Problem 23 · 2020 Math Kangaroo
Stretch
Number Theory
sum-constraint
Jonas and Elias went to the beach for their vacation, where they had ice cream every day. Each ice cream they had, had two or three balls. On the last day of vacation, Jonas and Elias had had 23 and 19 ice cream balls in total, respectively. At least how many days were they on vacation?
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Answer: C — 8
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Hint 1 of 3
To use up the fewest days, give the bigger eater (Jonas, 23 balls) the most balls each day.
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Hint 2 of 3
Even eating 3 balls every single day, count how many days it takes to reach 23.
Still stuck? Show hint 3 →
Hint 3 of 3
The same number of days has to work for both boys, so check that 19 also fits in that many days.
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Approach: give the most balls per day to use the fewest days, then check both totals fit
- To finish in as few days as possible, eat the biggest ice cream (3 balls) every day.
- Counting by 3 toward Jonas's 23: seven days give only 21 balls, which is not enough, so they need at least 8 days.
- Eight days really works for both: Jonas eats 3 balls on 7 days and 2 balls on 1 day (21 + 2 = 23), and Elias eats 3 balls on 3 days and 2 balls on 5 days (9 + 10 = 19).
- So they were on vacation at least 8 days, choice C.
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