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2020 Math Kangaroo

Problem 26

Problem 26 · 2020 Math Kangaroo Stretch
Fractions, Decimals & Percents Algebra & Patterns percent-multiplierwork-backward

Lady Josephine bought a pack of beans. The beans come mixed with impurities such as pebbles and sand, and the label says these impurities make up 8% of the contents of the package. Lady Josephine removes part of these impurities, reducing them to 4% of the contents of the package. What fraction of the total amount of impurities was removed from the package?

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Answer: B — 2548
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Hint 1 of 2
Start with a 100 g pack: 8 g impurities, 92 g good beans; removing impurities does not change the good beans.
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Hint 2 of 2
After removal the impurities are 4% of the new, smaller pack — solve for how much impurity was taken out.
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Approach: keep the good beans fixed
  1. Take a 100 g pack: 8 g impurities and 92 g good beans. Removing x g of impurity leaves a pack of (100 − x) g.
  2. Now (8 − x) is 4% of (100 − x): 8 − x = 0.04(100 − x), giving 4 = 0.96x, so x = 25/6 g.
  3. The fraction of the original impurities removed is (25/6) ÷ 8 = 25/48.
  4. The answer is 25/48, choice B.
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