Ratios and Proportions • Topic 4 of 7

Direct Variation

In direct variation, one quantity is a constant multiple of another: y = kx, where k is the constant of proportionality. As x increases, y increases in the same proportion, and the ratio y/x stays equal to k. To find k, divide a known y by its x; then you can predict any other value. The graph of y = kx is a straight line through the origin with slope k. Direct variation models cost-per-item, distance at constant speed, and "y is proportional to x" wording — all common SAT setups.

✅ Solved examples

1. y varies directly with x, and y = 12 when x = 3. Find k.
k = y/x = 12/3 = 4.
2. Using y = 4x, find y when x = 7.
y = 4 × 7 = 28.
3. If y is directly proportional to x and y = 20 when x = 5, find y when x = 8.
k = 20/5 = 4, so y = 4 × 8 = 32.
4. A car travels 150 miles in 3 hours at constant speed. How far in 5 hours?
Rate k = 150/3 = 50 mph, so distance = 50 × 5 = 250 miles.

✏️ Practice — try these, take hints as needed

1. y varies directly with x and y = 18 when x = 6. Find k.
k = y/x.
18 ÷ 6.
Compute.
3.
2. Using y = 5x, find y when x = 9.
Substitute x = 9.
5 × 9.
Compute.
45.
3. y ∝ x and y = 28 when x = 4. Find y when x = 10.
Find k = 28/4 = 7.
Then y = 7x.
Use x = 10.
70.
4. A worker earns $96 for 8 hours. At the same rate, how much for 5 hours?
Rate = 96/8 = $12 per hour.
Multiply by 5.
Compute.
$60.
5. y varies directly with x and y = 9 when x = 2. Find y when x = 6.
k = 9/2 = 4.5.
y = 4.5x.
x = 6.
27.

📝 Topic test — 8 questions

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