Expert: Jack Cheng - 11/29/2006

Question
One half the members of a junior math team are 9th graders. One fourth of the remaining members are 8th graders. One fifth of the rest are 7th graders. There are 12 6th graders on the team. How many members are on the team?

Answer
Hi Linda,

Let x be the number of members on the team.  Here's what I derived from what you gave me.

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(1/2)x = 9th graders
(1/4)(1/2)x = 8th graders
(1/5)(3/4)(1/2)x = 7th graders
12 = 6th graders

So x - 12 should be the number of students in the 7th, 8th and 9th grades.

x-12 = (1/2)x + (1/8)x + (3/40)x
-12 = (1/2)x + (1/8)x + (3/40)x  - x
-12 = (20/40)x + (5/40)x + (3/40)x - (40/40)x
-12 = -(12/40)x
 12 = (12/40)x
  x = 12 * (40/12) = 40

So there are 40 people in the team.

You can also work backwards. You know that at the end, you have 12 students (the 6th graders), and it's equaled to the (4/5) of the total 6th and 7th grade students, which is 15 (12 * 5/4). Since 7th graders are one fifth of it, it's 3 (notice that 12+3 equals 15). Using this same process, you can find out how many students are in each grade (20-9th, 5-8th, 3-7th, and 12-6th). Sum them up and you would get the same answer.

~ Jack