Most marks lost in quantitative reasoning are not lost to arithmetic. They are lost before the arithmetic begins, while the question is still a sentence. A number increased by thirty percent of itself equals 91 is one line of algebra once it is written down, and a trap while it is still English.
Translation is a skill with rules, and the rules are learnable. Apply them mechanically and a word problem stops being a test of reading and becomes a test of substitution.
Treat every word problem as three fixed steps.
The model omits two things you must supply. It assumes every quantity is in the same unit, and it assumes a single unknown; when a question has two, one unknown is expressed in terms of the other.
The keyword table. Learn it until reading a sentence produces symbols automatically.
| English | Symbol | Example |
|---|---|---|
| is, equals, was | = | x is 12 gives x = 12 |
| of | × | 30 percent of x gives 0.30 × x |
| per, for each | ÷ | km per hour gives km ÷ hours |
| more than, increased by | + | 5 more than x gives x + 5 |
| less than, decreased by | − | 5 less than x gives x − 5 |
| sum, total, together | + | the sum of x and y gives x + y |
| product, times | × | 3 times x gives 3x |
| quotient, ratio, split | ÷ | the quotient of x and 3 gives x ÷ 3 |
Percentages are multipliers. The word of is the reason. Thirty percent of itself means 0.30 times the number, so a number increased by thirty percent of itself becomes x + 0.30x, which is 1.30x. Translating the phrase directly is faster and safer than reasoning about it.
Unit consistency. If a rate is per hour and a time is in minutes, convert one before writing the equation. A mismatch here produces a wrong answer that still looks internally consistent.
Back-substitution. Every word problem is self-checking. Put the value you found back into the words and verify. In the example below, 70 increased by 30 percent of 70 is 70 + 21 = 91, exactly as stated.
A number increased by 30 percent of itself equals 91. Find the number.
Name the unknown: let the number be x. Translate: increased by 30 percent of itself is x + 0.30x. Equals 91 sets the whole thing to 91.
Back-substitute: 30 percent of 70 is 21, and 70 + 21 = 91. The sentence is satisfied, so the number is 70.
Notice where the difficulty actually sat. The equation 1.30x = 91 is trivial; the work was recognising that 30 percent of itself is a multiplier applied to x, not an amount added afterwards.
A number increased by 30 percent of itself equals 91. Find the number.
Which expression means seven less than three times a number?
The sum of two consecutive integers is 37. What is the smaller integer?
A car covers 180 km in 3 hours. What is its average speed in km per hour?
Without looking back: what does is translate to, what does of translate to, and what does per translate to?
The answers: is becomes equals, of becomes multiply, and per becomes divide.
In translation, is becomes , of becomes , and per becomes . In the worked example, the unknown number is .
A number is doubled and then 6 is added. The result equals 4 less than three times the number. Find the number.
Name the unknown x. Doubled and then 6 is added is 2x + 6. Four less than three times the number is 3x − 4, because the phrase reduces three times the number.
Back-substitute: 2(10) + 6 = 26, and 3(10) − 4 = 26. The two sides agree, so the number is 10.
The two traps are both translation errors. Four less than reverses, so it is 3x − 4 rather than 4 − 3x; and then 6 is added means the addition happens after the doubling, giving 2x + 6 rather than 2(x + 6).
You can now turn a sentence into an equation. The next lesson applies that skill to the most common commercial setting in the test: buying, marking up, discounting and selling, where the same percentage machinery reappears under the vocabulary of profit, loss and discount.