Exercise practice 1: Proportional Reasoning-1

3.1 Figure it Out: proportions and missing terms

This page answers the first Figure it Out set on textbook page 165 and the second set on pages 170 and 171. Every question and sub-part is solved with steps.

3.1.1 FIO 1, Q1: which proportion statements are true?

Statement: decide which of these six proportion statements are true, one at a time.

• First statement: 4 : 7 :: 12 : 21
• Second statement: 8 : 3 :: 24 : 6
• Third statement: 7 : 12 :: 12 : 7
• Fourth statement: 21 : 6 :: 35 : 10
• Fifth statement: 12 : 18 :: 28 : 12
• Sixth statement: 24 : 8 :: 9 : 3

Key step: a : b :: c : d is true exactly when ad = bc. Check each pair.

(i) 4 × 21 = 84 and 7 × 12 = 84, so true. (ii) 8 × 6 = 48 and 3 × 24 = 72, so false. (iii) 7 × 7 = 49 and 12 × 12 = 144, so false. (iv) 21 × 10 = 210 and 6 × 35 = 210, so true. (v) 12 × 12 = 144 and 18 × 28 = 504, so false. (vi) 24 × 3 = 72 and 8 × 9 = 72, so true.

Answer: (i), (iv) and (vi) are true.

3.1.2 FIO 1, Q2: three ratios proportional to 4 : 9

Key step: multiply both terms by the same number. By 2: 8 : 18. By 3: 12 : 27. By 4: 16 : 36.

Answer: 8 : 18, 12 : 27 and 16 : 36 all have simplest form 4 : 9.

3.1.3 FIO 1, Q3: missing terms for ratios proportional to 18 : 24

Key step: 18 : 24 in simplest form is 3 : 4. Use this form for each pair.

• First term 3: the ratio is 3 : 4, so the missing second term is 4.
• First term 12: 12 is 4 times 3, so the second term is 4 times 4 = 16.
• First term 20: 20 is 20/3 times 3, so the second term is 20/3 × 4 = 80/3 = 26⅔.
• First term 27: 27 is 9 times 3, so the second term is 9 times 4 = 36.

Answer: 3 : 4, 12 : 16, 20 : 80/3 and 27 : 36.

3.1.4 FIO 1, Q4: which rectangles are similar?

Statement: the chapter shows five rectangles A to E. Measure each with a scale, compare the ratios of width to height, and decide which rectangles are similar.

Key step: two rectangles are similar when the ratio of width to height is the same in simplest form. Measure each rectangle, write width : height, divide both terms by their HCF, and compare the resulting ratios. Rectangles with equal ratios are similar even when their sizes differ.

Answer: the method decides the question, and the particular pair depends on the printed sizes. In general, rectangles whose ratios reduce to the same pair of numbers are similar; a different simplest form means a different shape. The activity is about carrying out the measurement and comparison, not about guessing from appearance alone.

3.1.5 FIO 1, Q5: draw a smaller and a larger rectangle

Statement: given one rectangle, draw a smaller and a bigger one that keep the same width-to-height comparison, then compare with classmates.

Key step: measure the given rectangle and reduce its width : height. Then choose any convenient width and compute the matching height, or choose a height and compute the matching width.

Answer: drawings with the same simplest ratio are all correct even when their sizes differ, because size does not decide similarity; the ratio does. Classmates whose rectangles have a different simplest form drew a different shape, not a wrong method.

3.1.6 FIO 1, Q6: grey and coloured bricks in the wall patterns

Statement: the chapter shows two brick wall patterns. For each pattern, compare the number of grey bricks with the number of coloured bricks and reduce the result.

Key step: count the grey and coloured bricks in one repeating block of each pattern. Pattern (a) has 9 grey bricks and 6 coloured bricks per block. Pattern (b) has 16 grey bricks and 12 coloured bricks per block.

Answer: pattern (a) gives 9 : 6 = 3 : 2; pattern (b) gives 16 : 12 = 4 : 3.

3.1.7 FIO 1, Q7: ratios of head, torso, arms and legs

Statement: measure a friend's body parts (head, torso, arms, legs), write the three ratios listed in the chapter, then draw a figure that keeps those ratios.

Key step: measure each part in the same unit, write each ratio, and reduce it by the HCF. To draw the figure, choose a convenient head length and multiply by each reduced ratio to get the other lengths.

Answer: the ratios depend on the person measured; the drawing is correct when every part keeps the same ratio to the torso. A drawing with different ratios changes the proportions and looks unrealistic.

3.2 Figure it Out: real-world proportions

3.2.1 FIO 2, Q1: Earth's distance in a week

Statement: in a full year Earth's orbit is about 940 million kilometres long. What distance does a single week cover?

Key step: take one year as 52 weeks and set up the proportion 52 weeks : 940,000,000 km :: 1 week : x.

Answer: x = 940,000,000 ÷ 52 ≈ 18,076,923 km. Earth travels about 18 million kilometres in a week, which is a little over 1.8 crore kilometres.

3.2.2 FIO 2, Q2: the mason's bricks

Statement: a mason needs about 1,450 bricks for a 10-feet wall. The house plan needs outer walls plus one inner wall, and all walls have the same height and thickness. How many bricks are needed?

Key step: the number of bricks is proportional to the length of the wall. The plan labels ten wall segments: 12 feet, then 12 feet, then 12 feet, then 15 feet, then 9 feet, then 15 feet, then 9 feet, then 9 feet, then 9 feet and finally 6 feet. Their total is 108 feet. Set up 10 ft : 1,450 bricks :: 108 ft : x.

Answer: x = 1,450 × 108 ÷ 10 = 15,660 bricks.