Lesson 70: Solving Direct Variation Problems

New Concept

When two quantities vary directly, they are proportional. We use the letter $k$ for the constant of proportionality, and their relationship is defined as:

What’s next

This card is just the foundation. Soon, you'll master solving for unknown values using both the constant of proportionality and setting up direct proportions.

Property

When two quantities vary directly, their ratio is a constant, which we call the constant of proportionality, $k$.

Examples

If 2 CDs cost 30 dollars, the constant is $k = \frac{30}{2} = 15$. So, 4 CDs will cost $4 \times 15 = 60$ dollars.
Ellen charges 780 dollars for 60 people, so her rate is $k = \frac{780}{60} = 13$ dollars per person. For 100 people, she charges $100 \times 13 = 1300$ dollars.
A spring stretches 2 cm for a 30g weight, so $k = \frac{2}{30} = \frac{1}{15}$ cm/g. For a 75g weight, it stretches $75 \times \frac{1}{15} = 5$ cm.

Explanation

Think of this as a 'buy one, get one for a fixed price' deal, but for any amount! The relationship between two things stays perfectly consistent. Once you find the special constant, $k$, you can predict the outcome for any number. It's like having a secret formula to know the price of 100 CDs without counting!

Property

An alternate method for solving direct variation problems is to write a proportion by setting the two ratios equal to each other.

Examples

If 60 people cost 780 dollars, how much for 100? Set up the proportion: $\frac{780}{60} = \frac{x}{100}$. Cross-multiply to find $x = 1300$ dollars.
If Maggie drives 75 miles in 1.5 hours, how long for 100 miles? $\frac{75}{1.5} = \frac{100}{t}$. Cross-multiply to find $t = 2$ hours.
If Tom types 100 words in 2.5 minutes, how many in 15? $\frac{100}{2.5} = \frac{w}{15}$. Cross-multiply to find $w = 600$ words.

Explanation

This is the ultimate shortcut for direct variation problems. You don't even need to find the constant $k$! Just set up two fractions that you know are equal: one with the information you have, and one with the missing piece you want to find. Then, use the magic of cross-multiplication to solve for the unknown value.

Property

To make a prediction, first calculate the average rate of change. Then, multiply this rate by the remaining quantity to estimate the outcome.

Examples

In 3 minutes, 5 people were helped. The rate is $\frac{3}{5} = 0.6$ minutes per person. For the next 18 people, the wait is $18 \times 0.6 = 10.8$ minutes.
Mitchell traveled 3 miles in 15 minutes. His rate is $\frac{15}{3} = 5$ minutes per mile. To travel 5 more miles, it will take him $5 \times 5 = 25$ minutes.

Explanation

Life isn't always perfectly predictable, but you can make a solid estimate! First, figure out the 'speed' at which something is happening, like minutes per person in a line. Then, multiply that rate by how much is left to do. It’s like being a detective, using clues from the immediate past to solve a near-future mystery.