# Question: Children Playing Tennis

## Comment on Children Playing Tennis

Wouldn't the correct answer to this question actually be D? How can we know the ratio of girls to boys? It seems dubious to simply assume that half of the school is boys and the other half is girls.

### We're not assuming that there

We're not assuming that there are 100 boys and 100 girls; we're deducing it.

The big clue here is that 37% of the girls play tennis.

From this information, we can see that MANY of the possible values for the number of girls do not work.

For example, it wouldn't make any sense if there were 10 girls. The reason it wouldn't make sense is that if we find 37% of 10, we get 3.7. In other words, if there are 10 girls in total, then 3.7 of them play tennis.

Since the number tennis players must be an INTEGER, we can see that there can't be 10 girls at the school.

For the same reason, we know that there can't be 11 girls, since 37% of 11 = 4.07, and we can't have 4.7 tennis players.

In fact, there can be only ONE possible value for the number of girls at the school. There MUST be 100 girls, since all other possible values yield NON-INTEGER values for the number of girl tennis players.

Does that help?

### Yes, I think it does! Is the

Yes, I think it does! Is the crux of the solution, then, recognizing that for 37% of a number to be an integer, it must be a multiple of 100? Am I correct in believing that the only reason this is true of 37 is because it is a prime number?

### Not quite. The reason is

Not quite. The reason is because the fraction 37/100 does not simplify.

So, 37% of x = (37/100)(x) = (37x)/100
In order for (37x)/100 to be an integer, x must be divisible by 100

Likewise, 5% of x = (5/100)(x) = (1/20)(x) = x/20
In order for x/20 to be an integer, x must be divisible by 20

One last example: 25% of x = (25/100)(x) = (1/4)(x) = x/4
In order for x/4 to be an integer, x must be divisible by 4

Cheers,
Brent

### Hi,

Hi,
when we are told that the school has 200 children, will it be possible to assume that .37 of the 200 kids plus .10 of the kids play tennis ?. It says some boys and some girls so .47 of the total of the kids play tennis.

### That approach will work ONLY

That approach will work ONLY in cases in which the number of girls = the number of boys.
So, in this very specific question, the strategy works.

However, let's say the question stated that there are 200 children, and 25% of the girls play tennis and 40% of the boys play tennis.

Since we can't be certain that the number of girls = the number of boys, there are many possible cases that work. For example:

CASE A: There are 40 girls and 160 boys.
So, the number of tennis players = (25% of 40) + (40% of 160)
= 10 + 64
= 74

CASE B: There are 80 girls and 120 boys.
So, the number of tennis players = (25% of 80) + (40% of 120)
= 20 + 48
= 68

CASE C: There are 100 girls and 100 boys.
So, the number of tennis players = (25% of 100) + (40% of 100)
= 25 + 40
= 65

Notice that your approach works for Case C (the number of girls = the number of boys), but not the other cases.

Cheers,
Brent

Thank you

### Can we solve this using the

Can we solve this using the double matrix method? I was unable to arrive at the answer using double matrix

### Great question!

Great question!

Yes, we can use the Double Matrix Method for this question.
If we let B = the total number of boys, then...
(200 - B) = the total number of girls

So, if 10% of the B boys play tennis, then the number of boy tennis players = 10% of B = 0.1B.
Similarly, if 37% of the (200 - B) girls play tennis, then the number of girl tennis players = 37% of (200 - B) = 74 - 0.37B

So our matrix looks like this: https://imgur.com/ssaCai9

IMPORTANT: the number of boys must be a positive INTEGER, and the number of girls must be a positive INTEGER.
In other words, 0.1B (the number of boy tennis players) must be a positive integer, and (74 + 0.37B) must be a positive integer.
In order for both values to be positive integers, B must be a multiple of 100.
So, the possible values of B are 0, 100, 200, 300, etc.

Since the question tells us that there are SOME boys and SOME girls, we know that there is at least 1 boy, and at least 1 girl.
Since there are 200 children in total, it must be the case that B = 100, in which case..

The number of boy tennis players = (0.1)(100) = 10
The number of girl tennis players = 74 - 0.37(100) = 37

So the total number of tennis players = 10 + 37 = 47, in which case the answer is C