Using Equivalent Fractions to Add and Subtract Fractions

Key Concepts

• Estimate sum of two fractions
• Estimate difference of two fractions

Introduction

In this chapter, we will learn to estimate the sum and difference of two fractions.

Do you know why do you want to know how to estimate fractions?

It helps to quickly estimate the sum, difference, product, or quotient of an expression involving fractions.

Do you know when are you ever going to use this?

It helps in estimating the amount of money that you can make while doing a job.

It helps in estimating the distance between two locations that you want to visit.

It helps in approximating the amount of ingredients needed while preparing a recipe.

Benchmark Numbers:

A benchmark is nothing but, it is a number that is easy to use when you estimate. 

Sandra’s mom used to measure how much she grew each year. For each year, she marked Sandra’s height on the wall. Last year’s mark served as a benchmark to measure how much Sandra grew in a year.

We use the benchmarks 0, 1/2 , and 1 while estimating the sums and differences of any two fractions.

The following table helps you in setting the benchmark numbers:

The following image shows the estimation of 5/8 using benchmark numbers:

Let us see some more examples,

Example 1:

Estimate the fraction

Solution:

Drawing fraction models can help you decide how to round fractions.

Decide if 5/8 is closest to 0, 1/2 , or 1?

It is closest to 1/2.

Example 2:

Estimate the fraction 7/8.  

Solution:

In this example, the numerator, i.e., 7 is very close to the denominator, i.e., 8.

When the numerator and denominator are very close, we use benchmark 1.

Example 3:

Estimate the fractions 1/5, 2/11, 4/15, 5/11, 4/7, 9/20, 9/10, 13/19, and 6/7. 

Solution:

Estimate sum of two fractions

Let us learn how to estimate the sum of two fractions.

Example 1:

Gary is welding together two copper pipes to repair a leak. He will use the pipes shown. Is the new pipe closer to ½ foot or 1 foot long? Explain.

Solution:

Round each fraction to 0, 1/2 , or 1, whichever is closest as per the benchmark.

Step 1:

7/12 is closest to 1/2

4/5 is close to 1

Hence, 7/12 + 4/5 = 1/2 + 1

Step 2:

Then add the benchmarks.

1/2 + 1 = 1/2+1

Example 2:

Estimate the sum of 3/8 + 9/16 .

Solution:

Step 1:

Find 3/8 on the number line. Is 3/8 closer to 0, 1/2 , or 1?

Step 2:

Find 9/16 on the number line. Is 9/16 closer to 0, 1/2 , or 1?

Step 3:

Add to find the estimation.

1/2 + 1/2 = 1

Estimate difference of two fractions

Let us learn how to estimate the difference of two fractions.

Example 1:

Estimate the difference of 12/13 – 2/25  by replacing each fraction with 0,1/2 , or 1.

Solution:

Step 1:

Round each fraction to 0, 1/2 , or 1, whichever is closest as per the benchmark.

12/13 is closest to 1

2/25 is close to 0

Hence, 12/13 – 2/25=1 – 0

Step 2:

Then subtract the benchmarks.

1 – 0 = 1

Exercise

1. Name three fractions whose benchmark is 1/2.
2. Find 3/8 on the number line. Is 3/8 closer to 0 or 1/2?
3. Find 9/16 on the number line. Is 9/16 closer to 0 or 1/2? 
4. Estimate the sum of 2/5 + 3/4 by replacing each fraction with 0, 1/2 or 1.
5. Estimate the sum of 1/2+ 4/7 by replacing each fraction with 0,1/2 , or 1.
6. Estimate the sum of  7/15 + 6/10 by replacing each fraction with 0,1/2 , or 1.
7. Estimate the difference of 7/8 – 4/9 by replacing each fraction with 0,1/2 , or 1.
8. Estimate the difference of 7/8 – 5/6 by replacing each fraction with 0,1/2 , or 1.
9. Kelly estimated 5/8 + 1/9 by replacing 5/8 with 1 and 1/9 with 0. Her estimated sum was 1 + 0 = 1. Do you think Kelly’s estimate is accurate? Explain.
10. A recipe for strawberry lemonade calls for the following ingredients:
    1. 7/3 cup lemon juice

  2. 19/8 cup strawberry juice

  3. 5/4cup sugar

  4. 23/4 cups water

Estimate the amount of lemonade that can be made.

What have we learned

• Estimate fractions on the number line using benchmark numbers.
• Estimate the sum of two fractions.
• Estimate the difference of two fractions.

Concept Map