Class 7 › Mathematics › Ganita Prakash Part 1 Working with Fractions Class 7 Mathematics · Ganita Prakash Part 1 · Chapter 8
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View all 7 amazing Marks Booster kits → This chapter develops multiplication and division of fractions with meaning. Multiplying a fraction by a whole number repeats it (\(\tfrac{2}{5}\times 3 = \tfrac{6}{5}\)), and multiplying two fractions uses Brahmagupta's rule \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}\), 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: \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}\). 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.
Fraction
A number of the form \(\dfrac{a}{b}\) representing parts of a whole.
Multiplication of fractions
Brahmagupta's rule: \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}\).
Reciprocal
The fraction \(\dfrac{b}{a}\) of \(\dfrac{a}{b}\); a fraction times its reciprocal is \(1\).
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.
Q1 State Brahmagupta's rule for multiplying fractions. Answer Q2 What is \(\tfrac{3}{4}\times\tfrac{2}{5}\)? Answer Q3 What is the reciprocal of \(\tfrac{3}{5}\)? Answer Q4 How do you divide one fraction by another? Answer Q5 Evaluate \(\tfrac{2}{3}\div\tfrac{3}{5}\). Answer Q6 When is a product less than both the numbers multiplied? Answer Q7 When multiplying by a number between 0 and 1, what happens to the product? Answer Q8 When the divisor is between 0 and 1, how does the quotient compare to the dividend? Answer Q9 Why does the area of a rectangle illustrate fraction multiplication? Answer Q10 Simplify \(\tfrac{12}{7}\times\tfrac{5}{24}\). Answer Q1 1. Tenzin drinks \( \frac{1}{2} \) 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? Answer Q2 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. Answer Q3 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? Answer Q4 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 \( \frac{5}{6} \) hour later than the previous day. How many hours after 10 pm will the moon set on Thursday? Answer Q5 5. Multiply and then convert it into a mixed fraction: (a) \( 7 \times \frac{3}{5} \) Answer Q6 5. Multiply and then convert it into a mixed fraction: (b) \( 4 \times \frac{1}{3} \) Answer Q7 5. Multiply and then convert it into a mixed fraction: (c) \( \frac{9}{7} \times 6 \) Answer Q8 5. Multiply and then convert it into a mixed fraction: (d) \( \frac{13}{11} \times 6 \) Answer Q1 1. Find the following products. Use a unit square as a whole for representing the fractions: (a) \( \frac{1}{3} \times \frac{1}{5} \) Answer Q2 1. Find the following products. Use a unit square as a whole for representing the fractions: (b) \( \frac{1}{4} \times \frac{1}{3} \) Answer Q3 1. Find the following products. Use a unit square as a whole for representing the fractions: (c) \( \frac{1}{5} \times \frac{1}{2} \) Answer Q4 1. Find the following products. Use a unit square as a whole for representing the fractions: (d) \( \frac{1}{6} \times \frac{1}{5} \) Answer Q5 1. Find the following products. Use a unit square as a whole for representing the fractions: Now, find \( \frac{1}{12} \times \frac{1}{18} \) Answer Q6 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (a) \( \frac{2}{3} \times \frac{4}{5} \) Answer Q7 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (b) \( \frac{1}{4} \times \frac{2}{3} \) Answer Q8 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (c) \( \frac{3}{5} \times \frac{1}{2} \) Answer Q9 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations. (d) \( \frac{4}{6} \times \frac{3}{5} \) Answer Q1 1. A water tank is filled from a tap. If the tap is open for 1 hour, \( \frac{7}{10} \) of the tank gets filled. How much of the tank is filled if the tap is open for (a) \( \frac{1}{3} \) hour ____________ Answer Q2 1. A water tank is filled from a tap. If the tap is open for 1 hour, \( \frac{7}{10} \) of the tank gets filled. How much of the tank is filled if the tap is open for (b) \( \frac{2}{3} \) hour ____________ Answer Q3 1. A water tank is filled from a tap. If the tap is open for 1 hour, \( \frac{7}{10} \) of the tank gets filled. How much of the tank is filled if the tap is open for (c) \( \frac{3}{4} \) hour ____________ Answer Q4 1. A water tank is filled from a tap. If the tap is open for 1 hour, \( \frac{7}{10} \) of the tank gets filled. How much of the tank is filled if the tap is open for (d) \( \frac{7}{10} \) hour ____________ Answer Q5 1. A water tank is filled from a tap. If the tap is open for 1 hour, \( \frac{7}{10} \) 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? Answer Q6 2. The government has taken \( \frac{1}{6} \) 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 \( \frac{1}{3} \) 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? Answer Q7 2. The government has taken \( \frac{1}{6} \) 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 \( \frac{1}{3} \) 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? Answer Q8 2. The government has taken \( \frac{1}{6} \) 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 \( \frac{1}{3} \) 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? Answer Q9 2. The government has taken \( \frac{1}{6} \) 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 \( \frac{1}{3} \) 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? Answer Q10 3. Find the area of a rectangle of sides \( 3\frac{3}{4} \) ft and \( 9\frac{3}{5} \) ft. Answer Q11 4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is \( \frac{3}{4} \) m. Find the distance between the first and last sapling. [Hint: Draw a rough diagram with four saplings with distance between two saplings as \( \frac{3}{4} \) m] Answer Q12 5. Which is heavier: \( \frac{12}{15} \) of 500 grams or \( \frac{3}{20} \) of 4 kg? Answer Q1 1. Evaluate the following: \( 3 \div \frac{7}{9} \) Answer Q2 1. Evaluate the following: \( \frac{14}{4} \div 2 \) Answer Q3 1. Evaluate the following: \( \frac{2}{3} \div \frac{2}{3} \) Answer Q4 1. Evaluate the following: \( \frac{14}{6} \div \frac{7}{3} \) Answer Q5 1. Evaluate the following: \( \frac{4}{3} \div \frac{3}{4} \) Answer Q6 1. Evaluate the following: \( \frac{7}{4} \div \frac{1}{7} \) Answer Q7 1. Evaluate the following: \( \frac{8}{2} \div \frac{4}{15} \) Answer Q8 1. Evaluate the following: \( \frac{1}{5} \div \frac{1}{9} \) Answer Q9 1. Evaluate the following: \( \frac{1}{6} \div \frac{11}{12} \) Answer Q10 1. Evaluate the following: \( 3\frac{2}{3} \div 1\frac{3}{8} \) Answer Q11 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 \( \frac{1}{4} \) m for each bag and finished the lace. How many bags did she decorate? (i) \( 8 \times \frac{1}{4} \) (ii) \( \frac{1}{8} \times \frac{1}{4} \) (iii) \( 8 \div \frac{1}{4} \) (iv) \( \frac{1}{4} \div 8 \) Answer Q12 2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (b) \( \frac{1}{2} \) meter of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge? (i) \( 8 \times \frac{1}{2} \) (ii) \( \frac{1}{2} \div \frac{1}{8} \) (iii) \( 8 \div \frac{1}{2} \) (iv) \( \frac{1}{2} \div 8 \) Answer Q13 2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (c) A baker needs \( \frac{1}{6} \) kg of flour to make one loaf of bread. He has 5 kg of flour. How many loaves of bread can he make? (i) \( 5 \times \frac{1}{6} \) (ii) \( \frac{1}{6} \div 5 \) (iii) \( 5 \div \frac{1}{6} \) (iv) \( 5 \times 6 \) Answer Q14 3. If \( \frac{1}{4} \) kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis? Answer Q15 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 \( 1 \div \frac{1}{6} \), \( 1 \div \frac{1}{10} \), \( 1 \div \frac{1}{13} \), \( 1 \div \frac{1}{9} \), and \( 1 \div \frac{1}{2} \)”. What should the friend say? Answer Q16 5. Mira is reading a novel that has 400 pages. She read \( \frac{1}{5} \) of the pages yesterday and \( \frac{3}{10} \) of the pages today. How many more pages does she need to read to finish the novel? Answer Q17 6. A car runs 16 km using 1 litre of petrol. How far will it go using \( 2\frac{3}{4} \) litres of petrol? Answer Q18 7. Amritpal decides on a destination for his vacation. If he takes a train, it will take him \( 5\frac{1}{6} \) hours to get there. If he takes a plane, it will take him \( \frac{1}{2} \) hour. How many hours does the plane save? Answer Q19 8. Mariam's grandmother baked a cake. Mariam and her cousins finished \( \frac{4}{5} \) of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get? Answer Q20 9. Choose the option(s) describing the product of ( \( \frac{565}{465} \times \frac{707}{676} \)): (a) \( > \frac{565}{465} \) (b) \( < \frac{565}{465} \) (c) \( > \frac{707}{676} \) (d) \( < \frac{707}{676} \) (e) \( > 1 \) (f) \( < 1 \) Answer Q21 10. What fraction of the whole square is shaded? Answer Q22 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? Answer Q23 12. What is \( 1 - \frac{1}{2} \)? Answer Q24 12. What is \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \)? Answer Q25 12. What is \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{4}) \times (1 - \frac{1}{5}) \)? Answer Q26 12. What is \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{4}) \times (1 - \frac{1}{5}) \times (1 - \frac{1}{6}) \times (1 - \frac{1}{7}) \times (1 - \frac{1}{8}) \times (1 - \frac{1}{9}) \times (1 - \frac{1}{10}) \)? Make a general statement and explain. Answer
Brahmagupta's rule for multiplying fractions is:
A \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a+c}{b+d}\) B \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times b}{c\times d}\) C \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}\) D \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times d}{b\times c}\)
Flashcards & interactive practice 
