Ratios and Proportional Relationships: practice solutions, Grade 7 – download the PDF

1 Price per ounce ★★★

a) \( 4.20 \div 15 = 0.28 \). The unit price is $0.28 per ounce.

b) \( 22 \times 0.28 = 6.16 \). The 22-ounce box would cost $6.16.

2 Swimming speed ★★★

\( \dfrac{1/4}{1/5} = \dfrac{1}{4} \times 5 = \dfrac{5}{4} = 1.25 \).

Cara swims 1.25 miles per hour.

3 A proportional table ★★★

Compute \( \dfrac{y}{x} \): \( \dfrac{7.5}{3} = 2.5 \), \( \dfrac{15}{6} = 2.5 \), \( \dfrac{22.5}{9} = 2.5 \), \( \dfrac{30}{12} = 2.5 \).

All the quotients are equal, so the relationship is proportional with \( k = 2.5 \).

4 Not proportional ★★★

\( \dfrac{4}{1} = 4 \) but \( \dfrac{7}{2} = 3.5 \). The quotients are not the same, so the relationship is not proportional. (The values go up by 3 each time, but that is not enough.)

5 Printer speed ★★★

a) \( k = 18 \div 3 = 6 \) pages per minute.

b) \( y = 6x \).

c) \( 6x = 42 \), so \( x = 7 \). It takes 7 minutes.

6 True or false? ★★★

a) True. A straight line through the origin has a constant ratio \( \dfrac{y}{x} \).

b) False. A proportional graph must also be a straight line.

c) False. The line does not pass through \( (0, 0) \). (It would give \( y = 2x + 2 \).)

7 The point (1, r) ★★★

a) \( r = 14 \div 4 = 3.5 \), so the point is \( (1, 3.5) \).

b) One pound of apples costs $3.50.

c) \( 10 \times 3.5 = 35 \). Ten pounds cost $35.

8 A slow snail ★★★

a) \( \dfrac{2/3}{1/4} = \dfrac{2}{3} \times 4 = \dfrac{8}{3} \) feet per hour, about 2.67 ft/h.

b) \( \dfrac{8}{3} \times \dfrac{3}{2} = 4 \). The snail crawls 4 feet.

9 Scaling a recipe ★★★

a) \( \dfrac{9/4}{3/4} = \dfrac{9}{4} \times \dfrac{4}{3} = 3 \). One batch takes 3 cups.

b) \( 3 \times 2.5 = 7.5 \). He needs 7.5 cups of flour.

10 Fill in the table ★★★

\( k = \dfrac{14}{4} = 3.5 \), and \( \dfrac{24.5}{7} = 3.5 \), \( \dfrac{35}{10} = 3.5 \) confirm it.

The equation is \( y = 3.5x \). For \( x = 16 \): \( y = 3.5 \times 16 = 56 \).

11 Gas mileage ★★★

a) \( k = 96 \div 3 = 32 \), so \( y = 32x \).

b) \( 32 \times 7.5 = 240 \) miles.

c) \( 32x = 400 \), so \( x = 12.5 \) gallons.

12 Better buy ★★★

Store A: \( 13.50 \div 5 = 2.70 \) dollars per pound. Store B: \( 20.80 \div 8 = 2.60 \) dollars per pound.

Store B has the lower unit price.

13 Two runners ★★★

a) Ana: \( 30 \div 10 = 3 \) meters per second. Ben: \( 45 \div 10 = 4.5 \) meters per second.

b) Ana: \( y = 3x \). Ben: \( y = 4.5x \).

c) At 12 seconds, Ana has run 36 m and Ben 54 m. Ben is \( 54 - 36 = 18 \) meters ahead.

14 Mixing paint ★★★

There are \( 3 + 5 = 8 \) parts, and each part is \( 24 \div 8 = 3 \) liters.

Blue: \( 3 \times 3 = 9 \) liters. White: \( 5 \times 3 = 15 \) liters. Check: \( 9 + 15 = 24 \).

15 Jacket on sale ★★★

Sale price: \( 64 \times 0.85 = 54.40 \) dollars.

With tax: \( 54.40 \times 1.05 = 57.12 \) dollars.

The final cost is $57.12.

16 Fractions in a table ★★★

a) \( \dfrac{3/4}{1/2} = \dfrac{3}{2} \), \( \dfrac{9/4}{3/2} = \dfrac{9}{4} \times \dfrac{2}{3} = \dfrac{3}{2} \), \( \dfrac{15/4}{5/2} = \dfrac{15}{4} \times \dfrac{2}{5} = \dfrac{3}{2} \). Every quotient equals \( \dfrac{3}{2} \), so the relationship is proportional.

b) \( y = \dfrac{3}{2}x \). For \( x = 4 \): \( y = \dfrac{3}{2} \times 4 = 6 \).

17 Find the error ★★★

a) The quotients are \( \dfrac{5}{2} = 2.5 \), \( \dfrac{8}{4} = 2 \) and \( \dfrac{11}{6} \approx 1.83 \). They are not equal, so the relationship is not proportional. A constant difference does not make a relationship proportional.

b) With \( k = 2.5 \) we need \( y = 2.5 \times 4 = 10 \) and \( y = 2.5 \times 6 = 15 \).

18 Two pumps ★★★

Pump A: \( \dfrac{3/4}{1/2} = \dfrac{3}{2} \) tanks per hour. Pump B: \( \dfrac{5/8}{1/4} = \dfrac{5}{2} \) tanks per hour.

Together: \( \dfrac{3}{2} + \dfrac{5}{2} = 4 \) tanks per hour. One tank takes \( \dfrac{1}{4} \) hour, which is 15 minutes.

19 Markup then discount ★★★

a) Marked price: \( 40 \times 1.35 = 54 \) dollars. Final price: \( 54 \times 0.80 = 43.20 \) dollars.

b) \( 43.20 - 40 = 3.20 \), and \( \dfrac{3.20}{40} = 0.08 \). The final price is 8% more than the store paid.

20 Reading a map ★★★

a) \( \dfrac{3/4}{3/2} = \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{1}{2} \). One mile is \( \dfrac{1}{2} \) inch.

b) \( y = \dfrac{1}{2}x \).

c) \( \dfrac{1}{2}x = 4.375 \), so \( x = 8.75 \). The towns are 8.75 miles apart.

21 Rebuild from one point ★★★

a) \( k = 15 \div 6 = 2.5 \), so \( y = 2.5x \).

b) \( 2.5x = 40 \), so \( x = 16 \).

c) For \( x = 10 \) we get \( y = 25 \), not 26. The point is not on the graph.

22 Cyclist units ★★★

a) \( \dfrac{5/6}{1/18} = \dfrac{5}{6} \times 18 = 15 \) miles per hour.

b) \( 15 \times 5280 = 79{,}200 \) feet per hour, and \( 79200 \div 3600 = 22 \). She rides 22 feet per second.

c) \( 15 \times 1.609 \approx 24.1 \). Her speed is about 24.1 kilometers per hour.