This page will teach you how to set up and solve proportions in math. Below the introductory lesson, you will find a series of Grade 7 practice questions accompanied by step-by-step solutions to help you verify your work.
The table below shows the distance traveled by a car in kilometers (km) and the time in hours necessary to travel that distance. We assume that the car travels at a constant speed. How many hours are needed to travel 500 kilometers?
| Distance (km) | 50 | 100 | 150 | 500 |
|---|---|---|---|---|
| Time (hours) | 1 | 2 | 3 | $t$ |
Notice that if we divide the distance by the time for the first three columns, we obtain a constant rate:
$$\frac{50}{1} = 50, \quad \frac{100}{2} = 50, \quad \frac{150}{3} = 50$$
This rate is the constant speed of $50 \text{ km/hr}$. We say the distance is proportional to the time. Because the car maintains the same speed, this proportionality rule also works for the $500 \text{ km}$ distance:
$$\frac{150}{3} = \frac{500}{t}$$
The main difficulty in solving proportions is the presence of denominators (like $3$ and $t$). To solve the proportion, we need to eliminate these denominators. We do this by multiplying both sides of the equation by the product of the denominators ($3 \times t = 3t$).
Step 1: Multiply both sides by $3t$:
$$3t \times \frac{150}{3} = 3t \times \frac{500}{t}$$
Step 2: Simplify by canceling the common terms on each side:
$$t \times 150 = 3 \times 500$$
$$150t = 1500$$
Step 3: Divide both sides by $150$ to isolate $t$:
$$t = \frac{1500}{150} = \mathbf{10 \text{ hours}}$$
$$\frac{x}{2} = \frac{4}{8}$$
Multiply both sides of the proportion by the product of the denominators ($2 \times 8 = 16$):
$$16 \times \frac{x}{2} = 16 \times \frac{4}{8}$$
Simplify by canceling common factors:
$$8x = 2 \times 4$$
$$8x = 8$$
Divide both sides by $8$:
$$\frac{8x}{8} = \frac{8}{8} \implies \mathbf{x = 1}$$
$$\frac{3}{p} = \frac{1}{5}$$
Multiply both sides of the proportion by the product of the denominators ($5 \times p = 5p$):
$$5p \times \frac{3}{p} = 5p \times \frac{1}{5}$$
Simplify by canceling the common factors:
$$5 \times 3 = p \times 1$$
$$\mathbf{15 = p}$$
Multiply both sides of the proportion by the product of the denominators ($5 \times 15 = 75$):
$$75 \times \frac{31}{5} = 75 \times \frac{w}{15}$$
Simplify by calculating the divisions:
$$15 \times 31 = 5 \times w$$
$$465 = 5w$$
Divide both sides by $5$ to isolate $w$:
$$w = \frac{465}{5} \implies \mathbf{w = 93}$$
Alternative shortcut method: Since you only need to isolate $w$, you can simply multiply both sides by $15$:
$$15 \times \frac{31}{5} = w \implies 3 \times 31 = w \implies \mathbf{w = 93}$$
$$\frac{2k}{3} = \frac{20}{6}$$
Multiply both sides of the proportion by the product of the denominators ($3 \times 6 = 18$):
$$18 \times \frac{2k}{3} = 18 \times \frac{20}{6}$$
Simplify by performing the division first:
$$6 \times 2k = 3 \times 20$$
$$12k = 60$$
Divide both sides by $12$:
$$\frac{12k}{12} = \frac{60}{12} \implies \mathbf{k = 5}$$
$$\frac{3}{7} = \frac{y}{0}$$
No solution. Division by zero is undefined in mathematics, making it impossible for the fraction on the right to hold a valid numerical value.
$$\frac{1}{4} = \frac{0}{x}$$
No solution. If $x = 0$, the right side is undefined. For any other value where $x \neq 0$, the right side simplifies to $0$. However, the left side is $\frac{1}{4}$, which is not equal to $0$. Therefore, this proportion has no valid solution.
First, reduce the fraction on the right side to its simplest form:
$$\frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}$$
Rewrite the proportion:
$$\frac{m}{4} = \frac{1}{4}$$
Since the denominators are equal, the numerators must be equal as well. Therefore:
$$\mathbf{m = 1}$$
$$\frac{6}{14} = \frac{2t}{14}$$
Because the denominators on both sides are already equal to $14$, their numerators must be equal for the proportion to be true:
$$6 = 2t$$
Divide both sides by $2$ to isolate $t$:
$$\frac{6}{2} = \frac{2t}{2} \implies \mathbf{3 = t}$$