Ratios and Rate Reasoning: practice solutions, Grade 6 – download the PDF

1 Writing ratios ★★★

(a) Cats to dogs: \( 5:7 \).

(b) Dogs to cats: \( 7:5 \) (the order is reversed).

(c) There are \( 5 + 7 = 12 \) animals, so cats to all animals is \( 5:12 \). None of these can be simplified because the numbers share no common factor.

2 A winning season ★★★

Wins to losses is \( 9:3 \). The GCF of 9 and 3 is 3, so \( 9:3 = 3:1 \).

The team won 3 games for every game it lost.

3 Missing numbers ★★★

(a) \( 2 \times 4 = 8 \), so \( 3 \times 4 = 12 \): \( 2:3 = 8:12 \).

(b) \( 3 \times 10 = 30 \), so \( 2 \times 10 = 20 \): \( 2:3 = 20:30 \).

(c) \( 5 \times 7 = 35 \), so \( 4 \times 7 = 28 \): \( 5:4 = 35:28 \).

4 Simplest form ★★★

(a) GCF = 6: \( 18:24 = 3:4 \).

(b) GCF = 5: \( 45:20 = 9:4 \).

(c) GCF = 8: \( 16:40 = 2:5 \).

5 Typing speed ★★★

Divide the number of words by the number of minutes: \( 120 \div 4 = 30 \).

Noah types 30 words per minute.

6 True or false? ★★★

(a) True. \( 4:6 = 2:3 \) (divide by 2) and \( 10:15 = 2:3 \) (divide by 5). Both simplify to \( 2:3 \).

(b) False. \( 6:9 = 2:3 \) but \( 8:10 = 4:5 \). These simplest forms are different, so the ratios are not equivalent.

7 Notebook table ★★★

Multiply both rows of the first column (4 notebooks, $6) by 2, 3, 4 and 5.

Twenty notebooks cost $30.

8 Smoothie recipe ★★★

Strawberries to bananas is \( 3:2 \). Since \( 3 \times 4 = 12 \), multiply the bananas by 4: \( 2 \times 4 = 8 \).

You need 8 cups of banana slices.

9 Better deal at the store ★★★

Store A: \( 3.00 \div 12 = 0.25 \) dollar per ounce.

Store B: \( 4.40 \div 20 = 0.22 \) dollar per ounce.

Since \( 0.22 < 0.25 \), Store B has the lower unit price.

10 A cyclist’s speed ★★★

(a) \( 36 \div 3 = 12 \), so her speed is 12 miles per hour.

(b) In 5 hours she rides \( 12 \times 5 = 60 \) miles.

11 Printer on a double number line ★★★

Each 3-minute jump adds 12 pages: 0, 3, 6, 9, 12, 15 minutes match 0, 12, 24, 36, 48, 60 pages.

(a) At 9 minutes: 36 pages.

(b) 60 pages match 15 minutes: 15 minutes.

12 Trading cards ★★★

There are \( 3 + 5 = 8 \) boxes. One box is \( 40 \div 8 = 5 \) cards.

Rare cards: \( 3 \times 5 = 15 \). Common cards: \( 5 \times 5 = 25 \).

Check: \( 15 + 25 = 40 \). There are 15 rare and 25 common cards.

13 Inches and centimeters ★★★

Multiply 2.54 by 5, 10 and 20.

3 feet = \( 3 \times 12 = 36 \) inches, and \( 36 \times 2.54 = 91.44 \). So 3 feet is 91.44 centimeters.

14 Which paint is bluer? ★★★

Find the white paint that goes with 12 cans of blue. Mix A: \( 3:5 = 12:20 \). Mix B: \( 4:6 = 12:18 \).

For the same 12 cans of blue, Mix B uses less white (18 instead of 20), so Mix B is bluer.

Check with part-to-whole: \( \dfrac{3}{8} = 0.375 \) and \( \dfrac{4}{10} = 0.4 \), and \( 0.4 > 0.375 \).

15 A new student group ★★★

(a) There are \( 4 + 3 = 7 \) parts, so one part is \( 28 \div 7 = 4 \) students. Boys: \( 4 \times 4 = 16 \). Girls: \( 3 \times 4 = 12 \).

(b) Now there are \( 12 + 4 = 16 \) girls and still 16 boys. The ratio is \( 16:16 = 1:1 \).

16 Fuel economy ★★★

(a) Car A: \( 150 \div 5 = 30 \) miles per gallon. Car B: \( 210 \div 6 = 35 \) miles per gallon. Car B goes farther.

(b) \( 420 \div 35 = 12 \) gallons. Cost: \( 12 \times 3.60 = 43.20 \). The trip costs $43.20.

17 Mixing green paint ★★★

There are \( 2 + 3 = 5 \) parts. One part is \( 45 \div 5 = 9 \) liters.

Yellow: \( 2 \times 9 = 18 \) liters. Blue: \( 3 \times 9 = 27 \) liters. Check: \( 18 + 27 = 45 \).

18 A fast train ★★★

(a) \( 270 \div 3 = 90 \) kilometers per hour.

(b) 90 km = 90,000 m and 1 hour = 3,600 s, so \( 90000 \div 3600 = 25 \) meters per second.

(c) 2 h 40 min = \( 2\dfrac{2}{3} = \dfrac{8}{3} \) hours. Distance: \( 90 \times \dfrac{8}{3} = 240 \) kilometers.

19 Find the mistake ★★★

(a) Adding the same number to both terms does not keep a ratio the same. Equivalent ratios come from multiplying (or dividing) both terms by the same number. Here \( 4 \times 2 = 8 \) but \( 7 \times 2 = 14 \), not 11.

(b) \( x = 14 \), so \( 4:7 = 8:14 \). Also \( \dfrac{4}{7} \approx 0.571 \) and \( \dfrac{8}{11} \approx 0.727 \), which are different.

20 Baking with a fixed recipe ★★★

(a) One cup of sugar goes with \( 5 \div 2 = 2.5 \) cups of flour. For 3 cups of sugar: \( 2.5 \times 3 = 7.5 \) cups of flour.

(b) \( 17.5 \div 5 = 3.5 \) and \( 2 \times 3.5 = 7 \). She needs 7 cups of sugar. Check: \( 17.5:7 = 5:2 \).

21 Saving money ★★★

Maya has 7 boxes and Leo has 4 boxes, so the difference is \( 7 - 4 = 3 \) boxes, which equals $36.

One box is \( 36 \div 3 = 12 \) dollars. Maya: \( 7 \times 12 = 84 \) dollars. Leo: \( 4 \times 12 = 48 \) dollars.

Check: \( 84 - 48 = 36 \) and \( 84:48 = 7:4 \).