Chapter 8 exercise answers: Working with Fractions

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1. A week has 7 days, so drinking 1/2 glass every day means adding 1/2, seven times: 7 × 1/2.
2. 7 × 1/2 = 7/2 glasses in a week.
3. January has 31 days, so the same idea gives 31 × 1/2 = 31/2 glasses in January.

Answer7/2 glasses in a week; 31/2 glasses in January.

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1. 1 km of canal is built over 8 days, so cut that 1 km into 8 equal parts: one day's work is 1/8 km.
2. In a week of 5 working days, collect 5 of those parts: 5 × 1/8 = 5/8 km.

Answer1/8 km in one day; 5/8 km in a week.

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1. 5 litres shared equally among 3 families means each family gets 5/3 litres in a week.
2. Over 4 weeks, one family collects 4 lots of that share: 4 × 5/3 = 20/3 litres.

Answer5/3 litres a week; 20/3 litres in 4 weeks.

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1. From Monday to Thursday is 3 days later.
2. Each day adds 5/6 hour to the moonset time, so after 3 days the moon sets 3 × 5/6 hours after 10 pm.
3. 3 × 5/6 = 15/6 = 5/2 hours, which is 2 1/2 hours.

Answer5/2 hours (2 1/2 hours) after 10 pm.

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(a) 7 × 3/5

1. 7 × 3/5 = 21/5.
2. 21/5 = 4 1/5.

Answer21/5 = 4 1/5

(b) 4 × 1/3

1. 4 × 1/3 = 4/3.
2. 4/3 = 1 1/3.

Answer4/3 = 1 1/3

(c) 9/7 × 6

1. 9/7 × 6 = 54/7.
2. 54/7 = 7 5/7.

Answer54/7 = 7 5/7

(d) 13/11 × 6

1. 13/11 × 6 = 78/11.
2. 78/11 = 7 1/11.

Answer78/11 = 7 1/11

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(a) 1/3 × 1/5

1. Take a unit square. Cut it into 5 equal rows; one row is 1/5.
2. To take 1/3 of that row, cut the square into 3 equal columns as well. The square is now 5 × 3 = 15 equal pieces.
3. 1/3 of the row is exactly one of those pieces, so 1/3 × 1/5 = 1/15.

Answer1/15

(b) 1/4 × 1/3

1. Cut the unit square into 3 rows (for 1/3) and 4 columns (for 1/4): 3 × 4 = 12 equal pieces.
2. 1/4 of the 1/3 row is one piece, so 1/4 × 1/3 = 1/12.

Answer1/12

(c) 1/5 × 1/2

1. Cut the unit square into 2 rows (for 1/2) and 5 columns (for 1/5): 2 × 5 = 10 equal pieces.
2. 1/5 of the 1/2 row is one piece, so 1/5 × 1/2 = 1/10.

Answer1/10

(d) 1/6 × 1/5

1. Cut the unit square into 5 rows (for 1/5) and 6 columns (for 1/6): 5 × 6 = 30 equal pieces.
2. 1/6 of the 1/5 row is one piece, so 1/6 × 1/5 = 1/30.

Answer1/30

1/12 × 1/18 Now, find 1/12 × 1/18.

1. The same drawing would need 18 rows and 12 columns: 18 × 12 = 216 equal pieces, of which we take one.
2. So 1/12 × 1/18 = 1/(12 × 18) = 1/216. Each time, two unit fractions multiply to 1 over the product of their denominators.

Answer1/216

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(a) 2/3 × 4/5

1. Multiply the numerators and the denominators: 2/3 × 4/5 = (2×4)/(3×5) = 8/15.

Answer8/15

(b) 1/4 × 2/3

1. 1/4 × 2/3 = (1×2)/(4×3) = 2/12.
2. 2/12 = 1/6 in lowest terms.

Answer1/6

(c) 3/5 × 1/2

1. 3/5 × 1/2 = (3×1)/(5×2) = 3/10.

Answer3/10

(d) 4/6 × 3/5

1. 4/6 × 3/5 = (4×3)/(6×5) = 12/30.
2. 12/30 = 2/5 in lowest terms.

Answer2/5

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(a) 1/3 hour

1. Tank filled = time × rate = 1/3 × 7/10 = 7/30.

Answer7/30 of the tank

(b) 2/3 hour

1. 2/3 × 7/10 = 14/30 = 7/15.

Answer7/15 of the tank

(c) 3/4 hour

1. 3/4 × 7/10 = 21/40.

Answer21/40 of the tank

(d) 7/10 hour

1. 7/10 × 7/10 = 49/100.

Answer49/100 of the tank

(e) how long should the tap be running?

1. The tap fills 7/10 of the tank every hour, so ask: 7/10 × what time = 1 whole tank?
2. That time is 1 ÷ 7/10 = 1 × 10/7 = 10/7 hours.

Answer10/7 hours

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(a) What part of the original land did Krishna get?

1. The road takes 1/6 of the land, so 1 − 1/6 = 5/6 of the original land remains with Somu.
2. Krishna gets half of this remaining part: 1/2 × 5/6 = 5/12 of the original land.

Answer5/12 of the original land

(b) What part of the original land did Bora get?

1. Bora gets 1/3 of the same remaining part: 1/3 × 5/6 = 5/18 of the original land.

Answer5/18 of the original land

(c) What part of the original land did Somu keep for herself?

1. Somu keeps what is left of her 5/6 after both shares: 5/6 − 5/12 − 5/18.
2. Write each over 36: 30/36 − 15/36 − 10/36 = 5/36.
3. Check: road 6/36 + Krishna 15/36 + Bora 10/36 + Somu 5/36 = 36/36, the whole land.

Answer5/36 of the original land

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1. Write both sides as single fractions: 3 3/4 = 15/4 ft and 9 3/5 = 48/5 ft.
2. Area = length × breadth = 15/4 × 48/5 = 720/20 = 36 square ft.

Answer36 square ft.

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1. Four saplings in a row have 3 gaps between the first and the last one.
2. Each gap is 3/4 m, so the total distance is 3 × 3/4 = 9/4 m, which is 2 1/4 m.

Answer9/4 m (2 1/4 m).

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1. First bring both weights to the same unit: 4 kg = 4000 g.
2. 12/15 of 500 g = 400 g.
3. 3/20 of 4000 g = 600 g.
4. 600 g is more than 400 g, so 3/20 of 4 kg is the heavier weight.

Answer3/20 of 4 kg is heavier (600 g against 400 g).

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3 ÷ 7/9 3 ÷ 7/9

1. Multiply 3 by the reciprocal of 7/9: 3 × 9/7 = 27/7.

Answer27/7

14/4 ÷ 2 14/4 ÷ 2

1. Multiply 14/4 by the reciprocal of 2: 14/4 × 1/2 = 14/8 = 7/4.

Answer7/4

2/3 ÷ 2/3 2/3 ÷ 2/3

1. A number divided by itself is 1.

Answer1

14/6 ÷ 7/3 14/6 ÷ 7/3

1. 14/6 × 3/7 = 42/42 = 1.

Answer1

4/3 ÷ 3/4 4/3 ÷ 3/4

1. 4/3 × 4/3 = 16/9.

Answer16/9

7/4 ÷ 1/7 7/4 ÷ 1/7

1. 7/4 × 7 = 49/4.

Answer49/4

8/2 ÷ 4/15 8/2 ÷ 4/15

1. 8/2 = 4, so 4 × 15/4 = 15.

Answer15

1/5 ÷ 1/9 1/5 ÷ 1/9

1. 1/5 × 9 = 9/5.

Answer9/5

1/6 ÷ 11/12 1/6 ÷ 11/12

1. 1/6 × 12/11 = 12/66 = 2/11.

Answer2/11

3 2/3 ÷ 1 3/8 3 2/3 ÷ 1 3/8

1. Write both as single fractions: 3 2/3 = 11/3 and 1 3/8 = 11/8.
2. 11/3 × 8/11 = 8/3.

Answer8/3

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(a) How many bags did she decorate?

1. The 8 m of lace is used up 1/4 m per bag, so the number of bags is how many 1/4s fit into 8: 8 ÷ 1/4, option (iii).
2. 8 ÷ 1/4 = 8 × 4 = 32.

Answer(iii); 32 bags

(b) What is the length of the ribbon used for each badge?

1. 1/2 m of ribbon is shared equally among 8 badges, so each badge gets 1/2 ÷ 8, option (iv).
2. 1/2 ÷ 8 = 1/2 × 1/8 = 1/16.

Answer(iv); 1/16 m

(c) How many loaves of bread can he make?

1. The 5 kg of flour is used up 1/6 kg per loaf, so the number of loaves is how many 1/6s fit into 5: 5 ÷ 1/6, option (iii).
2. 5 ÷ 1/6 = 5 × 6 = 30.
3. Option (iv), 5 × 6, also gives 30. Dividing by 1/6 is the same as multiplying by 6, because each kilogram makes 6 loaves.

Answer(iii); 30 loaves

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1. 6 rotis are half of 12 rotis, so they need half the flour.
2. Half of 1/4 kg is 1/2 × 1/4 = 1/8 kg.

Answer1/8 kg.

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1. Dividing 1 by a unit fraction just gives its denominator: 1 ÷ 1/6 = 6, 1 ÷ 1/10 = 10, 1 ÷ 1/13 = 13, 1 ÷ 1/9 = 9, 1 ÷ 1/2 = 2.
2. Adding these: 6 + 10 + 13 + 9 + 2 = 40.

Answer40.

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1. In two days Mira read 1/5 + 3/10 = 2/10 + 3/10 = 5/10 = 1/2 of the novel.
2. The part left to read is 1 − 1/2 = 1/2 of the novel.
3. 1/2 of 400 pages = 200 pages.

Answer200 pages.

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1. Write 2 3/4 litres as a single fraction: 2 3/4 = 11/4 litres.
2. Distance = rate × petrol used = 16 × 11/4 = 176/4 = 44 km.

Answer44 km.

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1. Write the train time as a single fraction: 5 1/6 = 31/6 hours.
2. Hours saved = train time − plane time = 31/6 − 1/2 = 31/6 − 3/6 = 28/6 = 14/3 hours.

Answer14/3 hours (4 2/3 hours).

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1. The cake left after Mariam and her cousins ate is 1 − 4/5 = 1/5.
2. Sharing 1/5 of the cake equally among 3 friends gives each of them (1/5) ÷ 3 = 1/15 of the whole cake.

Answer1/15 of the whole cake.

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1. 565/465 is bigger than 1 (565 > 465), and 707/676 is bigger than 1 (707 > 676).
2. When both numbers being multiplied are bigger than 1, their product is bigger than 1 and bigger than each of them.
3. So the product is > 565/465, > 707/676, and > 1, without working out the exact value.

Answer(a), (c) and (e).

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1. The two mid-lines cut the whole square into 4 equal small squares, so the bottom-right small square is 1/4 of the whole.
2. Inside it, the two slanting lines run from its top corners to its centre, so they are halves of its two diagonals. The two full diagonals would cut the small square into 4 equal triangles, each 1/4 of it.
3. The shaded region is the left triangle (1/4 of the small square) together with the left half of the bottom triangle. The vertical line from the centre cuts that triangle into two equal halves, so this piece is 1/2 × 1/4 = 1/8 of the small square.
4. So the shaded region is 1/4 + 1/8 = 3/8 of the small square.
5. The small square is 1/4 of the whole, so the shaded part is 3/8 × 1/4 = 3/32 of the whole square.

Answer3/32 of the whole square.

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1. Read Fig. 8.7 from the bottom. There are four splitting points, and at each one the ants arriving divide equally among all the paths that leave it.
2. 1st point: 2 paths leave it, one to the mango tree and one onward. Each gets 1/2 of the colony.
3. 2nd point: 2 paths leave it, one to the mango tree and one onward. Each gets 1/2 × 1/2 = 1/4 of the colony.
4. 3rd point: 4 paths leave it: two to the mango tree, one straight to the sugarcane field and one on to the top point. Each gets 1/4 × 1/4 = 1/16 of the colony.
5. 4th point (top): 2 paths leave it, one to the mango tree and one to the sugarcane field. Each gets 1/16 × 1/2 = 1/32 of the colony.
6. Mango tree: 1/2 + 1/4 + 1/16 + 1/16 + 1/32 = 16/32 + 8/32 + 2/32 + 2/32 + 1/32 = 29/32.
7. Sugarcane field: 1/16 + 1/32 = 2/32 + 1/32 = 3/32. Check: 29/32 + 3/32 = 1, the whole colony.

Answer29/32 of the original group reached the mango tree and 3/32 reached the sugarcane field.

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1. 1 − 1/2 = 1/2.
2. Each bracket (1 − 1/k) equals (k−1)/k, so (1−1/2)×(1−1/3) = 1/2 × 2/3; the 2 cancels, leaving 1/3.
3. Carrying on the same cancelling, (1−1/2)×(1−1/3)×(1−1/4)×(1−1/5) = 1/2×2/3×3/4×4/5; every middle number cancels the one before it, leaving 1/5.
4. The same cancelling all the way to (1 − 1/10) leaves only the very first numerator (1) and the very last denominator (10): the product is 1/10.
5. In general, every numerator after the first cancels the denominator just before it, so the whole chain up to (1 − 1/n) collapses to 1/n.

Answer1/2; 1/3; 1/5; 1/10; and in general 1/n.

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