How to Solve Proportion Problems

Problems involving proportions aren’t too common on the SAT (or ACT, GMAT, or GRE), but they stump many students.

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One preliminary issue – you may encounter direct or indirect proportions on the test. If two variables are directly proportional, any pair of values will have the same ratio (e.g. 1 and 2, 3 and 6, etc.). Inverse proportionality is trickier – each pair of values will have the same product, as you will see.

Here are some ways to solve proportion problems:

1) The Algebra Method

This is the classical way of solving these problems, and it works for both direct and indirect proportions.

For direct proportions, use the formula y = kx.

Q. If y is proportional to x, and x = 14 when y = 2, what is the value of y when x = 5?

Take the values from the known pair (x = 14, y = 2), and substitute them into the formula
y = kx. You get 2 = 14k. Divide both sides of the equation by 14 to get k = 2/14 = 1/7. Now substitute the value of x for the second pair (x = 5), along with the value of k you just found (k = 1/7), and get y = (1/7)(5) = 5/7.

For indirect proportions, use the formula xy = k (remember, x and y always have the same product).

Q. If y is inversely proportional to x, and x = 12 when y = 3, what is the value of y
when x = 2?

Note that “inversely proportional” means the same thing as “indirectly proportional.” Again, just substitute the values of x and y for the first pair (x = 12, y = 3) into the proper formula xy = k and get (12)(3) = k (so k = 36). So when x = 2, 2y = 36, and y = 18.

If you’re very comfortable using algebra, this technique is all you need. However, many students find algebra tricky, and even good algebraists (yes, that’s a real word) tend to make careless errors, so read on.

2) Setting Up a Proportion

This method also involves algebra, but it’s more intuitive, and tends to result in fewer careless errors. The downside is that it only works for direct proportions. For comparison, I’ll use the same problem as before:

Q. If y is proportional to x, and x = 14 when y = 2, what is the value of y when x = 5?

Set up y1/x1 = y2/x2.  So 2/14 = y/5. Now just cross multiply: (2)(5) = 14y, 10 = 14y, y  = 10/14 = 5/7.

If you use the x values in the denominator, or both x values in the first fraction, you’ll probably still solve the problem correctly. Just make sure you order the variables logically.

3) Do the Same / Do the Opposite

This is a non-algebraic solution; many students come to prefer it. Let’s look at the previous problems:

Q. If y is proportional to x, and x = 14 when y = 2, what is the value of y when x = 5?

We can see what happens to x as we go from the first pair of values to the second – it goes from 14 to 5. In other words, multiply the first value by 5/14 to get the second value. Now just do the same to y. 2(5/14) = 10/14 = 5/7.

Q. If y is inversely proportional to x, and x = 12 when y = 3, what is the value of y when x = 2?

Here, x goes from 12 to 2. That is, x was divided by 6. Now do the opposite to y (i.e. multiply by 6). 3 X 6 = 18.

Let’s finish up with a couple of advanced problems.

Q. If y is proportional to x2, and y = 5 when x = 2, what is the value of y when x = 4?

It might help to draw 3 columns for y, x, and x2. Remember to use y and x2 for each pair of values. You can use any of the 3 methods described above. Here’s the 3rd (Do the Same):

For the first pair, y = 5, x2 = 4. For the second pair, x2 = 16, so we had to multiply the first value of x2 by 4. So we just do the same to y, and 5 X 4  = 20.

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Q. If 5 people working together can do a job in 8 hours, how long would it take 9 people to do the same job (assume that everyone works at the same rate)?

If you recognize that the values are inversely proportional (as the number of people goes up, the time required goes down), you can use method 1 or 3 above. Using Method 3 (Do the Opposite), the number of people goes from 5 to 9, so the first value was multiplied by 9/5 to get the second value. So just divide the initial time by 9/5: 8 X 5/9 = 40/9 hours.

But you shouldn’t panic if you don’t realize that this is a proportion problem. Just say that each person can do 1/5 of the job in 8 hours. Since 9 people will do 9/5 of the job in 8 hours, you want to multiply 8 hours by 5/9 to get one job done.

And if this is a multiple choice question, you could just estimate the answer and not even do the math (5 people in 8 hours, so 10 people would take 4 hours, and 9 people would take a little more than 4 hours).

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