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Working with Fractions — Class 7 Mathematics

Ganita Prakash Part 1 · Chapter 8

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Chapter summary

This chapter develops multiplication and division of fractions with meaning. Multiplying a fraction by a whole number repeats it (), and multiplying two fractions uses Brahmagupta's rule , which you also see as the area of a rectangle with fractional sides. Common factors can be cancelled before multiplying to keep numbers small. A key insight is how the size of a product depends on the factors: multiplying by a number between 0 and 1 makes the product smaller than the other number, while multiplying by a number greater than 1 makes it larger. Division is turned into multiplication using the reciprocal: . When the divisor is between 0 and 1 the quotient is bigger than the dividend; when it is greater than 1 the quotient is smaller. The chapter is rich in history — Brahmagupta, Bhaskaracharya's Lilavati, and the Shulbasutra — and applies fractions to many everyday sharing and measuring problems.

Multiplying fractions by whole numbers and fractionsBrahmagupta's rule and area modelCancelling common factors and lowest formReciprocals and dividing fractionsSize of products and quotients; word problems

Key ideas & terms

Fraction: A number of the form representing parts of a whole.
Multiplication of fractions: Brahmagupta's rule: .
Reciprocal: The fraction of ; a fraction times its reciprocal is .
Division of fractions: Multiplying the dividend by the reciprocal of the divisor.
Cancelling common factors: Dividing numerator and denominator by common factors before multiplying.
Lowest form: A fraction whose numerator and denominator share no common factor other than 1.

Extra questions & answers

State Brahmagupta's rule for multiplying fractions.

What is ?

What is the reciprocal of ?

Its reciprocal is , and .

How do you divide one fraction by another?

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

Evaluate .

When is a product less than both the numbers multiplied?

When both numbers are between 0 and 1, e.g. .

When multiplying by a number between 0 and 1, what happens to the product?

The product is smaller than the other number.

When the divisor is between 0 and 1, how does the quotient compare to the dividend?

The quotient is greater than the dividend.

Why does the area of a rectangle illustrate fraction multiplication?

A rectangle with fractional sides has area equal to the product of those sides.

Simplify .

Cancel the common factor 12: .

Figure it Out

1. Tenzin drinks glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

In one week there are 7 days. Milk in one day is glass. So milk in a week = glasses. January has 31 days. Milk in January = glasses.

2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make _ km of the water canal. If they work 5 days a week, they can make _ km of the water canal in a week.

Work in 1 day = km. Work in 5 days = km. So the blanks are and .

3. Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

Oil per family in one week = litres. In 4 weeks one family gets litres.

4. Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

From Monday to Thursday is 3 days later. Delay = hours after the Monday time. So it will set hours after 10 pm, i.e., at 12:30 am on Friday night.

5. Multiply and then convert it into a mixed fraction: (a)

5. Multiply and then convert it into a mixed fraction: (b)

5. Multiply and then convert it into a mixed fraction: (c)

5. Multiply and then convert it into a mixed fraction: (d)

Figure it Out

1. Find the following products. Use a unit square as a whole for representing the fractions: (a)

. The whole is divided into 3 rows and 5 columns giving 15 small rectangles, one is shaded.

1. Find the following products. Use a unit square as a whole for representing the fractions: (b)

1. Find the following products. Use a unit square as a whole for representing the fractions: (c)

1. Find the following products. Use a unit square as a whole for representing the fractions: (d)

1. Find the following products. Use a unit square as a whole for representing the fractions: Now, find

. Using .

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (a)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (b)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (c)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (d)

after cancelling.

Figure it Out

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (a) hour ____________

Filled part = of the tank.

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (b) hour ____________

of the tank.

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (c) hour ____________

of the tank.

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (d) hour ____________

of the tank.

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (e) For the tank to be full, how long should the tap be running?

Let time be hours, , so hours, about 1 hour 26 minutes.

2. The government has taken of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and of it to her son Bora. After giving them their shares, she keeps the remaining land for herself. What part of the land remains with Somu now?

Remaining before distribution = of the original land.

2. The government has taken of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and of it to her son Bora. After giving them their shares, she keeps the remaining land for herself. (a) What part of the original land did Krishna get?

Krishna gets half of the remaining : of the original land.

2. The government has taken of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and of it to her son Bora. After giving them their shares, she keeps the remaining land for herself. (b) What part of the original land did Bora get?

Bora gets of the remaining : of the original land.

2. The government has taken of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and of it to her son Bora. After giving them their shares, she keeps the remaining land for herself. (c) What part of the original land did Somu keep for herself?

Somu kept of the original land.

3. Find the area of a rectangle of sides ft and ft.

ft, ft. Area = square feet.

4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is m. Find the distance between the first and last sapling. [Hint: Draw a rough diagram with four saplings with distance between two saplings as m]

With 4 saplings there are 3 intervals. Distance = m between the first and last sapling.

5. Which is heavier: of 500 grams or of 4 kg?

of 500 g = g. of 4 kg = g = 600 g. So of 4 kg is heavier.

Figure it Out

1. Evaluate the following:

, so .

1. Evaluate the following:

, . So .

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (a) Maria bought 8 m of lace to decorate the bags she made for school. She used m for each bag and finished the lace. How many bags did she decorate? (i) (ii) (iii) (iv)

Correct expression is (iii) . Simplifying: . She decorated 32 bags.

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (b) meter of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge? (i) (ii) (iii) (iv)

Correct expression is (iv) . Simplifying: meter per badge.

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (c) A baker needs kg of flour to make one loaf of bread. He has 5 kg of flour. How many loaves of bread can he make? (i) (ii) (iii) (iv)

Correct expression is (iii) , which is same as (iv) . Simplifying: . He can make 30 loaves.

3. If kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?

Flour per roti = kg. For 6 rotis: kg.

4. Pāṭīgaṇita, a book written by Sridharacharya in the 9th century CE, mentions this problem: “Friend, after thinking, what sum will be obtained by adding together , , , , and ”. What should the friend say?

, , , , . Sum = .

5. Mira is reading a novel that has 400 pages. She read of the pages yesterday and of the pages today. How many more pages does she need to read to finish the novel?

Read yesterday = pages. Read today = pages. Total read = 200 pages. Remaining = pages.

6. A car runs 16 km using 1 litre of petrol. How far will it go using litres of petrol?

litres. Distance = km.

7. Amritpal decides on a destination for his vacation. If he takes a train, it will take him hours to get there. If he takes a plane, it will take him hour. How many hours does the plane save?

Time saved = hours.

8. Mariam's grandmother baked a cake. Mariam and her cousins finished of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get?

Remaining cake = . Each friend gets of the whole cake.

9. Choose the option(s) describing the product of ( ): (a) (b) (c) (d) (e) (f)

Both factors are greater than 1, so their product is greater than each factor and greater than 1. Correct options are (a), (c) and (e).

10. What fraction of the whole square is shaded?

As shown in the figure, the shaded region is of half of of the whole. So area = or directly of the whole square.

11. A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig. 8.7) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?

Tracing the equal splits in the diagram, the group that reaches the mango tree receives of the original colony and the group that reaches the sugarcane field receives of the original colony, together making the whole.

12. What is ?

. All intermediate numerators and denominators cancel.

12. What is ? Make a general statement and explain.

This product is . In general, , because and telescoping cancellation leaves .

Working with Fractions — Class 7 Mathematics

CBSE Class 7 community

Chapter summary

Key ideas & terms

Extra questions & answers

State Brahmagupta's rule for multiplying fractions.

What is ?

What is the reciprocal of ?

How do you divide one fraction by another?

Evaluate .

When is a product less than both the numbers multiplied?

When multiplying by a number between 0 and 1, what happens to the product?

When the divisor is between 0 and 1, how does the quotient compare to the dividend?

Why does the area of a rectangle illustrate fraction multiplication?

Simplify .

Figure it Out

1. Tenzin drinks glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make ___ km of the water canal. If they work 5 days a week, they can make ___ km of the water canal in a week.

3. Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

4. Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

5. Multiply and then convert it into a mixed fraction: (a)

5. Multiply and then convert it into a mixed fraction: (b)

5. Multiply and then convert it into a mixed fraction: (c)

5. Multiply and then convert it into a mixed fraction: (d)

Figure it Out

1. Find the following products. Use a unit square as a whole for representing the fractions: (a)

1. Find the following products. Use a unit square as a whole for representing the fractions: (b)

1. Find the following products. Use a unit square as a whole for representing the fractions: (c)

1. Find the following products. Use a unit square as a whole for representing the fractions: (d)

1. Find the following products. Use a unit square as a whole for representing the fractions: Now, find

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (a)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (b)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (c)

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (d)

Figure it Out

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (a) hour ____________

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (b) hour ____________

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (c) hour ____________

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (d) hour ____________

1. A water tank is filled from a tap. If the tap is open for 1 hour, of the tank gets filled. How much of the tank is filled if the tap is open for (e) For the tank to be full, how long should the tap be running?

3. Find the area of a rectangle of sides ft and ft.

4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is m. Find the distance between the first and last sapling. [Hint: Draw a rough diagram with four saplings with distance between two saplings as m]

5. Which is heavier: of 500 grams or of 4 kg?

Figure it Out

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

1. Evaluate the following:

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (a) Maria bought 8 m of lace to decorate the bags she made for school. She used m for each bag and finished the lace. How many bags did she decorate? (i) (ii) (iii) (iv)

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (b) meter of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge? (i) (ii) (iii) (iv)

2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (c) A baker needs kg of flour to make one loaf of bread. He has 5 kg of flour. How many loaves of bread can he make? (i) (ii) (iii) (iv)

3. If kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?

4. Pāṭīgaṇita, a book written by Sridharacharya in the 9th century CE, mentions this problem: “Friend, after thinking, what sum will be obtained by adding together , , , , and ”. What should the friend say?

5. Mira is reading a novel that has 400 pages. She read of the pages yesterday and of the pages today. How many more pages does she need to read to finish the novel?

6. A car runs 16 km using 1 litre of petrol. How far will it go using litres of petrol?

7. Amritpal decides on a destination for his vacation. If he takes a train, it will take him hours to get there. If he takes a plane, it will take him hour. How many hours does the plane save?

8. Mariam's grandmother baked a cake. Mariam and her cousins finished of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get?

9. Choose the option(s) describing the product of ( ): (a) (b) (c) (d) (e) (f)

10. What fraction of the whole square is shaded?

11. A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig. 8.7) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?

12. What is ?

12. What is ?

12. What is ?

12. What is ? Make a general statement and explain.

2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make _ km of the water canal. If they work 5 days a week, they can make _ km of the water canal in a week.