Question 1
A tortoise walks 1/4 km per hour. How far does it walk in 3 hours?
Steps only
Multiply the hourly distance by 3.
3×1/4=3/4.
Answer
3/4 km.
CLASS 7 · GANITA PRAKASH PART I · CHAPTER 8
Working with Fractions
Multiply and divide fractions, interpret rates and solve fraction problems with pictures and clear steps.
45 worked answersAll 1 exercise setsAll 12 end-of-chapter questions
What you will learn
- Multiply whole numbers and fractions using equal groups and area.
- Divide fractions by using reciprocals and interpret quotients in context.
- Apply fraction operations to measurement, sharing and rates.
45 worked answers · short steps first, explanations when you need them
8.1 Multiplying fractions
Textbook pages 174–185
Question 2
Aaron walks 3 km per hour. How far does he walk in 1/5 hour and 2/5 hour?
Steps only
Multiply 3 by each time fraction.
Answer
3×1/5=3/5 km; 3×2/5=6/5=1 1/5 km.
Question 3
Five grandchildren each receive 2/3 acre. Find the total land distributed.
Steps only
Add five equal shares, or multiply 5×2/3.
Answer
10/3=3 1/3 acres.
Question 4
Internet costs ₹8 per hour. Find the cost of 1 1/4 hours.
Steps only
Convert 1 1/4 to 5/4.
Multiply by the hourly rate: 5/4×8.
Answer
₹10.
Question 5
Tenzin drinks 1/2 glass daily. How many glasses in 7 days and in January?
Steps only
Multiply 1/2 by the number of days.
Answer
7 days: 7/2=3 1/2 glasses. January: 31/2=15 1/2 glasses.
Question 6
Workers make 1 km of canal in 8 days. How much in one day and five working days?
Steps only
Divide 1 km into 8 equal daily amounts.
Multiply the daily amount by 5.
Answer
1/8 km per day; 5/8 km in five days.
Question 7
Three families share 5 litres equally each week. Find each family’s weekly and four-week share.
Steps only
Divide 5 by 3.
Multiply the weekly share by 4.
Answer
5/3=1 2/3 litres per week; 20/3=6 2/3 litres in four weeks.
Question 8
The Moon sets 5/6 hour later each day. From Monday 10 pm, how much later by Thursday?
Steps only
Monday to Thursday has three daily intervals.
Multiply 3×5/6=15/6=2 1/2 hours.
Answer
2 1/2 hours later, so approximately 12:30 am on Friday.
Question 9
Multiply and convert to mixed numbers: 7×3/5; 4×1/3; 9×6/7; 13×6/11.
Steps only
Multiply the whole number by the numerator, then divide by the denominator.
Answer
21/5=4 1/5; 4/3=1 1/3; 54/7=7 5/7; 78/11=7 1/11.
Question 10
Find 1/3×1/5, 1/4×1/3, 1/5×1/2, 1/6×1/5 and 1/12×1/18.
Steps only
Multiply the numerators and multiply the denominators.
Answer
1/15; 1/12; 1/10; 1/30; 1/216.
Question 11
Find 2/3×4/5; 1/4×2/3; 3/5×1/2; 4/6×3/5.
Steps only
Multiply across, then reduce common factors.
Answer
8/15; 2/12=1/6; 3/10; 12/30=2/5.
Question 12
State and use the rule for multiplying a/b by c/d. Evaluate 5/12×7/18 and 12/7×5/24.
Steps only
Multiply numerators and denominators.
Cancel common factors before multiplying when possible.
Answer
(a/b)(c/d)=ac/bd. 5/12×7/18=35/216; 12/7×5/24=5/14.
Question 13
How far does Aaron walk in 5/4 hours at 3 km/hour? How far does a tortoise walk in 1/2 hour at 1/4 km/hour?
Steps only
Multiply rate by time.
Answer
15/4=3 3/4 km; 1/8 km.
Question 14
Compare the product with its factors when both are above 1, both are between 0 and 1, or one is each.
Steps only
Multiplying by a factor above 1 enlarges a positive number.
Multiplying by a positive factor below 1 makes it smaller.
Answer
Both >1: product exceeds both. Both between 0 and 1: product is below both. One below 1 and one above 1: the product lies between the two.
Question 15
Does changing the order of two fractions change their product?
Steps only
The numerator and denominator products are unchanged when factors are swapped.
Answer
No: a/b×c/d=c/d×a/b.
8.2 Dividing fractions
Textbook pages 186–193
Question 1
What is the reciprocal of c/d, where c and d are nonzero?
Steps only
Find the number that multiplies c/d to make 1.
Answer
d/c, because (c/d)(d/c)=1.
Question 2
Evaluate 1÷2/3; 3÷2/3; 1/5÷1/2; 2/3÷3/5.
Steps only
Keep the first fraction, change division to multiplication, and flip the divisor.
Answer
3/2; 9/2; 2/5; 10/9.
Question 3
State the rule for dividing fractions and explain why it works.
Steps only
If x÷(c/d)=a/b, then (c/d)×(a/b)=x.
Multiplying by d/c reverses multiplication by c/d.
Answer
a/b÷c/d=(a/b)×(d/c)=ad/bc, provided c≠0.
Question 4
Can a quotient be greater than its dividend? Compare 6÷1/4 and 1/8÷1/4.
Steps only
Divide by 1/4 by multiplying by 4.
Answer
Yes. 6÷1/4=24 and 1/8÷1/4=1/2; both exceed their dividends.
Question 5
A tap fills 7/10 of a tank in one hour. How much fills in 1/3, 2/3, 3/4, 7/10 hour? How long for a full tank?
Steps only
Multiply 7/10 by each time fraction.
For a full tank, divide 1 by the rate 7/10.
Answer
7/30; 14/30=7/15; 21/40; 49/100 of the tank. Full tank: 1÷7/10=10/7=1 3/7 hours.
Question 6
A road takes 1/6 of a plot. Somu gives half the remainder to Krishna and one-third of the remainder to Bora. What shares of the original plot do they receive, and what remains for Somu?
Steps only
After the road, 5/6 remains.
Krishna gets 1/2×5/6; Bora gets 1/3×5/6.
Subtract both gifts from 5/6.
Answer
Krishna: 5/12; Bora: 5/18; Somu keeps 5/36 of the original plot.
Question 7
Find the area of a rectangle with sides 3 3/4 ft and 9 3/5 ft.
Steps only
Convert to improper fractions: 15/4 and 48/5.
Multiply side lengths.
Answer
(15/4)(48/5)=36 square feet.
Question 8
Four saplings are in a line, with 3/4 m between neighbours. Find first-to-last distance.
Steps only
Four saplings create three equal gaps.
Multiply 3×3/4.
Answer
9/4=2 1/4 m.
Question 9
Which is heavier: 12/15 of 500 g or 3/20 of 4 kg?
Steps only
12/15×500=400 g.
Convert 4 kg to 4,000 g; multiply by 3/20.
Answer
3/20 of 4 kg=600 g, so it is heavier.
Question 10
Four fountains fill a cistern in 1 day, 1/2 day, 1/4 day and 1/5 day respectively. How many cisterns can they fill together in one day?
Steps only
Their individual rates are 1, 2, 4 and 5 cisterns per day.
Add rates.
Answer
1+2+4+5=12 cisterns per day.
Question 11
The textbook’s two square diagrams ask what fraction of each whole is shaded.
Steps only
Count equal parts in the whole square and count shaded parts.
Reduce the resulting fraction.
Answer
Figure (a): 3/8; figure (b): 1/16.
Question 12
Evaluate 3÷7/9; 14/4÷2; 2/3÷2/3; 14/6÷7/3; 4/3÷3/4; 7/4÷1/7; 8/2÷4/15; 1/5÷1/9; 1/6÷11/12; 3 2/3÷1 3/8.
Steps only
For each, keep the first number, change division to multiplication, and flip the second number.
Simplify each product.
Answer
3×9/7=27/7; 14/4×1/2=7/4; 2/3×3/2=1; 14/6×3/7=1; 4/3×4/3=16/9; 7/4×7=49/4; 4×15/4=15; 1/5×9=9/5; 1/6×12/11=2/11; 11/3×8/11=8/3=2 2/3.
8.3 Fraction applications
Textbook pages 190–197
Question 1
Leena uses 1/4 litre of milk for 5 cups of tea. How much milk is in each cup?
Steps only
Divide the total amount equally among 5 cups.
1/4÷5=1/4×1/5.
Answer
1/20 litre per cup.
Question 2
Cover 7 1/2 square units with square bricks of side 1/5 unit. How many bricks are needed?
Steps only
One brick area=(1/5)(1/5)=1/25.
Total area=15/2.
Divide total area by area per brick.
Answer
(15/2)÷(1/25)=375/2=187 1/2 square-brick areas. This shows the given area is not an exact whole number of such bricks; 188 bricks would be needed to cover it completely.
Question 3
Using the four fountain rates 1, 2, 4 and 5 cisterns per day, how long will they take to fill one cistern together?
Steps only
Together they fill 12 cisterns in one day.
Time for one cistern is the reciprocal rate.
Answer
1/12 day.
Question 4
Maria has 8 m of lace and uses 1/4 m per bag. How many bags can she decorate?
Steps only
Number of bags=total lace÷lace per bag.
8÷1/4=8×4.
Answer
32 bags.
Question 5
Half a metre of ribbon makes 8 badges. How much ribbon is used for each?
Steps only
Share 1/2 m equally among 8.
Answer
1/2÷8=1/16 m per badge.
Question 6
Each loaf needs 1/6 kg flour. How many loaves can be made from 5 kg?
Steps only
Divide flour available by flour per loaf.
Answer
5÷1/6=30 loaves.
Question 7
If 1/4 kg flour makes 12 rotis, how much flour makes 6 rotis?
Steps only
Six is half of twelve, so use half the flour.
Answer
1/8 kg.
Question 8
Evaluate 1÷1/6 + 1÷1/10 + 1÷1/13 + 1÷1/9 + 1÷1/2.
Steps only
Dividing 1 by 1/n gives n.
Add the resulting whole numbers.
Answer
6+10+13+9+2=40.
Question 9
A 400-page novel: Mira read 1/5 yesterday and 3/10 today. How many pages remain?
Steps only
Read fraction=1/5+3/10=1/2.
Remaining fraction=1/2.
Find half of 400.
Answer
200 pages remain.
Question 10
A car travels 16 km per litre. How far on 2 3/4 litres?
Steps only
Convert 2 3/4=11/4.
Multiply 16×11/4.
Answer
44 km.
Question 11
A train takes 5 1/6 hours and a plane takes 1/2 hour. How much time does the plane save?
Steps only
Subtract 1/2 from 5 1/6.
Use sixths: 31/6−3/6=28/6.
Answer
4 2/3 hours.
Question 12
Friends ate 4/5 of a cake. The remaining cake is split equally among three people. What fraction does each get?
Steps only
Remaining fraction=1−4/5=1/5.
Divide by 3.
Answer
1/5÷3=1/15 of the whole cake each.
Question 13
An ant group splits equally at each fork in the printed path. What fractions reach the mango tree and sugarcane field?
Steps only
Multiply the branch fractions along each route and add routes reaching the same food.
Answer
Mango tree: 29/32; sugarcane field: 3/32.
Question 14
Evaluate (1−1/2)(1−1/3)…(1−1/n).
Steps only
Rewrite each factor as (k−1)/k.
The product is (1/2)(2/3)(3/4)…((n−1)/n); cancel matching factors.
Answer
1/n. For n=2,3,5,10 the values are 1/2, 1/3, 1/5, 1/10.
Question 15
Find the shaded fraction in the textbook’s end figure.
Steps only
Count the equal smallest regions in the whole and the shaded set.
The figure’s exact subdivision is image-based.
Answer
3/32 of the whole square.
Question activity-dramma
A donation is 1/5 of 1/16 of 1/4 of 1/2 of 2/3 of 3/4 of a dramma. If 1 dramma equals 1,280 cowrie shells, how many cowries is that?
Steps only
Multiply the fractions: (3/4)(2/3)(1/2)(1/4)(1/16)(1/5).
Cancel common factors, then convert the fraction of a dramma to cowries.
Answer
The product is 1/1,280 dramma. Since 1 dramma is 1,280 cowries, the donation is 1 cowrie.
Question activity-coin-conversion
Using the stated assumptions 1 gold dinar=12 silver drammas, 1 dramma=4 copper panas, and 1 pana=30 cowrie shells, write the value of 1 cowrie in panas and in gold dinars.
Steps only
One pana is divided into 30 cowries, so one cowrie is 1/30 pana.
One pana is 1/4 of a dramma, and one dramma is 1/12 of a gold dinar. Multiply these fractions.
Answer
1 cowrie=1/30 pana=(1/30)(1/4)(1/12)=1/1,440 gold dinar.
Question 9-product
Choose all statements that describe the product (56/45)×(70/67): (a) greater than 56/45; (b) less than 56/45; (c) greater than 70/67; (d) less than 70/67; (e) greater than 1; (f) less than 1.
Steps only
56/45>1 and 70/67>1.
Multiplying a positive number by a factor greater than 1 makes it larger.
Answer
The product is greater than each factor and greater than 1. Choose (a), (c), and (e).