Dear Math Central,
Hi, I'm struggling with this problem and am wondering if you could give me a hand with it. I'm a student in 8th grade and this is a middle-school problem.
Thanks very much!
Jean

QUESTION:
An equal number of juniors and seniors responded to the question, "Do you like math?" Each respondent said yes or no. If 70% of those who said yes were seniors and 80% of those who said no were juniors, what percentage of the seniors polled yes?

Hi Jean,

I drew a table to help me understand what is being asked. There are two kinds of people, juniors and seniors and two kinds of answers, yes and no so my table looks like

I don't know any numbers to put in the table yet but I do now a few facts.

• The sum of the numbers in the first column (the number of juniors) is the same as the sum of the numbers in the second column (the number of senior).
  
• In the first row (the yes row) I don't know the total but I do know that 70% of this total are seniors and hence 30% are juniors.

At this point I realize that I need some algebraic notation to keep track so let Y be the number of students who responded yes. Then the number of seniors who responded yes is 70% of Y and the number of juniors who responded yes is 30% of Y. Hence I can fill in some values in my table

	juniors	seniors	Total
yes	0.3 Y	0.7 Y	Y
no
Total

Likewise, in the no row if I let N be the number of students who responded no then 80% of N are juniors and hence 20% of N are seniors. Thus

	juniors	seniors	Total
yes	0.3 Y	0.7 Y	Y
no	0.8 N	0.2 N	N
Total

Now you can use the first fact that the number of juniors and the number of seniors are equal. Hence

0.3 Y + 0.8 N = 0.7 Y + 0.2 N

Simplify to get

3N = 2Y

You were asked what percentage of seniors polled yes. From the table this is

 ^{0.7 Y}/_{(0.7 Y + 0.2 N)} 100

Use 3N = 2Y  to eliminate N from this expression and simplify.

Penny