Problem 18 · 1996 AJHSME
Medium
Fractions, Decimals & Percents
successive-percent
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
Show answer
Answer: A — 1920 dollars.
Show hints
Hint 1 of 2
Tempting trap: 'up 20% then down 20% must cancel back to $2000.' It doesn't — the cut is taken off the bigger raised amount, so 20% of more is more than 20% of less. Multiply the changes instead of adding/cancelling them.
Still stuck? Show hint 2 →
Hint 2 of 2
A raise multiplies by 1.2, a cut multiplies by 0.8. Apply them in turn: 2000 × 1.2 × 0.8. Notice 1.2 × 0.8 < 1, so she ends up below where she started.
Show solution
Approach: percent changes multiply (and don't cancel)
- Don't add the percents — multiply the factors. Raise: 2000 × 1.2 = 2400. Cut: 2400 × 0.8 = 1920. So her salary is $1920.
- Why it's not $2000: the 20% cut comes off the larger $2400, removing $480, while the raise only added $400. The combined factor is 1.2 × 0.8 = 0.96, a net 4% loss — an up-then-down (or down-then-up) by the same percent ALWAYS leaves you a little lower.
- Why this transfers: chain percent changes by multiplying their factors. Equal up/down percents never return to the start; the result is start × (1 − p²) for a p-fraction change either way.
Mark:
· log in to save