Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

1061. There are candies in three boxes. In the first box, there are 8 candies less than in the other two together, and in the second box, there are 12 less than in the first and third together. How many candies are in the third box?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

△\triangle Let the number of candies in the first, second, and third boxes be denoted by a,ba, b, and cc respectively. We obtain the system of equations:

{a=b+c−8b=a+c−12 \left\{\begin{array}{l} a=b+c-8 \\ b=a+c-12 \end{array}\right.

Adding these equations term by term:

a+b=(a+b)+2c−20 a+b=(a+b)+2 c-20

Then

2c−20=0,c=10 2 c-20=0, \quad c=10

Answer: 10.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.