 2/3 A repeating decimal such as 0.\overline{66} indicates a pattern that simplifies to 2/3 as a fraction.
  • Small Decimal Value: 0.01 = 1/100 The decimal 0.01 signifies one hundredth, directly converting to the fraction 1/100.
  • Large Decimal to Fraction: 2.75 = 11/4 A decimal with a whole number part, like 2.75, converts to a mixed number, then to an improper fraction, 11/4.
  • Decimal with Multiple Digits: 0.125 = 1/8 Understanding 0.125 represents one hundred twenty-five thousandths simplifies to the fraction 1/8.

These examples illustrate the direct relationship between decimals and fractions, emphasizing the importance of place value and simplification in conversions.

Writing Decimal Numbers in the Decimal Place Value Chart

decimal place value chart

The Decimal Place Value Chart is a tool used to understand and visualize the position and value of digits in a decimal number. It helps in breaking down the number into its constituent parts, such as tenths, hundredths, thousandths, etc., enhancing clarity and comprehension.

  • Hundredths: 8
  • Thousandths: 9 The digit 7 is in the tenths place, 8 in the hundredths, and 9 in the thousandths, illustrating the value of each decimal place.
  • Hundredths: 4 The number 3 is in the ones place, showing the whole number part, while 4 is in the hundredths place, indicating a fractional part.
  • Hundredths: 0
  • Thousandths: 0 This demonstrates how zeros in the hundredths and thousandths places indicate precision but do not add to the decimal’s value.
  • Thousandths: 3 This example highlights the importance of zeros preceding the 3 in indicating its place value as thousandths.
  • Hundredths: 4
  • Thousandths: 5 Each digit’s place value is clearly defined, showing both the whole number and the decimal parts.
  • Hundreds: 1
  • Thousandths: 1 This shows a large whole number with a very small decimal part, emphasizing the chart’s range.
  • Hundredths: 9
  • Thousandths: 9
  • Ten-thousandths: 9 Demonstrating the decimal places extending beyond thousandths, useful for high precision numbers.

Understanding the Decimal Place Value Chart

The Decimal Place Value Chart is instrumental in teaching the concept of decimals, offering a visual representation of how each digit’s position affects its value. It underscores the relationship between places to the left and right of the decimal point, facilitating a deeper understanding of decimals.

  • Identifying Place Value: Recognizing that each step to the left or right changes the digit’s value tenfold is fundamental. For example, moving from the tenths to the hundredths place divides the value by 10.
  • Comparing Decimals: Using the chart to compare two decimals, such as 0.5 and 0.05, illustrates the importance of place value in determining which is larger.
  • Adding and Subtracting Decimals: Aligning decimals on the chart ensures that digits are correctly lined up by their place value for accurate addition or subtraction.
  • Multiplying Decimals: Understanding how multiplication affects the place value and the overall number of digits in the product.
  • Dividing Decimals: Visualizing the shift in place value as decimals are divided, highlighting how division impacts the decimal’s position and value.
  • Rounding Decimals: The chart helps in determining which digit to look at when rounding to a certain place value, making it easier to understand the rounding process.
  • Converting Decimals to Fractions: By identifying the place value of the last digit in a decimal, one can convert it to an equivalent fraction accurately.

This comprehensive guide outlines the significance and utility of the Decimal Place Value Chart in teaching and understanding decimals, providing a solid foundation for arithmetic operations and conceptual clarity.

Decimal Problems

Decimal problems involve scenarios where operations on decimal numbers are required to find solutions. These can range from basic arithmetic to more complex applications in real-world contexts, helping students to understand decimals’ practicality and develop their numerical skills.

  • Calculating Money: Problem: If you have $5.75 and you spend $2.45, how much money do you have left? Explanation: This problem teaches subtraction with decimals, reflecting a common real-life scenario.
  • Measuring Lengths: Problem: A rope is 3.5 meters long. If you cut off 1.25 meters, how long is the remaining piece? Explanation: This requires subtraction and introduces decimals in the context of measurement.
  • Multiplying Quantities: Problem: If one liter of paint covers 2.5 square meters, how much area would 3 liters cover? Explanation: This illustrates multiplication with decimals, applying it to calculate area coverage.
  • Dividing Portions: Problem: A pie is cut into 8 equal slices. If each slice is 0.125 of the whole pie, how many slices do you get from half a pie? Explanation: It involves division and multiplication, teaching fractions to decimals conversion.
  • Temperature Changes: Problem: If the temperature rises from 17.5°C to 20.1°C, what is the increase? Explanation: This scenario requires subtraction and introduces decimals in a scientific context.
  • Speed Calculation: Problem: If a car travels 150.5 kilometers in 3 hours, what is its average speed in kilometers per hour? Explanation: This problem teaches division with decimals, calculating rate of speed.
  • Mixing Solutions: Problem: If you mix 250.75 ml of water with 100.25 ml of solution, what is the total volume of the mixture? Explanation: It involves addition with decimals, applying it to volume measurements.
  • How do you read decimals? Explanation: Decimals are read by stating the whole number part followed by “point” for the decimal point, and then reading each digit individually in the fractional part.
  • How do you convert fractions to decimals? Explanation: To convert a fraction to a decimal, divide the numerator by the denominator using long division or a calculator.
  • What is the difference between a terminating and a non-terminating decimal? Explanation: A terminating decimal has a finite number of digits after the decimal point, while a non-terminating decimal has an infinite number of digits without repeating patterns.
  • How do you add and subtract decimals? Explanation: To add or subtract decimals, align the decimal points and add or subtract the numbers as you would with whole numbers, ensuring digits are correctly lined up.
  • How do you multiply decimals? Explanation: Multiply the numbers ignoring the decimal points, then count the total number of digits to the right of the decimal points in the original numbers and place the decimal point in the product so that it has the same number of digits to the right.
  • How do you divide decimals? Explanation: To divide by a decimal, you can multiply both the dividend and the divisor by the same power of 10 to make the divisor a whole number, then proceed with the division as usual, placing the decimal point in the quotient accordingly.

Decimals play a vital role in mathematics and everyday life, offering precision in measurements, calculations, and financial transactions. Understanding their properties, operations, and applications enhances numerical literacy and problem-solving skills. This guide has explored various aspects of decimals, from basic concepts to real-world applications, aiming to provide a comprehensive understanding that empowers both teachers and students to navigate the world of decimals with confidence and clarity.

fractions and decimals problem solving

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PROBLEM SOLVING WITH FRACTIONS DECIMALS AND PERCENTAGES WORKSHEET

Problems with fractions.

(1)  A fruit merchant bought mangoes in bulk. He sold 5/8  of the mangoes. 1/16 of the mangoes were spoiled. 300 mangoes remained with him. How many mangoes did he buy? 

(2)  A family requires 2 1/2 liters of milk per day. How much milk would family require in a month of 31 days?  

(3)  A ream of paper weighs 12 1/2 kg.  What is the weight per quire ?

(4)   It was Richard's birthday. He distributed 6 kg of candies to his friends. If he had given 1/8  kg of candies to each friend, how many friends were there ?

(5)  Rachel bought a pizza and ate 2/5 of it. If he had given 2/3 of the remaining to his friend, what fraction of the original pizza will be remaining now ?

Answer Key :

(1)  960 mangoes

(2)   77 1/2 liter

(3)  5/8 kg

(4)   48 friends

(5)  1/5

Fraction Word Problems Mixed Operations

(1)  Linda walked 2 1/3 miles on the first day and 3 2/5   miles on the next day. How many miles did she walk in all ?                Solution

(2)   David ate 2 1/7 pizzas and he gave 1 3/14    pizzas to his mother. How many pizzas did David have initially ?

(3)   Mr. A has 3 2/3 acres of land. He gave 1 1/4 acres of land to his friend. How many acres of land does Mr. A have now ?          Solution

(4)   Lily added 3 1/3 cups of walnuts to a batch of trail mix. Later she added 1 1/3 cups of almonds. How many cups of nuts did Lily put in the trail mix in all? 

(5)   In the first hockey games of the year, Rodayo played 1 1/2 periods and 1 3/4 periods. How many periods in all did he play ?         Solution

(6)   A bag can hold 1 1/2 pounds of flour. If Mimi has 7 1/2 pounds of flour, then how many bags of flour can Mimi make ?        Solution

(7)   Jack and John went fishing Jack caught 3 3/4 kg of fish and while John  caught 2 1/5 kg of fish. What is the total weight of the fish they caught?

(8)   Amy has 3 1/2 bottles in her refrigerator. She used 3/5 bottle in the morning 1 1/4 bottle in the afternoon. How many bottles of milk does Amy have left over ?  

(9)   A tank has 82 3/4 liters of water. 24 4/5 liters of water were used and the tank was filled with another 18 3/4 liters. What is the final volume of the water in the tank ?

(10)   A trader prepared 21 1/2 liters of lemonade. At the end of the day he had 2 5/8 liters left over. How many liters of lemonade was sold by the Trader? 

Answer key :

Problems on Decimals

(1)  A chemist mixed 6.35 grams of one compound with 2.45 grams of another compound. How many grams were there in the mixture.      Solution

(2)   If the cost of a pen is $10.50, a book is $25.75 and a bag is $45.50, the  find the total cost of 2 books, 3 pens and 1 bag.         Solution

(3)    John wants to buy a bicycle that cost $ 450.75. He has saved $ 125.35. How much more money must John save in order to have enough money to buy the bicycle ?

(4)   Jennifer bought 6.5 kg of sugar. she used 3750 grams. How many kilograms of sugar were left ?

(5)   The inner radius of a pipe is 12.625 mm and the outer radius is 18.025 mm. Find the thickness of the pipe.          Solution

(6)   A copy of English book weighs 0.45 kg. What is the weight of 20 copies ?          Solution

(7)   Find the weight of 25.5 meters of copper wire in kilograms, if one meter weighs 10 grams.          Solution

(8)   Robert paid $140 for 2.8 kg of cooking oil. How much did 1 kg of the cooking oil cost ?         Solution

(9)   If $20.70 is earned in 6 hours, how much money will be earned in 5 hours ?            Solution

(10)   A pipe is 76.8 meters long. What will the greatest number of pieces of pipe each 8 meters long that can be cut from this pipe ?          Solution

Answers Key :

fractions and decimals problem solving

Problems on Percentage

(1)  In a particular store the number of TV's sold the week of Black Friday was 685. The number of TVs sold the following week was 500. TV sales the week following Black Friday were what percent less than TV sales the week of Black Friday ?

(A)  17%   (B)  27%   (C)  37%   (D)  47%

(2)  In March, a city zoo attracted 32000 visitors to its polar bear exhibit. In April, the number of visitors to the exhibit increased by 15%. How many visitors did the zoo attract to its polar bear exhibit in April ?

(A)  32150   (B)  32480   (C)  35200  (D)  36800

(3)  A charity organization collected 2140 donations last month. With the help of 50 additional volunteers, the organization collected 2690 donations this month. To the nearest tenth of a percent, what was the percent increase in the number of donations the charity organization collected ?

(A) 20.4%   (B)  20.7%    (C)  25.4%   (D)  25.7%

(4)  The discount price of a book is 20% less than the retail price. James manages to purchase the book at 30% off the discount price at the special book sale. What percent of the retail price did James pay ?

(A)  42%   (B)  48%    (C)  50%   (D)  56%

(5)  Each day, Robert eats 40% of the pistachios left in his jar at the time. At the end of the second day, 27 pistachios remain. How many pistachios were in the jar at the start of the first day ?

(A)  75   (B)  80   (C)  85  (D)  95

(6) Joanne bought a doll at a 10 percent discount off the original price of $105.82. However, she had to pay a sales tax of x% on the discounted price. If the total amount she paid for the doll was $100, what is the value of x ?

(A)  2   (B)  3   (C)  4  (D)  5

(7)  In 2010, the number of houses built in Town A was 25 percent greater than the number of houses built in Town B. If 70 houses were built in Town A during 2010, how many were built in Town B ?

(A)  56   (B)  50    (C)  48   (D)  20

(8)  Over two week span, John ate 20 pounds of chicken wings and 15 pounds of hot dogs. Kyle ate 20 percent more chicken wings and 40 percent more hot dogs. Considering only chicken wings and hot dogs, Kyle ate approximately x percent more food, by weight, than John, what is x (rounded to the nearest percent) ?

(A)  25   (B)  27    (C)  29   (D)  30

(9) Due to deforestation, researchers, expect the deer population to decline by 6 percent every year. If the current deer population is 12000, what is the approximate expected population size in 10 years from now ?

(A)  25000   (B)  48000    (C)  56000   (D)  30000

(10)  In 2000 the price of a house was $72600. By 2010 the price of the house has increased to 125598.

(A)  70%    (B)  62%    (C)  73%    (D)  90%

fractions and decimals problem solving

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Performing Fundamental Operations With Fractions and Decimals

Performing Fundamental Operations With Fractions and Decimals

The four fundamental operations are the heart of arithmetic.

If we can perform these mathematical operations with integers , we can also perform them on fractions and decimals. Many real-life world problems that whole numbers cannot address can be answered using the operations of these rational numbers .

This review gives you the most comprehensive guide to performing the four fundamental operations with fractions and decimals. Most importantly, you’ll also learn how they can be applied to solve real-life word problems. 

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Table of Contents

Part i: operations on fractions, addition and subtraction of similar fractions.

As you can recall, similar fractions have the same denominator . The rules for adding and subtracting similar fractions are the same. To add or subtract similar fractions, follow these steps:

  • Add or subtract the numerators of the given fractions and use the sum or difference as the numerator of the resulting fraction.
  • Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.
  • Reduce the answer to its lowest terms, if possible.

To summarize: To add or subtract similar fractions , you first need to add or subtract the numerator, then copy the denominator. Afterward, simplify your answer to its lowest terms.

Example 1: ⅗ + ⅕.

Step 1: Add the numerators of the given fractions and use the sum or difference as the numerator of the resulting fraction.

operations on fractions and decimals 1

We add the numerators of the given fractions (i.e., 3 and 1) and put the answer as the numerator of the resulting fraction.

Step 2: Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.

operations on fractions and decimals 2

The denominator of the given fractions is 5. Hence, we will use 5 as the denominator of the common fraction.

Step 3: Reduce the answer to its lowest terms, if possible.

⅘ is a fraction in the lowest terms. Hence, no need to simplify it further. Therefore, our final answer is ⅘.

Let’s try to answer more examples:

Example 2: Add ¼ and 2/4.

operations on fractions and decimals 3

Example 3: Subtract 3⁄21 from 10⁄21.

Note that the problem asked us to subtract 3⁄21 from 10⁄21 . This means that the minuend (the number being subtracted from) is 10⁄21 while the subtrahend (the number being subtracted) is 3⁄21.

operations on fractions and decimals 4

Example 4 : Bea ate 2⁄8 of the pie that her mother prepared. Meanwhile, Bea’s brother ate 4⁄8 of the same pie that Bea ate. What is the total fraction of the pie eaten by Bea and her brother?

We can answer this question by adding the fraction of the pie eaten by Bea and the fraction of the pie eaten by her brother. Since 2⁄8 and 4⁄8 are similar fractions, we can use our steps for adding similar fractions.

Step 1 : Add the numerators of the given fractions and use the sum as the numerator of the resulting fraction .

operations on fractions and decimals 5

Step 2 : Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.

operations on fractions and decimals 6

Step 3 : Reduce the answer to its lowest terms, if possible.

6⁄8 is not in its lowest terms yet since 6 and 8 have a Greatest Common Factor (GCF) of 2. Hence, we divide both 6 and 8 by 2.

operations on fractions and decimals 7

Therefore, Bea and her brother ate 6⁄8 or ¾ of the pie.

Transforming Dissimilar Fractions to Similar Fractions

Do you still remember what dissimilar fractions are? These are fractions with different denominators. Before adding or subtracting dissimilar fractions, you should transform them into similar fractions. But how is that possible?

These are the steps on how to transform dissimilar fractions into similar fractions:

  • Find the Least Common Multiple (LCM) of the denominators. The number that you will obtain is the Least Common Denominator (LCD). Use the LCD as the new denominator of the fractions.
  • Divide the LCM you have obtained by the denominator of the first fraction. Multiply the resulting number by the numerator. The number that you will get is the numerator for the new fraction. 
  • Apply Step 2 for the second fraction.

Example : Transform the fractions ⅗ and ⅓ into similar fractions.

Let us apply all the steps previously discussed.

Step 1 : Find the denominators’ Least Common Multiple (LCM). The number that you will obtain is the Least Common Denominator (LCD). Use the LCD as the new denominator of the fractions.

The Least Common Multiple of 5 and 3 is 15 (we colored it purple in the list). 15 will be our Least Common Denominator (LCD).

operations on fractions and decimals 8

We will use 15 as the denominator of our fractions. We leave the numerators of the fractions blank because we need to compute them in the next step.

operations on fractions and decimals 9

Step 2 : Divide the LCM you obtained by the first fraction’s denominator. Multiply the resulting number by the numerator. The number that you will obtain is the numerator for the new fraction.

Let us apply this step to ⅗.

operations on fractions and decimals 10

The LCD we have obtained is 15. We divide the LCD by the denominator of ⅗. Thus, 15 ÷ 5 = 3. Afterward, we multiply 3 by the numerator of ⅗. Hence, 3 x 3 = 9. Therefore, the new numerator is 9. 

Step 3 : Apply Step 2 for the second fraction .

We will do the same thing we performed on ⅗ for the second fraction, which is ⅓. We divide the LCD of 15 by the denominator of ⅓, which is 3. Thus, 15 ÷ 3 = 5. Afterward, we multiply 5 by the numerator of ⅓. Hence, 5 x 1 = 5. The new numerator for the second fraction is 5.

operations on fractions and decimals 11

When we transform the fractions ⅗ and ⅓ into similar fractions, we have 9⁄15 and 5⁄15.

Transforming dissimilar fractions into similar fractions is essential in adding and subtracting dissimilar fractions. This means you should master the method presented above before proceeding to the next section of this review.

Addition and Subtraction of Dissimilar Fractions

Here are the steps on how to add or subtract dissimilar fractions:

  • Change the given dissimilar fractions into similar fractions (refer to the section above for the steps on transforming dissimilar fractions to similar fractions).
  • Proceed with the steps on addition or subtraction of similar fractions.
  • Reduce the resulting fraction to its lowest terms, if possible.

Let us try the steps above for the examples below.

Example 1: What is the sum of ⅑ and ⅔?

operations on fractions and decimals 12

The LCD of 3 and 9 is 9. Hence, we used it as the new denominator of our fractions. Afterward, we performed the steps for changing dissimilar fractions into similar fractions. In Step 2, we just added the numerators: 1 + 6 = 7 and copied the denominator 9. Thus, we obtained a fraction of 7⁄9.

7⁄9 is already in its lowest terms, so there is no need to simplify it further. Hence, the final answer is 7⁄9.

Example 2: Compute for ⅓ – ¼.

operations on fractions and decimals 13

The LCD of 3 and 4 is 12. Thus, we used it as the new denominator of the fractions. Afterward, we applied the steps for transforming dissimilar fractions into similar fractions. Therefore, we obtained 4⁄12 and 3⁄12. In Step 2, we subtracted the numerators: 4 – 3 = 1 and then copied the denominator 12. Thus, we obtained a fraction of 1⁄12.

Since 1⁄12 is already in its lowest terms, there is no need to simplify it further. Therefore, the final answer is 1⁄12.

Addition and Subtraction of Mixed Numbers

You are now familiar with adding and subtracting similar or dissimilar fractions. How about mixed numbers or those combinations of a whole number and a proper fraction? Can we also add or subtract them? Of course, we can.

Here are the steps you need to follow if you are adding or subtracting mixed numbers:

  • Add or subtract the whole numbers. The resulting number is the whole number part of the sum or difference.
  • Add or subtract the proper fractions. If the given fractions are similar, add or subtract the numerators, then copy the denominator. If the given fractions are dissimilar, make the fractions similar first.
  • Combine the whole number you obtained from Step 1 and the proper fraction from Step 2 to arrive at a mixed number.
  • Reduce the proper fraction to its lowest terms, if possible.

Example: Add 1⅓ and 4⅖.

Step 1: Add the whole numbers. The resulting number is the whole number part of the sum.

The whole number parts of 1⅓ and 4⅖ are 1 and 4, respectively. Adding the whole numbers:

Therefore, 5 is the whole number part of our sum.

Step 2: Add the proper fractions. If the given fractions are similar, add the numerators, then copy the denominator. If the given fractions are dissimilar , make the fractions similar first.

The proper fractions of 1⅓ and 4⅖ are ⅓ and ⅖, respectively. These proper fractions are dissimilar,, so we must first transform them into similar fractions.

If we transform ⅓ and ⅖ into similar fractions, we will have (refer to our previous section to review how to transform dissimilar fractions into similar fractions):

Now, we add similar fractions: 

5⁄15 + 6⁄15 = 11⁄15

Step 3: Combine the whole number you obtained from Step 1 and the proper fraction you got from Step 2 to arrive at a mixed number.

The whole number that we have obtained from Step 1 is 5. Meanwhile, the proper fraction we have obtained from Step 2 is 11⁄15. Combining them, we have 5 11⁄15.

Step 4 : If possible, reduce the proper fraction to its lowest terms.

Since 11⁄15 is in its lowest terms, we do not need to simplify it.

Therefore, 1⅓ + 4⅖ = 5 11⁄15.

Multiplication of Fractions

Multiplying fractions is much easier than adding or subtracting fractions because you do not have to consider whether the fractions are similar or dissimilar. To multiply fractions, all you have to do is follow these three steps:

  • Multiply the numerators of the given fractions. The resulting number is the numerator of the product (or answer).
  • Multiply the denominators of the given fractions. The resulting number is the denominator of the product (or answer).
  • If possible, reduce the product (or answer) to its lowest terms.

We can summarize these three steps this way: Multiply numerator by numerator and then denominator by denominator. Afterward, reduce the product to its lowest terms.

Example 1 : Multiply ¾ by ⅕.

Step 1: Multiply the numerators of the given fractions. The resulting number is the numerator of the product (or answer).

operations on fractions and decimals 14

The numerators of the given fractions are 3 and 1. When we multiply them, we will obtain 3 x 1 = 3. Hence, 3 is the numerator of our resulting fraction.

Step 2: Multiply the denominators of the given fractions. The resulting number is the denominator of the product (or answer).

operations on fractions and decimals 15

Step 3: If possible, reduce the product (or answer) to its lowest terms.

3⁄20 is a fraction that is already in the lowest terms. Hence, no need to simplify it further.

Therefore, our final answer is 3⁄20.

Let us have more examples:

Example 2: Multiply 5⁄9 by 2⁄4.

operations on fractions and decimals 16

Therefore, the product is 5⁄18.

Example 3: What is ⅖ of 50?

The word “of” is a signal word for the multiplication of fractions. Hence, the question above can also be interpreted as ⅖ × 50.

But how do we multiply a fraction by a whole number or vice versa?

The answer is simple! Just put a denominator of 1 for the whole number:

Afterward, proceed with the steps on multiplying fractions.

⅖ × 50⁄1 = 100⁄5

Note that we can simplify 100⁄5 as 20⁄1.

If the denominator of a fraction is 1, it means that the fraction is equal to the whole number indicated in the numerator.

Therefore, 20⁄1 = 20

Hence, ⅖ of 50 is equal to 20.

Example 4: What is ¾ of 100?

This question can be solved using the same method we used for the previous example. Again, the word “of” is a signal word for multiplying fractions.

Let us start by putting a denominator of 1 for 100:

Multiply the numerators as well as the denominators:

           ¾ × 100⁄1 = 300⁄4

We can simplify 300⁄4 as 75⁄1, which is equal to 75.

Hence, ¾ of 100 is equal to 75.

Multiplying Fractions Through Cancellation Method.

We can make the process of multiplying fractions quicker through the cancellation method. In this method, we “cancel” numbers with common factors so we can arrive at the product already in its lowest terms.

Example 1 : Multiply 4⁄20 by ⅝.

operations on fractions and decimals 17

Using the cancellation method, the answer is 1⁄8.

Example 2 : What is 3⁄7 of 49?

operations on fractions and decimals 18

Again, the word “of” is a signal word for multiplying fractions.

Therefore, 3⁄7 of 49 is equal to 21.

Division of Fractions

We are now on the fourth mathematical operation on fractions – division. However, before we proceed to the actual process of dividing fractions, let me introduce you first to the concept of the reciprocal or multiplicative inverse of a number.

Reciprocal or Multiplicative Inverse of a Number

The reciprocal or multiplicative inverse of a fraction is the fraction that, when multiplied by the original fraction, the result is 1. This definition sounds confusing and too technical, so let me provide you with an easier way to grasp this concept.

Let’s use the fraction ⅚ as an example. The reciprocal of this fraction can be obtained by interchanging the positions of the numerator and the denominator. Therefore, the reciprocal of ⅚ is simply 6⁄5.

Easy, right? Now, can you determine the reciprocal of the following:

⅘, ⅝, and 25.

Here are the answers:

The reciprocal of ⅘ is 5⁄4.

The reciprocal of ⅝ is 8⁄5.

Meanwhile, the reciprocal of 25 is 25⁄1.

Let us go back to the definition of the reciprocal. The reciprocal or multiplicative inverse of a fraction is the fraction that, when multiplied by the original fraction, the result is 1 . This means that when you multiply a fraction by its reciprocal, the result is 1. For instance, when you multiply ⅘ by 5⁄4, you will obtain 1.

Now that you know what the reciprocal is, you are prepared to proceed with the steps to divide fractions.

How To Divide Fractions

Here are the steps you need to follow so you will be able to divide fractions:

  • Transform the second fraction (the divisor) into its reciprocal (turn the fraction upside down).
  • Multiply the first fraction by the reciprocal of the second fraction.
  • Reduce the obtained fraction to its lowest terms, if possible.

Example 1: What is ⅚ divided by 6⁄4?

operations on fractions and decimals 19

Therefore, ⅚ ÷ 6⁄4 = 5⁄9

Example 2: Divide 3⁄7 by ½.

Step 1: Transform the second fraction (the divisor) into its reciprocal (turn the fraction upside down).

The second fraction (the divisor) is ½. Its reciprocal is 2⁄1.

Step 2: Multiply the first fraction by the reciprocal of the second fraction.

3⁄7 × 2⁄1 = 6⁄7

Step 3: If possible, reduce the obtained fraction to its lowest terms.

6⁄7 is already in its lowest terms. Hence, we do not need to simplify it.

Therefore, 3⁄7 ÷ ½ = 6⁄7

Multiplication and Division of Mixed Numbers

You already learned how to perform multiplication and division on fractions. This time, let us discuss how to perform the same operations with mixed numbers.

When multiplying or dividing mixed numbers, you first have to transform the given mixed numbers into improper fractions. Afterward, proceed with the steps on multiplying or dividing fractions. 

Therefore, I suggest you review the steps on how to transform mixed numbers into improper fractions so you can multiply or divide mixed numbers with ease.

Let’s have some examples:

Example 1:  Multiply 1⅔ by ⅖.

The first thing you have to do is to transform the given mixed number into an improper fraction.

1⅔ is a mixed number. If you transform it into an improper fraction, you have 5⁄3.

Afterward, you may now proceed with multiplying 5⁄3 by ⅖.

operations on fractions and decimals 20

Lastly, we can reduce 10⁄15 to its lowest terms:

operations on fractions and decimals 21

Therefore, 1⅔ × 2⁄5 = ⅔

Example 2: Divide 8⅗ by 9.

Start by transforming the given mixed number into an improper fraction.

Now, let’s proceed to divide 43⁄5 by 9 . The reciprocal of  9 is ⅑ .

operations on fractions and decimals 22

Therefore, 8⅗ ÷ 9 = 43⁄45 

Part II: Operations on Decimals

If we can add, subtract, multiply, and divide fractions, we can perform these operations with decimal numbers. In this section, let’s discuss how to perform these mathematical operations with decimals.

Addition and Subtraction of Decimals

To add decimal numbers, follow these steps:

  • Align the decimal numbers vertically, with the decimal points lined up.
  • Add zeros at the end of some decimal numbers so the decimals will be the same length.
  • Add or subtract the digits and put the decimal point in the final answer.

Example 1:  Delly bought a pencil worth ₱8.25 and an eraser worth ₱4.105. How much is the total amount of items that Delly bought?

Solution: We can answer this problem by adding the given amounts, decimal numbers.

operations on fractions and decimals 23

To solve this problem, we started by aligning the given decimal numbers. Afterward, we added a zero at the end of 8.25 so that it would be the same length as 4.105. Lastly, we performed column addition from right to left (just like with whole numbers) and put the decimal point by bringing it down.

Therefore, 8.25 + 4.105 = 12.355

Example 2: Letty loves jogging. On Monday, she jogged a distance of 3.258 km. Meanwhile, on Tuesday, she jogged a distance of 4.15 km. What is the total distance covered by Letty on Monday and Tuesday?

We can answer this problem by adding the given distances, decimal numbers.

operations on fractions and decimals 24

Therefore, Letty covered a total distance of 7. 408 km on Monday and Tuesday.

Example 3: Berto has 2.598 liters of alcohol. He used 0.52 liters to disinfect his furniture. How many liters of alcohol were left?

We can solve this problem by subtracting 0.52 from 2.598

operations on fractions and decimals 25

Therefore, 2.078 liters of alcohol were left.

Example 4: What is the difference between 9.453 and 7.38?

operations on fractions and decimals 26

Thus, the difference between 9.453 and 7.38 is 2.073.

Multiplication of Decimals

When multiplying decimal numbers, you first have to ignore the decimal point and multiply the digits just like whole numbers. Then, put the decimal point in the answer.  The resulting number must have as many decimal places as the number of decimal places the original decimals have. 

To understand the method mentioned above, let us have some examples:

Example 1: Multiply 5.45 by 1.2

We start our calculation by ignoring the decimal point and multiplying the numbers like whole numbers.

operations on fractions and decimals 27

We have obtained 6540 from Step 1, but it is not the final answer yet. We need to put the decimal point somewhere in its digits. 

operations on fractions and decimals 28

5.45 has two digits at the right of its decimal point. Thus, it has two decimal places. Meanwhile, 1.2 has one digit at the right of its decimal point. Therefore, it has one decimal place. The total number of decimal places we now have is three (two from 5.45 and one from 1.2). Thus, the final answer must have three decimal places.

To determine where we should put our decimal point in 6540, count three digits from the right, then put the decimal point. Hence, the decimal point should be at 6.540

Thus, the answer is 6.540 or 6.54

Division of Decimals

To divide decimal numbers, you may follow these steps:

  • Move the decimal point of the divisor (the second decimal) to the right until it becomes a whole number.
  • Move the decimal point in the dividend (the first decimal) to the right in the same places you move the decimal point in the divisor.
  • Divide normally just like whole numbers using the new decimals obtained from Steps 1 and 2 and put the decimal point to the final answer.

Let us apply these steps to our example below:

Example: Divide 32.95 by 0.5

Step 1 : Move the decimal point of the divisor (the second decimal) to the right until it becomes a whole number.

operations on fractions and decimals 29

We can move one decimal place to the right of 0.5 so that it becomes a whole number (which is 5). 

Step 2 : Move the decimal point in the dividend (the first decimal) to the right in the same number of places you move the decimal point in the divisor.

operations on fractions and decimals 30

Step 3: Divide normally like whole numbers using the new decimals obtained from Steps 1 and 2 and put the decimal point to the final answer.

operations on fractions and decimals 31

Therefore, 32.95 ÷ 0.5 = 65.9

Next topic:  Percentage

Previous topic: Fractions and Decimals

Return to the main article:  The Ultimate Basic Math Reviewer

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in Civil Service Exam , College Entrance Exam , LET , NAPOLCOM Exam , NMAT , PMA Entrance Exam , Reviewers , UPCAT

Last Updated May 2, 2023 06:18 AM

fractions and decimals problem solving

Jewel Kyle Fabula

Jewel Kyle Fabula is a Bachelor of Science in Economics student at the University of the Philippines Diliman. His passion for learning mathematics developed as he competed in some mathematics competitions during his Junior High School years. He loves cats, playing video games, and listening to music.

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1: Whole Numbers, Fractions, Decimals, Percents and Problem Solving

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  • Page ID 45726
  • 1.1: Introduction to Decimal Calculations
  • 1.2: Place Value in Decimals
  • 1.3: Adding and Subtracting Decimals
  • 1.4: Multiplying and Dividing Decimals
  • 1.5: Convert Between Decimals and Fractions
  • 1.6: Introduction to Percent Calculations
  • 1.7: Solving Problems Using Ratios
  • 1.8: Writing Fractions and Decimals as Percents
  • 1.9: Solving Problems Using Percents
  • 1.10: Percent Increase and Decrease
  • 1.11: Why It Matters- Whole Numbers, Fractions, Decimals, Percents, and Problem Solving
  • 1.12: Putting It Together- Whole Numbers, Fractions, Decimals, Percents, and Problem Solving
  • 1.13: Assignment- Whole Numbers, Fractions, Decimals, Percents, and Problem Solving
  • 1.14: Introduction to Whole Number Calculations
  • 1.15: Place Value in Whole Numbers
  • 1.16: Rounding Whole Numbers
  • 1.17: Adding, Subtracting, Multiplying, and Dividing Whole Numbers
  • 1.18: Introduction to Fraction Calculations
  • 1.19: Convert Between Types of Fractions
  • 1.20: Adding and Subtracting Fractions
  • 1.21: Multiplying and Dividing Fractions

IMAGES

  1. Fractions Decimals Percents

    fractions and decimals problem solving

  2. Fractions, Decimals, and Percents Word Problems

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  3. Fractions and Decimals: Measure and Money Problem Solving Lesson 1

    fractions and decimals problem solving

  4. Problem-Solving Investigation: Equivalent fractions: add and subtract (Year 6 Decimals

    fractions and decimals problem solving

  5. Decimals to Fractions Puzzle

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  6. 4th Grade Freebie Converting Fractions and Decimals Center

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VIDEO

  1. Decimals to Fractions

  2. FRACTIONS

  3. GCSE Maths: Fractions Step-By-Step. Higher and Foundation. AQA, Edexcel, OCR

  4. WORD PROBLEMS WITH FRACTIONS AND DECIMALS / MATH TUTORIAL TAGALOG

  5. Multiplying Fractions And Decimals

  6. Word Problems

COMMENTS

  1. Solving Problems that Include Fractions and Decimals

    1. Why is it important to line up the decimal point when adding or subtracting decimal numbers? 2. When regrouping 1 one and 3 tenths into 13 tenths, why do you cross out the original 3 in the tenths place and replace it with 13?

  2. Converting fractions to decimals (practice)

    Course: 7th grade > Unit 2. Lesson 2: Converting fractions to decimals. Rewriting decimals as fractions: 2.75. Rewriting decimals as fractions challenge. Worked example: Converting a fraction (7/8) to a decimal. Fraction to decimal: 11/25. Fraction to decimal with rounding. Converting fractions to decimals.

  3. Fractions, decimals, & percentages

    Practice Rewriting decimals as fractions challenge Get 5 of 7 questions to level up! Converting fractions to decimals Get 3 of 4 questions to level up! Adding & subtracting rational numbers Learn Comparing rational numbers Adding & subtracting rational numbers: 79% - 79.1 - 58 1/10 Adding & subtracting rational numbers: 0.79 - 4/3 - 1/2 + 150%

  4. Decimals and Fractions

    To convert any decimal number to a fraction, we ignore the decimal point and divide the number by the place value of the last digit in the fractional part of the number. Then we simplify the fraction so obtained. The decimals and fractions can be conveniently interchanged to each other.

  5. Common fractions and decimals (video)

    Well, to go from four to 100, you have to multiply by 25. So let's multiply the numerator by 25 to get an equivalent fraction. So one times 25 is 25. So 1/4 is equal to 25/100 and we can represent that in decimal notation as 25/100 which we could also consider 2/10 and 5/100. Now, let's do 1/2. Same exact idea.

  6. Fractions, Decimals and Percentages

    Fractions, Decimals and Percentages - Short Problems Fractions, Decimals and Percentages - Short Problems This is part of our collection of Short Problems. You may also be interested in our longer problems on Fractions, Decimals and Percentages. Printable worksheets containing selections of these problems are available here. Mean Sequence

  7. Decimal Fraction Worksheets

    Decimal Fraction Worksheets. Addition :: Subtraction :: Multiplication :: Division :: Conversion. The following worksheets all use "Decimal Fractions", in other words tenths, hundredths, etc. This makes them a little easier to work with. But remember to simplify your answer (example: 50 100 becomes 1 2 ).

  8. Math Exercises & Math Problems: Fractions and Decimals

    Fractions and decimals. Calculate the sum, difference, product and quotient of fractions and convert a decimal number to a fraction on Math-Exercises.com.

  9. Relationship Between Fractions and Decimals

    Step 2: Make the dividend greater than the numerator by placing 0 next to the digit and to the quotient. Now we have 30 as a new dividend. (30>8). Step 3: In the quotient, place decimal after 0 and start the division. Step 4: Multiply 8 with a number so that the product is less than equal to 30. 8 times 3 is 24.

  10. Fraction

    Solution: Given, One number = 38.46 Product of two numbers = 658.17 The other number = 658.17÷38.46 =

  11. Decimals, Fractions and Percentages

    To convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above). Example: To convert 80% to a fraction. Start with: 80%. First convert to a decimal (=80/100): 0.8. Write down the decimal "over" the number 1:

  12. Problem Solving using Fractions (Definition, Types and Examples

    When we divide something into equal pieces, each part becomes a fraction of the whole. For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction \ (\frac {1} {2}\). Similarly, if it is divided into 4 equal parts, then each part is one ...

  13. Exploring Fractions

    Fractions can refer to objects, quantities or shapes, thus extending their complexity. In order to be able to develop their understanding and then generalise about fractions, children need to explore many representations and uses over a significant period of time.

  14. Rates and percentages

    7th grade Unit 2: Rates and percentages 800 possible mastery points Mastered Proficient Familiar Attempted Not started Quiz Unit test About this unit In these tutorials, we'll look at how rates and percentages relate to proportional thinking. We'll also solve interesting word problems involving percentages (discounts, taxes, and tip calculations).

  15. Decimals and Fractions Practice Questions

    Next: Percentages and Fractions Practice Questions GCSE Revision Cards. 5-a-day Workbooks

  16. Solving Decimal Word Problems

    Explain why or why not. Analysis: We must compare and order these decimals to help us solve this problem. Specifically, we need to determine if the third decimal is between the first two. Step 1: Let's start by writing one decimal beneath the other in their original order. We will place an arrow next to 1.4691 so that we can track its value.

  17. Fractions, Decimals & Percentages

    Age 7 to 14 Challenge Level A task involving the equivalence between fractions, percentages and decimals which depends on members of the group noticing the needs of others and responding. Fractions and Percentages Card Game Age 11 to 16 Challenge Level Can you find the pairs that represent the same amount of money?

  18. Math Practice Problems

    Fractions to Decimals - Sample Math Practice Problems The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program.

  19. Decimal Fractions Questions with Solutions (Complete Explanation)

    Decimal Fractions Questions are designed to help students learn and practise the concepts of decimals and fractions in maths. These questions cover various topics such as converting decimals to fractions, comparing and ordering decimals and fractions, and solving word problems involving decimals and fractions. By solving these questions, students can improve their skills and confidence in maths.

  20. Solving problems

    To calculate the area of the rectangle, we must work out how many square centimetres (\ (\text {cm}^ {2}\)) will fit in the space. We use the formula for area of a rectangle: Siyavula's open Mathematics Grade 8 textbook, chapter 14 on Decimal fractions covering Solving problems.

  21. Decimals

    Repeating Decimal to Fraction: 0.\overline{3} = 1/3 A repeating decimal like 0.\overline{3} signifies an infinite series of 3s, equivalent to the fraction 1/3. ... Understanding their properties, operations, and applications enhances numerical literacy and problem-solving skills. This guide has explored various aspects of decimals, from basic ...

  22. Problem Solving with Fractions Decimals and ...

    Problem Solving with Fractions Decimals and Percentages Worksheet. (1) A fruit merchant bought mangoes in bulk. He sold 5/8of the mangoes. 1/16 of the mangoes were spoiled. 300 mangoes remained with him. How many mangoes did he buy?

  23. Performing Fundamental Operations With Fractions and Decimals

    Performing Fundamental Operations With Fractions and Decimals The four fundamental operations are the heart of arithmetic. If we can perform these mathematical operations with integers, we can also perform them on fractions and decimals.

  24. 1: Whole Numbers, Fractions, Decimals, Percents and Problem Solving

    1.18: Introduction to Fraction Calculations. 1.19: Convert Between Types of Fractions. 1.20: Adding and Subtracting Fractions. 1.21: Multiplying and Dividing Fractions. 1: Whole Numbers, Fractions, Decimals, Percents and Problem Solving is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.