Problem 16 · 2020 Math Kangaroo
Hard
Algebra & Patterns
substitutionwork-backward
Ana, Bia and Cris have, together, 100 reais. They go to the movies and each one pays her own entrance fee. Before paying, Ana had twice as much as each of her friends. After paying the fee, Ana now has three times what the other two friends have together. How much did the movie entrance cost?
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Answer: E — R$ 20
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Hint 1 of 3
Split the 100 reais into equal shares first: Ana's pile is as big as both friends' piles put together.
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Hint 2 of 3
After they pay, picture Ana's leftover money as three equal piles next to the two friends' leftover piles together.
Still stuck? Show hint 3 →
Hint 3 of 3
Compare how much Ana lost to how much the two friends lost in total.
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Approach: share the money into equal parts, then compare the leftovers in piles
- Before paying, Ana has as much as both friends together, so split 100 into four equal piles of 25: Ana holds two piles (50) and each friend holds one pile (25).
- After everyone pays the same ticket, Ana's leftover is three times the two friends' leftover together; so think of Ana's leftover as 3 small piles and the two friends' leftover together as 1 small pile, four small piles in all.
- The four friends-and-Ana started with 100 reais and spent 3 tickets, and those leftovers split evenly: trying the choices, a 20-real ticket leaves Ana 30 and each friend 5 (the two friends have 10 together, and 30 is three times 10).
- So the ticket cost 20 reais, choice E.
For older kids (algebra)
Let each friend start with x, so Ana has 2x and 2x + x + x = 100 gives x = 25. After paying f each, 50 − f = 3((25 − f) + (25 − f)) leads to 5f = 100, so f = 20.
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