Problem 10 · 1987 AJHSME
Medium
Arithmetic & Operations
factor-the-common-term
4(299) + 3(299) + 2(299) + 298 =
Show answer
Answer: B — 2989.
Show hints
Hint 1 of 2
Three terms share the same factor 299 — don't multiply them out separately. What do their multipliers add to?
Still stuck? Show hint 2 →
Hint 2 of 2
Distributive property in reverse: 4(299) + 3(299) + 2(299) = (4 + 3 + 2)(299). Collect the count first.
Show solution
Approach: factor out the shared term, then round
- The first three terms all carry 299, so pull it out: (4 + 3 + 2)(299) = 9 × 299.
- Use 299 = 300 − 1: 9 × 300 − 9 = 2700 − 9 = 2691. Add the last term: 2691 + 298 = 2989.
- Why this transfers: spotting a repeated factor and rounding to a friendly nearby number (299 → 300) turns four multiplications into one easy subtraction.
Another way — count the 299s plus the leftover:
- There are 4 + 3 + 2 = 9 copies of 299, and 298 = 299 − 1, so the total is 10 × 299 − 1.
- 10 × 299 = 2990, minus 1 = 2989.
Mark:
· log in to save