Getting from English to algebra — and the two phrases that come out backwards.
The problem
Five less than twice a number is 17. Find the number.
"Five less than twice a number" is 2n − 5, not 5 − 2n — the phrase names the amount subtracted before the thing it comes off. So 2n − 5 = 17 and n = 11. D solves 2n + 5 = 17, reading "less" as "more".
| Operation | Words that signal it |
|---|---|
| + | sum, more than, increased by, total, plus, added to |
| − | difference, less than, decreased by, minus, fewer |
| × | product, times, twice, of, doubled, tripled |
| ÷ | quotient, per, ratio of, divided by |
| = | is, was, equals, gives, results in, will be |
“Is” is the one people forget to look for — it marks where the equals sign goes.
This single phrase causes more errors than the rest combined.
x − 5
start at x, take 5 away
5 − x
that is “5 minus x”
Same with “subtracted from” — 5 subtracted from x is x − 5. “More than” is safe, because addition does not care about order.
“x divided into 20” means 20 ÷ x, not x ÷ 20. Any phrase with from, into or than is worth reading twice.
a number → x
name it
twice a number → 2x
multiply
five less than that → 2x − 5
reversed, as in Step 2
is 17 → 2x − 5 = 17
the equation
x = 11
+ 5, then ÷ 2
The word “the” usually signals a whole quantity being acted on.
| Phrase | Expression |
|---|---|
| twice the sum of x and 3 | 2(x + 3) |
| twice x, plus 3 | 2x + 3 |
| the sum of x and 3, squared | (x + 3)² |
| the sum of x squared and 3 | x² + 3 |
| 3 less than the product of 4 and x | 4x − 3 |
“The sum of x and 3” is one object. Anything done to it wraps round the whole bracket, which is exactly what the brackets record.
| Situation | Let x be… | The others |
|---|---|---|
| consecutive integers | the first | x + 1, x + 2 |
| consecutive even or odd | the first | x + 2, x + 4 |
| two numbers summing to 30 | one of them | 30 − x |
| a two-digit number | tens digit t, units u | value = 10t + u |
| ages in 5 years | age now | x + 5 |
| ₱5 and ₱10 coins | number of ₱5 coins | worth 5x + 10 × (number of ₱10 coins) |
7, 9, 11 — the gap is 2, not 1. Using x, x + 1, x + 2 for odd numbers is a standard trap, and it gives an answer that is not odd.
A number subtracted from 20 is 3 more than the number. Find the number.
20 − x = x + 3
“subtracted from 20”: 20 comes first
17 = 2x
+ x, − 3
x = 172 = 8.5
check: 20 − 8.5 = 11.5 = 8.5 + 3 ✓
Different: the unknown is the thing taken away, so it sits second — and the answer need not be a whole number.
Writing x − 20 = x + 3 leaves −20 = 3 and the false verdict “no such number”. When a translation collapses like that, re-read the “from” before trusting it.
A number exceeds two-fifths of itself by 18. Find the number.
x − 25x = 18
“exceeds by”: bigger − smaller = gap
5x − 2x = 90
× 5 on every term
3x = 90, x = 30
check: 30 − 12 = 18 ✓
Different: there is no “is” — “exceeds … by” hides a subtraction and an equals sign in one phrase.
Twice the sum of a number and 3 is 4 more than twice the number. Find the number.
2(x + 3) = 2x + 4
“the sum” gets brackets
2x + 6 = 2x + 4
expand
6 = 4
the x’s cancel: no such number
Different: a correct translation can still have no answer. Had the sentence said “6 more”, every number would work.
The units digit of a two-digit number is 3 more than the tens digit, and the number is 4 times the sum of its digits. Find the number.
tens digit t, units digit t + 3
name the digits, not the number
value = 10t + (t + 3) = 11t + 3
the tens digit is worth ten
digit sum = t + (t + 3) = 2t + 3
—
11t + 3 = 4(2t + 3) = 8t + 12
“is 4 times the sum”
3t = 9 → t = 3, units digit 6
the number is 36
Reversing the digits swaps their place values: 36 becomes 10(6) + 3 = 63, and 63 − 36 = 27 = 9 × (6 − 3). In letters the reversed number is 10u + t, and subtracting: (10u + t) − (10t + u) = 9u − 9t = 9(u − t), always a multiple of 9.
Division with a remainder: dividend = quotient × divisor + remainder. 75 ÷ 12 is 6 remainder 3 because 75 = 6 × 12 + 3. So “divided by the sum of its digits, the quotient is 6 and the remainder is 5” is the equation 10t + u = 6(t + u) + 5.
Writing the number as t + u (or as “tu”, which reads as t × u) is the classic slip. A two-digit number is 10t + u; only the digit sum is t + u.
Translation items usually ask “which equation represents…”, with distractors that reverse “less than” or drop the brackets. Fastest check: pick a number such as 10, read the sentence aloud with it, and see which equation gives the same result.
Translating back into words is the only reliable check on a translation.
Translate, then solve
Seven more than three times a number is 34.
Three times the number, then seven more: 3n + 7 = 34, so 3n = 27 and n = 9. B is 3n, not n — stopping one step short, just before the division, is the commonest error in this topic.
