8. Working with Fractions
Answers to all Figure it Out and in-text questions on multiplying and dividing fractions, with word problems. CBSE / NCERT Class 7 Ganita Prakash.
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8.1 Multiplication of Fractions
Distance = 25 × 3 = 2 × 35 = 65 km = 115 km.
1. Tenzin drinks 12 glass of milk every day. How much in a week? In January?
In a week: 7 × 12 = 72 = 312 glasses.
In January (31 days): 31 × 12 = 312 = 1512 glasses.
2. A team makes 1 km of a canal in 8 days.
In one day: 18 km. In a week of 5 days: 5 × 18 = 58 km.
3. 5 litres of oil shared equally by 3 families every week.
Each family in a week: 5 ÷ 3 = 53 = 123 litres.
In 4 weeks: 4 × 53 = 203 = 623 litres.
4. The Moon sets 56 hour later each day. It set at 10 pm on Monday. When will it set on Thursday?
Thursday is 3 days after Monday. Delay = 3 × 56 = 156 = 212 hours.
So, the Moon sets 212 hours after 10 pm, that is, at 12:30 am.
5. Multiply and convert into a mixed fraction.
(a) 7 × 35 = 215 = 415
(b) 4 × 13 = 43 = 113
(c) 97 × 6 = 547 = 757
(d) 1311 × 6 = 7811 = 7111
32 divided into 4 equal parts is 38. Then 5 × 38 = 158.
1. Find the products (use a unit square).
(a) 13 × 15 = 115 (the square is cut into 5 rows and 3 columns; 1 of 15 parts)
(b) 14 × 13 = 112
(c) 15 × 12 = 110
(d) 16 × 15 = 130
112 × 118 = 1216
2. Find the products.
(a) 23 × 45 = 815
(b) 14 × 23 = 212 = 16
(c) 35 × 12 = 310
(d) 46 × 35 = 1230 = 25
1. A tap fills 710 of a tank in 1 hour. How much is filled in:
(a) 13 hour: 13 × 710 = 730
(b) 23 hour: 23 × 710 = 1430 = 715
(c) 34 hour: 34 × 710 = 2140
(d) 710 hour: 710 × 710 = 49100
(e) Time to fill the tank = 1 ÷ 710 = 107 = 137 hours.
2. The government took 16 of Somu’s land.
Land left with Somu = 1 − 16 = 56.
(a) Krishna: 12 × 56 = 512 of the original land.
(b) Bora: 13 × 56 = 518 of the original land.
(c) Somu keeps 56 − 512 − 518 = 3036 − 1536 − 1036 = 536.
3. Area of a rectangle of sides 334 ft and 935 ft
154 × 485 = 72020 = 36 sq ft.
4. Four saplings, 34 m apart. Distance between the first and last?
There are 3 gaps. Distance = 3 × 34 = 94 = 214 m.
5. Which is heavier: 1215 of 500 g or 320 of 4 kg?
1215 × 500 g = 400 g. 320 × 4000 g = 600 g.
So, 320 of 4 kg is heavier.
Both numbers greater than 1 (e.g. 43 × 4 = 163): the product is greater than both.
Both between 0 and 1 (e.g. 12 × 35 = 310): the product is less than both.
One between 0 and 1, one greater than 1 (e.g. 34 × 5 = 154): the product lies between them.
• When one of the numbers is between 0 and 1, the product is less than the other number.
• When one of the numbers is greater than 1, the product is greater than the other number.
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8.2 Division of Fractions
If the divisor is greater than 1, the quotient is less than the dividend (6 ÷ 3 = 2).
If the divisor is between 0 and 1, the quotient is greater than the dividend (6 ÷ 14 = 24).
For the divisor, use divisor × quotient = dividend. If the quotient is greater than 1, the dividend is greater than the divisor. If the quotient is between 0 and 1, the dividend is less than the divisor.
8.3 Some Problems Involving Fractions
Number of bricks = 152 ÷ 125 = 152 × 25 = 3752 = 18712.
In a day: 1 ÷ 1 = 1, 1 ÷ 12 = 2, 1 ÷ 14 = 4, 1 ÷ 15 = 5 times.
Together: 1 + 2 + 4 + 5 = 12 times a day. So, they fill it once in 112 of a day (2 hours).
(a) The band lies between the diagonal and a parallel line through the midpoints of the left and bottom sides.
Half the square lies below the diagonal. From it, remove the corner triangle of area 12 × 12 × 12 = 18.
Shaded part = 12 − 18 = 38.
(b) The top-left quarter (14 of the square) has a tilted square joining the midpoints of its sides. The two shaded corner triangles make 14 of the quarter.
Shaded part = 14 × 14 = 116.
1 copper pana = 112 × 14 = 148 gold dinar.
1 cowrie shell = 130 copper pana.
1 cowrie shell = 130 × 148 = 11440 gold dinar.
1. Evaluate.
3 ÷ 79 = 3 × 97 = 277; 144 ÷ 2 = 74; 23 ÷ 23 = 1; 146 ÷ 73 = 1;
43 ÷ 34 = 169; 74 ÷ 17 = 494; 82 ÷ 415 = 15;
15 ÷ 19 = 95; 16 ÷ 1112 = 211; 323 ÷ 138 = 113 × 811 = 83 = 223
2. Choose the expression and simplify.
(a) (iii) 8 ÷ 14 = 32 bags
(b) (iv) 12 ÷ 8 = 116 m for each badge
(c) (iii) 5 ÷ 16 = 30 loaves
3. 14 kg flour makes 12 rotis. Flour for 6 rotis?
6 rotis are half of 12. So, flour = 12 × 14 = 18 kg.
4. Sridharacharya’s sum
1 ÷ 16 + 1 ÷ 110 + 1 ÷ 113 + 1 ÷ 19 + 1 ÷ 12 = 6 + 10 + 13 + 9 + 2 = 40.
5. Pages left for Mira
Yesterday: 15 × 400 = 80. Today: 310 × 400 = 120.
Pages left = 400 − 80 − 120 = 200.
6. A car runs 16 km on 1 litre. How far on 234 litres?
16 × 114 = 44 km.
7. How many hours does the plane save?
516 − 12 = 316 − 36 = 286 = 423 hours.
8. How much cake did each friend get?
Cake left = 1 − 45 = 15. Each friend = 15 ÷ 3 = 115 of the cake.
9. Product of 565465 × 707676
Both fractions are greater than 1. So, the product is greater than each of them and greater than 1.
Correct options: (a), (c) and (e).
10. What fraction of the whole square is shaded?
The shaded part is in the bottom-right quarter. It is 38 of that quarter (a rectangle of half the quarter, minus a small triangle of 18 of the quarter).
Shaded part = 38 × 14 = 332 of the whole square.
11. What fraction of the ants reached each food source?
The splits give 12, 14, 116, 116 and 132 to the mango tree, and 116 and 132 to the sugarcane field.
Mango tree: 1632 + 832 + 232 + 232 + 132 = 2932.
Sugarcane field: 232 + 132 = 332.
12. Products of (1 − 12) × (1 − 13) × …
1 − 12 = 12
12 × 23 = 13
12 × 23 × 34 × 45 = 15
Up to (1 − 110): 110
General statement: (1 − 12) × (1 − 13) × … × (1 − 1n) = 1n. Each numerator cancels the denominator before it, so only 1 on top and n at the bottom are left.
Puzzle: Non-attacking Queens
4 × 4: put the queens in rows 1 to 4 at columns 2, 4, 1, 3.
8 × 8: one answer is columns 1, 5, 8, 6, 3, 7, 2, 4 for rows 1 to 8. No two queens share a row, column or diagonal.