# Dividing Fractions: Using the Shortcut

In this lesson we divide fractions using the shortcut (multiply by the reciprocal of the divisor or 'flip and multiply'). The lesson also includes many word problems. The previous lesson introduces the reciprocal numbers and explores them conceptually.

The video below explains the reciprocal numbers and the "rule" for dividing fractions conceptually.

SHORTCUT: instead of dividing, multiply by the reciprocal of the divisor.

This shortcut works always, whether the numbers involved are whole numbers or fractions.

 2 5 ÷ 7 9 ↓ ↓ 2 5 × 9 7 = 18 35

 Check: 18 35 × 7 9 = 2 5
 7 ÷ 9 10 ↓ ↓ 7 × 10 9 = 70 9 =  7 7 9

 Check: 70 9 × 9 10 = 7 1 =  7
 10 11 ÷ 5 ↓ ↓ 10 11 × 1 5 = 10 55 = 2 11

 Check: 2 11 × 5  = 10 11
Notice: when you check the problems, you will need to use the original divisor, not the “flipped” one.

1. Solve. Change mixed numbers to fractions before dividing. Check each division by multiplication.

 a. 9 10 ÷ 2 5

Check:

 b. 3 7 ÷ 4 3
 c. 2 11 ÷ 2 3
 d.  1 7 8 ÷ 3 4

Check:

 e.  2 1 15 ÷  1 3 5
 f.  5 10 11 ÷  6

2. Write a division sentence for each problem, and solve it.

 a. How many times does go into   ?

 b.  How many times does go into  ?

One other meaning of division is equal sharing, or “dividing equally among so many people.”
In this, the divisor needs to be a whole number.
 Example 1. Divide 8/10 of a pie among four people. Each person gets 2/10. The division is 8 10 ÷ 4  = 2 10 .
 Example 2. Solve 6 7 ÷ 4. We have six slices (each slice being a seventh) and four people. Each person gets

one slice, first of all, and then we have 2 slices left. We split those. So, each person gets 1 1/2 slices. In

fraction terms, the 1/2 slice is a fourteenth-part and the 1 slice becomes 2/14. All totaled, each person gets
3/14 of the whole.

3. Solve these problems reasoning logically. Write a division sentence for each problem.

 a. There is 1 4/6 of the pizza left over and two people     share it equally. How much does each one get? b. There is 9/10 of the cake left over and three people     share it equally. How much does each one get?

4. The picture shows how much pie is left. That is divided among a certain number of people.
How much does each person get? Write a division sentence.

a. Divide among three people: b. Share among three people: c. Divide among six people: d. Divide between two people: e. Divide among five people: f. Share among four people: 5. Three people share a 1/4-kg chocolate bar equally.
How many kilograms will each of them get?

6. How many 3/8-foot long pieces can you cut out
of 11 feet of ribbon?
How long is the piece that is left over?

7. Five siblings inherited a plot of land that measures 2 4/10 acres.
If they divide the plot equally, what portion of an acre will each one get?

8. Among many other ingredients, a recipe calls for 2/3 cup of wheat flour. In her pantry Sarah had plenty of all
the other ingredients, but only a little wheat flour. How many batches of the recipe can she make if she has ...

a. 1/3 cup of wheat flour?

b. 1 cup of wheat flour?

9. An airport takes up a rectangular area that is 2 1/8 miles
long and 1/2 mile wide. What is its area?

10. An airport runway is two miles long, and takes up 1/16 square
mile in area. How wide is it, in miles? In feet?

To solve an equation involving fractions, you use the same
solution steps as if the equation had whole numbers.

In this case, we divide both sides by 4.

4x =
 3 5
÷ 4
x =
 3 5 ÷ 4

x =
 3 5 × 1 4 = 3 20

11. Solve the equations.

 a. 8x  = 1 2
 b. 3x  = 3 4
 c. 2 3 x  = 1 5
 d. 2 3 x  =  6

12. Solve these easy division problems!

 a.   1 ÷ 3 4
 b.   1 ÷ 3 2
 c.   1 ÷ 11 7
 d.   1 ÷ 2 1 4

13. How many 2/3 cup servings can you get out of 5 cups of ice cream?

14. Sam planted tomatoes in his garden, which is a rectangular area of 2 1/2 square
meters. If the length of the area is 5 meters, how wide is the area?

15. The sides of a rectangle are in a ratio of 2:3, and its perimeter is 1 1/4 inches.

a. What are the lengths of its sides?

b. Draw the rectangle.

16. Mary’s vegetable garden is 6 1/2 feet by 6 1/2 feet.

a. Find its area in square feet.

b. Find its area in square inches.
(Hint: Change the lengths of the sides into inches.)

c. Mary divided her garden into quarters in order to plant four different
vegetables. What is the area of one of those quarters in square feet?

This lesson is taken from Maria Miller's book Math Mammoth Fractions 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

#### Math Mammoth Fractions 2

A self-teaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.