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Solving Direct Proportionality Problems on the ACT

The ACT will probably include some math problems that involve direct proportionality. Direct proportionality refers to a connection between two variables based on either multiplication or division, where the variables tend to rise and fall together. That is, as one increases or decreases, the other does the same.

Two variables, x and y, are directly proportional when the following equation is true for some constant k:

image0.png

In practical terms, direct proportionality simply means that as the value of one variable changes, the other value also must change so that any resulting fraction x/y remains constant.

Example 1

Two variables, a and b, are directly proportional. If a = 6, then b = 18. Which of the following must be true?

(A)    If a = 1, then b = 6

(B)    If a = 3, then b = 9

(C)    If a = 12, then b = 12

(D)    If a = 18, then b = 6

(E)    If a = 100, then b = 200

The fraction a/b is a constant, and

image1.png

Thus, any combination of a and b must make a fraction equivalent to 1/3. The only such combination is a = 3 and b = 9, because:

image2.png

So the correct answer is Choice (B).

Example 2

Two variables, x and y, are directly proportional such that if x = 3, then y = 5. What is the value of x when y = 15?

(F)    1

(G)    2

(H)    6

(J)    9

(K)    13

When x = 3 and y = 5:

image3.png

Thus, any value of x/y must also produce the fraction 3/5. Substitute 15 for y into the preceding equation:

image4.png

Cross-multiply and solve for x:

image5.png

Therefore, the correct answer is Choice (J).

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