Math 1 - Ratio and Proportion Worksheet Solutions

Ratio and Proportion Worksheet Solutions
Here are the step-by-step solutions for the Grade 6 Mathematics worksheet.

1. Express each of the following ratios in the simplest form:

(a) 5.6 m to 28 cm

Convert to same units: $5.6 \text{ m} = 560 \text{ cm}$

Ratio: $560 : 28$

Divide both by 28: 20 : 1

(b) dozen to a score

1 dozen = 12; 1 score = 20

Ratio: $12 : 20$

Divide both by 4: 3 : 5

(c) 6 hours to a day

1 day = 24 hours

Ratio: $6 : 24$

Divide both by 6: 1 : 4

(d) 20 litres to 0.75 litres

Ratio: $20 : 0.75 \rightarrow 2000 : 75$

Divide both by 25: 80 : 3

2. Compare the following ratios:

(a) 5 : 7 and 4 : 3

Convert to fractions: $\frac{5}{7}$ and $\frac{4}{3}$

Cross-multiply: $5 \times 3 = 15$ and $4 \times 7 = 28$

Since $15 < 28$, $4 : 3$ is greater than $5 : 7$.

(b) 1/3 : 1/4 and 1/5 : 1/4

First ratio: $\frac{1}{3} \div \frac{1}{4} = \frac{4}{3}$

Second ratio: $\frac{1}{5} \div \frac{1}{4} = \frac{4}{5}$

Comparing $\frac{4}{3}$ and $\frac{4}{5}$: Since the numerators are the same, the fraction with the smaller denominator is larger.

$1/3 : 1/4$ is greater than $1/5 : 1/4$.

3. Which of the following statements are true?

(a) 25 : 35 = 45 : 55

Simplifying: $5 : 7 = 9 : 11$. False.


(b) 105 : 30 = 49 : 14

Simplifying LHS (divide by 15): $7 : 2$

Simplifying RHS (divide by 7): $7 : 2$

True.

4. Determine if the following ratios form a proportion:

(a) 25 cm : 1 m = $40 : $160

LHS: $25 \text{ cm} : 100 \text{ cm} = 1 : 4$

RHS: $40 : 160 = 1 : 4$

Yes, they form a proportion.

(b) 32 m : 64 m = 7 seconds : 14 seconds

LHS: $32 : 64 = 1 : 2$

RHS: $7 : 14 = 1 : 2$

Yes, they form a proportion.

5. Find the value of x in each of the following:
(Assuming these are in proportion: $a:b = c:d$)

(a) 4, 5, x, 480 $\rightarrow \frac{4}{5} = \frac{x}{480} \rightarrow 5x = 1920 \rightarrow$ x = 384

(b) 9, 21, 33, x $\rightarrow \frac{9}{21} = \frac{33}{x} \rightarrow 9x = 693 \rightarrow$ x = 77

(c) x, 30, 60 (Assuming mean proportion $x:30 = 30:60$) $\rightarrow 60x = 900 \rightarrow$ x = 15

(d) 6, x, 24 (Assuming mean proportion $6:x = x:24$) $\rightarrow x^2 = 144 \rightarrow$ x = 12

6. The ratio of cats to dogs is 3:5. If there are 12 cats, how many dogs are there?

$\frac{3}{5} = \frac{12}{\text{dogs}}$

$3 \times \text{dogs} = 60 \rightarrow$ 20 dogs

7. A car travels 30 miles in 45 minutes. What is the ratio of distance traveled to time taken in miles per minute?

Ratio = $30 : 45$

Simplified: 2 : 3 (or 0.67 miles/minute)

8. A recipe calls for 4 eggs for every 2 cups of milk. How many eggs are needed if you use 10 cups of milk?

Ratio = $4 : 2$ (which is $2 : 1$)

If milk is 10, eggs = $10 \times 2 = $ 20 eggs

9. The ratio of red balls to blue balls is 3:4. If there are 21 red balls, how many blue balls are there?

$\frac{3}{4} = \frac{21}{\text{blue}}$

$3 \times \text{blue} = 84 \rightarrow$ 28 blue balls

10. The ratio of boys to girls in a school is 5:8. If there are 400 boys, how many girls are there?

$\frac{5}{8} = \frac{400}{\text{girls}}$

$5 \times \text{girls} = 3200 \rightarrow$ 640 girls

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