Problems on Numbers

This topic is not arithmetic; it is translation. Give the unknown a name, write one line per sentence, and the algebra is trivial.

The standard translations:

1. Sum and difference given. larger = (sum + difference)/2, smaller = (sum − difference)/2. Two lines, no equations.
2. Consecutive numbers. Call them x−1, x, x+1 — the middle one is always sum ÷ 3. For consecutive even or odd numbers use x, x+2, x+4.
3. A two-digit number. Write it as 10a + b, and its reverse as 10b + a. Their difference is always 9(a − b) and their sum is always 11(a + b) — that one fact solves most of these.
4. Sum and product given. a² + b² = (a+b)² − 2ab and (a−b)² = (a+b)² − 4ab.

The single most common mistake is mis-translating "less than". "5 less than 3 times a number" is 3x − 5, but "3 times a number less than 5" is 5 − 3x. Whenever you see "less than", the quantity written after it comes first in the subtraction. Read the sentence twice before writing the equation — everything after that is safe.

larger = (S+D)/2 · smaller = (S−D)/2 · two-digit = 10a+b, reverse = 10b+a · difference = 9(a−b) · a²+b² = (a+b)² − 2ab

The sum of two numbers is 84 and their difference is 12. Find them. → larger = (84 + 12)/2 = 48, smaller = (84 − 12)/2 = 36 and 48. Sanity check: 48 + 36 = 84 and 48 − 36 = 12 — both conditions hold, so no algebra was needed at all.