Question

Among the 7th graders in a middle school, three quarters of them play at least one sport. Among the 7th graders who play sports, half of them play more than one sport. Among the 7th graders who play more than one sport, one third of them play more than two sports. Among the 7th graders who play more than two sports, one quarter of them play more than three sports. Sixteen 7th graders play more than three sports at the school. How many 7th graders are in the school

Answer 1

512

Explanation:

Let the number of 7th graders in the school is x.

As three-quarters of them play at least one sport, so, the number of 7th graders who play sport =
`\frac 3 4 x`.

Among the 7th graders who play sports, half of them play more than one sport. So, the number of 7th graders who play more than one sport
`= \frac 1 2* \frac 3 4 x=\frac 3 8 x`.

Among the 7th graders who play more than one sport, one-third of them play more than two sports. So, the number of 7th graders who play more than two sport
`= \frac 1 3* \frac 3 8 x=\frac 1 8 x`.

Among the 7th graders who play more than two sports, one-quarter of them play more than three sports.

So, the number of 7th graders who play more than three sport
`= \frac 1 4* \frac 1 8 x=(1)/(32) x`.

7th graders play more than three sports at the school= 16

`\Rightarrow (1)/(32) x=16`

`\Rightarrow x =16 * 32`

`\Rightarrow x = 512`

Hence, the total numbers of 7th graders in the school are 512.