Std 5 Mathematics · Chapter 5

5. Fractions

Solutions to Problem Sets 17 to 23 — equivalent fractions, comparing, adding and subtracting fractions, and fraction of a collection. Maharashtra State Board (Balbharati), Grade Five Mathematics, Part One.

Key Ideas

Equivalent fractions: multiply or divide the numerator and the denominator by the same non-zero number. 25 = 2 × 35 × 3 = 615

Like fractions have the same denominator; unlike fractions have different denominators. To make unlike fractions like, change both to a common multiple of the denominators.

Comparing: with the same denominator, the bigger numerator is bigger. With the same numerator, the bigger denominator is smaller. Otherwise, first make the denominators equal.

Adding / subtracting like fractions: add or subtract the numerators and keep the denominator. For unlike fractions, first make them like fractions.

Fraction of a number: divide the number by the denominator, then multiply by the numerator. 34 of 20 = (20 ÷ 4) × 3 = 15

Problem Set 17

1Write the proper number in the box.

12 = 1020 (2 × 10 = 20, so 1 × 10 = 10)
34 = 1520 (3 × 5 = 15, so 4 × 5 = 20)
911 = 1822 (9 × 2 = 18, so 11 × 2 = 22)
1040 = 28 (40 ÷ 5 = 8, so 10 ÷ 5 = 2)
1426 = 713 (26 ÷ 2 = 13, so 14 ÷ 2 = 7)
23 = 46 (6 ÷ 2 = 3, so 4 ÷ 2 = 2)
15 = 420 (4 ÷ 4 = 1, so 20 ÷ 4 = 5)
25 = 1025 (25 ÷ 5 = 5, so 10 ÷ 5 = 2)

2Find an equivalent fraction with denominator 18, for each of the following fractions.

12 = 1 × 92 × 9 = 918
23 = 2 × 63 × 6 = 1218
46 = 4 × 36 × 3 = 1218
29 = 2 × 29 × 2 = 418
79 = 7 × 29 × 2 = 1418
53 = 5 × 63 × 6 = 3018

3Find an equivalent fraction with denominator 5, for each of the following fractions.

615 = 6 ÷ 315 ÷ 3 = 25
1025 = 10 ÷ 525 ÷ 5 = 25
1230 = 12 ÷ 630 ÷ 6 = 25
610 = 6 ÷ 210 ÷ 2 = 35
2135 = 21 ÷ 735 ÷ 7 = 35

4From the fractions given below, pair off the equivalent fractions: 23, 57, 511, 79, 1418, 1533, 1827, 1014

23 and 1827 (dividing 18 and 27 by 9 gives 23)
57 and 1014 (dividing 10 and 14 by 2 gives 57)
511 and 1533 (dividing 15 and 33 by 3 gives 511)
79 and 1418 (dividing 14 and 18 by 2 gives 79)

5Find two equivalent fractions for each of the following fractions.

79 = 1418 = 2127 (multiplying by 2 and by 3)
45 = 810 = 1215 (multiplying by 2 and by 3)
311 = 622 = 933 (multiplying by 2 and by 3)

Any fraction obtained by multiplying the numerator and denominator by the same number is correct, so other answers are also possible.

Problem Set 18

QConvert the given fractions into like fractions.

34, 58 — common denominator 8: 3 × 24 × 2 = 68; 58. Like fractions: 68, 58
35, 37 — common denominator 35: 3 × 75 × 7 = 2135; 3 × 57 × 5 = 1535. Like fractions: 2135, 1535
45, 310 — common denominator 10: 4 × 25 × 2 = 810; 310. Like fractions: 810, 310
29, 16 — common denominator 18: 2 × 29 × 2 = 418; 1 × 36 × 3 = 318. Like fractions: 418, 318
14, 23 — common denominator 12: 1 × 34 × 3 = 312; 2 × 43 × 4 = 812. Like fractions: 312, 812
56, 45 — common denominator 30: 5 × 56 × 5 = 2530; 4 × 65 × 6 = 2430. Like fractions: 2530, 2430
38, 16 — common denominator 24: 3 × 38 × 3 = 924; 1 × 46 × 4 = 424. Like fractions: 924, 424
16, 49 — common denominator 18: 1 × 36 × 3 = 318; 4 × 29 × 2 = 818. Like fractions: 318, 818

The smallest common multiple of the denominators is used here. Any other common multiple (for example 16 or 24 in (1)) also gives correct like fractions. In (4), the fractions are read from the textbook page as 29 and 16.

Problem Set 19

QWrite the proper symbol from <, > or = in the box.

37 = 37 (same denominator, compare numerators)
38 > 28 (same denominator, compare numerators)
211 < 1011 (same denominator, compare numerators)
515 = 1030 (1030 and 1030)
58 > 59 (same numerator, the smaller denominator is bigger)
47 > 411 (same numerator, the smaller denominator is bigger)
1011 > 1013 (same numerator, the smaller denominator is bigger)
15 > 19 (same numerator, the smaller denominator is bigger)
56 > 18 (2024 and 324)
512 > 16 (512 and 212)
78 = 1416 (1416 and 1416)
49 = 49 (same denominator, compare numerators)
518 > 19 (518 and 218)
23 > 47 (1421 and 1221)
37 < 59 (2763 and 3563)
411 > 15 (2055 and 1155)

Problem Set 20

1Add:

15 + 35 = 1 + 35 = 45
27 + 47 = 2 + 47 = 67
712 + 212 = 7 + 212 = 912 = 34
29 + 79 = 2 + 79 = 99 = 1
315 + 415 = 3 + 415 = 715
27 + 17 + 37 = 2 + 1 + 37 = 67
210 + 410 + 310 = 2 + 4 + 310 = 910
49 + 19 = 4 + 19 = 59
58 + 38 = 5 + 38 = 88 = 1

2Mother gave 38 of one guava to Meena and 28 of the guava to Geeta. What part of the guava did she give them altogether?

38 + 28 = 3 + 28 = 58

Mother gave them 58 of the guava altogether.

3The girls of Std V cleaned 34 of a field while the boys cleaned 14. What part of the field was cleaned altogether?

34 + 14 = 3 + 14 = 44 = 1

The whole field (1) was cleaned altogether.

Problem Set 21

1Subtract:

57 − 17 = 5 − 17 = 47
58 − 38 = 5 − 38 = 28 = 14
79 − 29 = 7 − 29 = 59
811 − 511 = 8 − 511 = 311
913 − 413 = 9 − 413 = 513
710 − 310 = 7 − 310 = 410 = 25
912 − 212 = 9 − 212 = 712
1015 − 315 = 10 − 315 = 715

2710 of a wall is to be painted. Ramu has painted 410 of it. How much more needs to be painted?

710 − 410 = 7 − 410 = 310

310 of the wall still needs to be painted.

Problem Set 22

1Add:

18 + 34 = 18 + 3 × 24 × 2 = 18 + 68 = 1 + 68 = 78
221 + 37 = 221 + 3 × 37 × 3 = 221 + 921 = 2 + 921 = 1121
25 + 13 = 2 × 35 × 3 + 1 × 53 × 5 = 615 + 515 = 6 + 515 = 1115
27 + 12 = 2 × 27 × 2 + 1 × 72 × 7 = 414 + 714 = 4 + 714 = 1114
39 + 35 = 3 × 59 × 5 + 3 × 95 × 9 = 1545 + 2745 = 15 + 2745 = 4245 = 1415

2Subtract:

310 − 120 = 3 × 210 × 2 − 120 = 620 − 120 = 6 − 120 = 520 = 14
34 − 12 = 34 − 1 × 22 × 2 = 34 − 24 = 3 − 24 = 14
614 − 27 = 614 − 2 × 27 × 2 = 614 − 414 = 6 − 414 = 214 = 17
46 − 35 = 4 × 56 × 5 − 3 × 65 × 6 = 2030 − 1830 = 20 − 1830 = 230 = 115
27 − 14 = 2 × 47 × 4 − 1 × 74 × 7 = 828 − 728 = 8 − 728 = 128

Where the answer can be divided further, its simplest form is also shown. Both forms are correct.

Problem Set 23

1What is 13 of each of the collections given below?

13 of 15 pencils = 15 ÷ 3 = 5 pencils
13 of 21 balloons = 21 ÷ 3 = 7 balloons
13 of 9 children = 9 ÷ 3 = 3 children
13 of 18 books = 18 ÷ 3 = 6 books

2What is 15 of each of the following?

15 of 20 rupees = 20 ÷ 5 = 4 rupees
15 of 30 km = 30 ÷ 5 = 6 km
15 of 15 litres = 15 ÷ 5 = 3 litres
15 of 25 cm = 25 ÷ 5 = 5 cm

3Find the part of each of the following numbers equal to the given fraction.

23 of 30 = (30 ÷ 3) × 2 = 10 × 2 = 20
711 of 22 = (22 ÷ 11) × 7 = 2 × 7 = 14
38 of 64 = (64 ÷ 8) × 3 = 8 × 3 = 24
513 of 65 = (65 ÷ 13) × 5 = 5 × 5 = 25

Activities

1Colour the Hats. Colour 13 of the hats red and 35 of the hats blue. How many hats have you coloured red? How many blue? How many are still not coloured?

There are 15 hats in the picture.

Red: 13 of 15 = 15 ÷ 3 = 5 hats
Blue: 35 of 15 = (15 ÷ 5) × 3 = 9 hats
Not coloured: 15 − (5 + 9) = 15 − 14 = 1 hat

15 is a common multiple of 3 and 5, so both fractions give whole numbers of hats. Count the hats in your own book once to confirm.

2Make a Magic Spinner. What fraction of the disc is each of the coloured parts? What colour does it appear to be when it spins fast?

The disc is divided into 6 equal parts, so each coloured part is 16 of the disc.

When the disc spins fast, the six colours mix in our eyes and the disc appears whitish (nearly white or light grey).

★The Clever Poet — fill in the blanks and answer the question.

First guard: 110 of 100 = 100 ÷ 10 = 10 lashes
Second guard: 25 of 100 = (100 ÷ 5) × 2 = 40 lashes
Third guard: half (12) of 100 = 100 ÷ 2 = 50 lashes

Together: 10 + 40 + 50 = 100 lashes. In fractions, 110 + 25 + 12 = 110 + 410 + 510 = 1010 = 1, the whole prize.

The poet’s clever idea: The guards had demanded shares of his prize: 110, 25 and then 12 (the poet offered half instead of 14). These three shares add up to exactly the whole prize. So the poet asked the king for a “prize” of 100 lashes of the whip. All 100 lashes then went to the three greedy guards, and none to the poet. In this way he exposed their greed without accusing them himself, and the king appreciated his cleverness.

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