How to Solve Math Word Problems About Proportions and …



Solving Math Word Problems Involving Proportions and Ratios

Step 1

Learn to write proportions as fractional equations with matching numerators and denominators.

For example, if the ratio of green to red apples is 2 to 1, you can write G/R = 2/1. Both numerators must represent the same type of object, and both denominators must represent the same type. Here, G and 2 stand for green apples, and R and 1 stand for red apples.

Step 2

Learn the cross multiplication trick for solving equations where two fractions are set equal to each other. For example, if 5/(x+3) = 7/(2x-3), you multiply 5 by (2x-3), and 7 by (x+3) and set the two quantities equal to each other.

The figure at left shows why this mathematical operation is often called "cross multiplication." Click thumbnail to enlarge.

Example #1:

Right now, Rose has 3 times more video games than Nick, i.e., the ratio of Rose's games to Nick's games is 3 to 1. If each one buys 10 more games tomorrow, the ratio of Rose's games to Nick's games will be 5 to 2. How many games does Nick have right now?

First, let x be the number that Nick has right now, and 3x be the number that Rose has. Then tomorrow Nick will have x+10, and Rose will have 3x+10. Thus, (3x+10)/(x+10) = 5/2

When you cross multiply and solve for x, you get x=30.

Example #2:

Try this problem with three quantities:

The ratio of socks to underwear is 7 to 2 and the ratio of underwear to hats is 5 to 3. What is the ratio of socks to hats?

We know S/U = 7/2 and U/H = 5/3 and we want to find the value of S/H. Notice if you multiply the two fractions S/U and U/H, the U's cancel out and you are left with S/H, as desired. Since (7/2)(5/3) = 35/6, we have 35 socks for every 6 hats.

(Note, in this problem we do not use cross multiplication since we do not set two fractions equal to each other.)

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