🇺🇸 AMC 8 ⇄ switch contest
2008 AMC 8

Problem 7

Problem 7 · 2008 AMC 8 Easy
Fractions, Decimals & Percents equivalent-fractions

If 35 = M45 = 60N, what is M + N?

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Answer: E — 127.
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Hint 1 of 2
All three fractions equal 35 — spot how each denominator (or numerator) was scaled and copy that scaling.
Still stuck? Show hint 2 →
Hint 2 of 2
Equal fractions are linked by one multiplier; find it instead of cross-multiplying from scratch.
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Approach: scale the known fraction
  1. For M/45: the denominator jumped from 5 to 45, that's ×9, so the top scales the same way: M = 3 × 9 = 27. Scaling is faster than writing 3·45 = 5M.
  2. For 60/N: the numerator jumped from 3 to 60, that's ×20, so N = 5 × 20 = 100.
  3. Sum: 27 + 100 = 127.
  4. Why this transfers: equivalent fractions are one fraction in disguise — multiply top and bottom by the same number and you can fill any blank.
Another way — cross-multiply:
  1. 3 · 45 = 5M ⇒ M = 27; and 3N = 5 · 60 ⇒ N = 100.
  2. Sum: 27 + 100 = 127.
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