How to Solve Percentage Word Problems

So, given translating word problems from English into Math is generally such a tricky, hazardous task; the fact that short, percentage word problems very often can be correctly translated into math equations using transliteration is, in my opinion, a piece of very good news.

Here is how to do it:

1) Make sure the problem is given in really short phrases. Do not attempt this technique with problems longer than 50 words.

2) Percentage problems often involve three main quantities: a reference total, a part, and the corresponding percent. Make sure to correctly identify them within the problem, meaning, be clear on what is a percentage of what. The “part” is a percentage of the “reference total.”
In the phrase “B is p% of A,” B is the part, p is the percent, and A is the reference total.

3) If needed, or possible, rephrase the problem into simpler, more straightforward forms. You want to ultimately reduce the problem to one of the simplest formats:
“How much is p% of A?”
“B is what percent of A?”; or equivalently “What percent of A is B?”
“B is p% of what amount?”

Here is how to translate each of these simple format phrases into an equation:

“How much is p% of A?” translates into:
X = (p/100)(A)

“B is what percent of A?” translates into:
B = (X / 100)(A)

“B is p% of what amount?” translates into:
B = (p/100)(X)

Once you have the problem reduced to one of these formats through your math resources, you just plug the given numbers into their corresponding places, and solve for X, as shown in the example problems below.

4) Change key phrases like “what is,” or “how much,” into a variable letter like “X,” indicating an unknown quantity you want to solve for though it must be said that this is too far-fetched for 6th graders.

5) When the word “of” comes right after a percentage (or a fraction), change it into multiplication (either the “times” symbol or a couple of parentheses indicating multiplication).

6) Change keyword/phrases like “is,” “was,” “are,” “amounts to,” “constitute,” “is equivalent to,” or “equals,” into the equal sign: “=”

Example problem 1:

What is 20% of 140?

X = (0.20)(140)

X = 28

“What” became “X”; “is” became “=”; “20%” became “0.20”; and the word “of” became the multiplication times 140 indicated by the parentheses. Key is that Math becomes a bit of FUN again.

Example problem 2:

If 150,000 people live in Town A, and 37,500 of them are between the ages of 20 and 30, what is the percentage of Town A’s residents between the ages of 20 and 30?

Here 150,000 is the reference total, town A’s total population. The number 37,500 is the part, not everybody but just those between ages 20 and 30. The problem is not giving us the percent but asking for it, that is the problem: to find the percent. So we can rephrase this problem into the simpler form:

150,000 is the total population. What percent of 150,000 is represented by 37,500?

and then we can translate the question into the following equation:

(X / 100)(150,000) = 37,500

X = (37,500)(100) / 150,000

X = 25 (meaning 37,500 is 25% of 150,000)

“What percent” became “X/100”; “of” became the multiplication indicated by the parentheses; and “is represented by” became the equal sign “=” So make sure you’ll open your child’s’ mind to the logic and fun of Math.

Example problem 3:

Mark spent $33.50 cash on groceries. This is 15% of the cash amount he had when he left home. How much he had when he left home?

Here the reference total is the amount he had when he left home, and we don’t know how much that is, because the problem is not giving us that information. Actually, that is the question, the amount the problem is asking for, so we use “X” to represent it.

The original amount is X; and $33.50 is 15% of X.

$33.50 = (15/100)(X) = (0.15)(X)

X = 33.50 / 0.15

“Is” became “=”; “15%” became 0.15; “of” became the multiplication indicated by the parentheses; and “the amount he had when he left home” became “X.”

I hope these problems throw some light into this process of translating short, percentage word problems into mathematical equations.