Question

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^\text{th}$ -graders to $6^\text{th}$ -graders is $5:3,$ and the ratio of $8^\text{th}$ -graders to $7^\text{th}$ -graders is $8:5.$ What is the smallest number of students that could be participating in the project?

$16$

$40$

$55$

$79$

$89$

Video solution:

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Written solution:

The number of $8^\text{th}$ -graders must be a multiple of $5$ and $8,$ which means that it is at least $40.$ Using this number, we get that the number of $6^\text{th}$ -graders is $40 \cdot \dfrac{3}{5} = 24.$ Similarly, the number of $7^\text{th}$ -graders is $40 \cdot \dfrac{5}{8} = 25.$

The total number of students is therefore $40 + 24 + 25 = 89.$

Thus, E is the correct answer.

2013 AMC 8 Problem 16

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Written solution:

Problem 16 in Other Years