Problem 17 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Tim, Tom and Jim are triplets. Their brother Carl is exactly 3 years younger. All four are having their birthdays today. How old can the four brothers be altogether?
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Answer: A — 53
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Hint 1 of 3
The three triplets are all the same age, and Carl is 3 years younger than that age.
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Hint 2 of 3
Imagine giving Carl 3 extra birthdays so all four are the same age: then the total would be 4 equal ages.
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Hint 3 of 3
So the real total is 3 less than some multiple of 4.
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Approach: make all four ages equal, then adjust by 3
- If Carl were the same age as the triplets, all four ages would be equal, so the total would be 4 times one age, a multiple of 4.
- Carl is actually 3 years younger, so the real total is 3 less than a multiple of 4.
- Checking the choices, only \(53 + 3 = 56 = 4 \times 14\) is a multiple of 4, so the total is 53, choice (A).
With algebra
If each triplet is \(t\), Carl is \(t-3\) and the total is \(3t+(t-3)=4t-3\); setting \(4t-3=53\) gives \(t=14\).
Mark:
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