When they are short, English to Math simple translation works
Percentage word problems are important because they have many real life applications, as in discount sales, interest rates, calculating tips, commissions, taxes, and so on. These problems also show up in a wide variety of school exams, and standardized tests.
Percentage word problems constitute an exceptional category among word problems, in that you can actually translate them word by word into a mathematical equation (not all but most of the short ones). Further, this translation is rather easy, once you get the hang of the problem-solving process we describe below.
A word of caution is in order here. Be aware that word-by-word translation almost never works. If you speak two or more languages, you already know this. Each language has its own grammar rules. Whenever you translate a long English sentence into Spanish, French, Russian, German, or any other language, if you do a verbatim, word-by-word translation, chances are the result will mean something different in the target language than the original English phrase. Sometimes it may sound funny, or weird, or it may even be total nonsense in the other language. That is why word-by-word translation has its own name. It’s called transliteration, because it is not real translation.
Math as a translation target language is no different in this respect. Actually, as you may have noticed, it is harder to translate an English word problem into mathematical equations than it is to translate it into another natural language you know. In math, every single little symbol makes a difference, many times a big difference. If you skip a symbol, add an extra symbol, or switch two symbols, it is very likely you will end up with a meaningless expression, or solving a totally different problem altogether.
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, 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.
5) When the word “of” comes right after a percentage (or a fraction), change it into multiplication (either the “times” symbol, or a couple parentheses indicating multiplication).
6) Change key word / 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.
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 “=”
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.
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