Direct Variation

Direct variation is written as the equation y = kx, where k is the constant of variation (or rate). This relationship means that when x doubles, y doubles too. When x is cut in half, y is cut in half as well.

Let's see how this works in real situations. When the Acela Express train travels 455 miles in 7 hours, we can find how far it goes in 4 hours by finding the constant k first. We divide 455 by 7 to get k = 65, meaning the train travels at 65 miles per hour. Then we multiply: y = 65(4) = 260 miles in 4 hours.

The same approach works for calculating pay. If Heather earns $174 for 24 hours of work, her rate is $7.25 per hour (found by dividing 174 by 24). So for 40 hours, she'll earn y = 7.25(40) = $290.

Quick Tip: To solve any direct variation problem, first find the constant k using a known pair of values, then use that k to find any unknown values.

For calorie burning, if a 150-pound person burns 207 calories swimming for 30 minutes, the rate is 6.9 calories per minute. To burn 520 calories, they need to swim for approximately 75.4 minutes (520 ÷ 6.9). Direct variation makes calculating these proportional relationships straightforward!