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Subtract Fractions with Like Denominators

Key Concepts

  • Subtract fractions using fraction strips
  • Subtract fractions using numberline
  • Subtract fractions directly.

Introduction: 

In the previous lesson, we subtracted fractions with like denominators by decomposing the first number (minuend) into unit fractions and then separating the second number (subtrahend) from that.  

For example 

Find 3/5−1/5

We decomposed 3/5 into unit fractions and removed the second fraction from it.  

In this lesson, we use the relationship between addition and subtraction to find the difference of two fractions with like denominators.  

The idea of decomposing the first fraction into two fractions can be used to find the fraction difference. 

Let us look at this example. 

Find 7/8−4/8

Here, the denominator tells us in how many parts the whole has been divided. 

Since we need to subtract 4/8, let us decompose 7/8 as a sum of 4/8+3/8. 

That is 7/8= 4/8+3/8

If 4/8  is removed, we are left with 3/8. 

7/8−4/8=3/8

We can understand that subtraction of fractions with like denominators is similar to the addition of fractions with like denominators.  

Different models like fraction strips, number lines or area figures can be used to add or subtract fractions with like denominators. The simplest way of finding the sum or difference of fractions with like denominators is either to add or subtract the numerators and copy the denominator once. 

Example 1: 

Leah and Josh live in the same direction from school and on the same side of Forest road. Leah’s house is 8/10 mile from the school. Josh’s house is 5/10 mile from the school. How much farther does Leah have to walk home when she reaches Josh’s house? 

Solution: 

Step 1: 

While solving these kinds of questions, first, we need to check what information is given to us? 

It is given that Leah’s house is 8/10 miles from school. 

Josh’s house is 5/10 miles from school. 

Step 2: 

Then we need to check what are we asked to do? 

We need to find how much farther Leah has to walk home when she reaches Josh’s house. 

Step 3: 

Then we need to analyze what operation we need to use to find how much farther Leah has to walk home when she reaches Josh’s house. 

We need to subtract because we need to find the difference between the total and the part. 

Step 4: 

Now we need to decide what tools can be used to find the difference. 

We can use fraction strips, number line, or simply paper and pencil to directly subtract the numerators and write the denominator once. 

Let us try using all the tools one by one. 

  1. Using fraction strips 

Let the total distance be 1 whole mile which is divided into 10 equal parts. 

Leah’s house is 8/10 miles from school.  

  Distance of school from Leah’s house. 

Distance of school from Josh’s house. 

The difference in the distance between Leah’s house to Josh’s house is  

The additional distance Leah has to walk home when she reaches Josh’s house                                                                  

that is 1/10+1/10+1/10= 3/10 miles.

  1. Using number line: 

Divide the space between 0 and 1 into 10 equal parts.  

Now, mark 8/10, the distance from Leah’s house from school. 

Move backwards 5 steps, as the distance from Josh’s house to school is 5/10. 

The last point at which we stop moving backwards is the difference between the distance of the school from Leah’s house to Josh’s house. 

That is 3/10 miles 

  1. Direct method: 

Distance of the school from Leah’s house = 8/10  miles 

Distance of the school from Josh’s house = 5/10  miles 

Difference = 8/10 – 5/10= 3/10 miles  

Leah has to walk home another 3/10miles when she reaches Josh’s house. 

Example 2: 

Kayla used 4/10 of her allowance to buy yoghurt and 5/10 to go skating. What fraction of her allowance does Kayla have left? Explain.   

Solution:  

Let us consider the total allowance Kayla got as 1. 

Which is divided into 10 equal parts = 10/10

Fraction of allowance Kayla used to buy yoghurt = 4/10

Fraction of allowance Kayla used to go skating = 5/10

Total allowance used =

4/10 +5/10  = 4+5/10= 9/10 

Fraction of allowance Kayla have left = 10/10−9/10=1/10

Example 3: 

Draw a number line to represent 5/6−3/6. 

5/6−3/6=2/6

fractions can be added or subtracted by locating a fraction on the number line and then moving to the right to add or moving to the left to subtract. 

Exercise:

  1. Shira has 10 books on her bookshelf. Of those, 5 of them are about horses. The rest are about nature. What fraction of Shira’s books are about nature?
  2. Flora needs an additional 2/8 cup of flour to make her dough. The dough recipe calls for 6/8 cups of flour. How many cups of flour does Shira have?
  3. Jeff ran 5/8 of a mile and Julie ran 2/8 of a mile. How much farther did jeff run than Julie?
  4. Alistair combined two buckets of water. One held 2/9 gallon of water and the other held 4/9 gallon of water. How much water did Alistair have in all?
  5. Donald picked two tomatoes from his garden. One weighed 5/6 pound and the other weighed 3/6 pound. Find the difference in weight between these two tomatoes.
  6. Megan’s driveway is 14/15 mile long. Megan’s driveway is 2/15 mile longer than Matt’s driveway. How long is Matt’s driveway?
  7. Jessi spent 3/4 hour on her homework . She spent 1/4 hour one her math and the rest of the time on her spelling. How long did Jessi spend on her spelling home work?
  8. The waterfall trail is 3/4 mile long. The mountain trail is 3/4 mile longer than waterfall trail. The mountain trail is 1/4 mile longer than the lake trail. How long is the lake trail?
  9. Maritta backed a chicken pot pie. She serves 2/3 of the pie at dinner. How much of the pie remains?
  10. An engineer was supposed to draw a line exactly 7/10 cm long. An error was made, and he drew the line 9/10 cm long. How much longer than needed was the line the engineer drew? Write an equation and solve

Concept Map:

What have we learned:

  • Subtraction of fractions with like denominators using any model
  • Subtraction of fractions using equation of addition
  • Adding and Subtracting fractions with like denominators directly

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