How is this even possible

Question

of 40 kids, 24 sing, 16 play the drums, and 10 do neither. The ratio of the number who both sing and play drums to the number who do neither is

a)1:4

b)3:5

c)4:5

d)1:1

what I want to know how 24 can sing 16 can play drums and still 10 can do neither. 24+16 is already 40

Answer 1

There are 40 kids in all. That there are 10 kids who do neither means that there are 30 who do something. 

 

Draw a Venn diagram. 

 

One circle can be for the kids who sing.

 

Another circle can be for the kids who play the drums.

 

Picture those circles overlapping.

 

The total number of kids that fit into that diagram is 30. Those are the kids who do something.

 

However, if we add the 24 and the 16 we get 40. How is that possible when there can only be 30 kids in there? Some portion must fall in that overlapping region. 

 

Think of something analogous: Like, if I put a ruler down on the table, and then right on top of it put another ruler starting at the 6 in mark, the total length of those rulers combined would be 18 inches. How is that possible when I know there are 12 inches to a ruler and that there must then be 24 inches between two rulers? Because they overlap for 6 of those. The combined length minus the actual length when they overlap tells you how much one covers the other. (24 - 18 = 6)

 

Here with the students, the combined number participating in singing and drumming minus the number of students total tells you how many do both of those activities. 

 

So 40 is the number of students doing drumming plus the number doing singing, subtract from that 30 and that is the number doing both of those things. 

 

It is 10. 

 

The number not doing anything is also 10.

 

Take the ratio of 10 and 10 and you get 1.

Answer 2

The answer is d, 1:1.

 

How do we figure this out? I tried to make a two-column graphical display. Each row represents a student and the columns represent whether or not they sing and/or plays the drums. First of all, you know that 10 students do neither. So, I constructed the first 10 rows as "N     N". I then filled out the rest of the singing column. I made the next 6 rows in the singing column N, and then the last 24 as Y. Now, when I filled out the rest of the Drums column, I had to start with Y's. If I started with N's, I would create more "N   N", but I am already maxed out at 10. So, I had to lay down 16 Y's followed by 14 N's.

 

Now, we can count how many rows are "Y   Y", representing students who both sing and play the drums. I count 10 of those. We already know that there are 10 "N   N", or students who do not sing or play the drums. So, the ratio of 10:10, or otherwise 1:1.  

 

Sing(Y/N)  Drums(Y/N)

N              N

N              N

N              N

N              N

N              N

N              N

N              N

N              N

N              N

N              N

N              Y

N              Y

N              Y

N              Y

N              Y

N              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              Y

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

Y              N

 

             

Answer 3

There exists a group of individuals who can sing and play drums.

 

Let A be the group of kids who can sing, and B be the group of kids who can play the drums

 

AuB is an unknown amount that we have to find to solve this problem. (u denotes a "union", or a logical or operator)

 

Let !A be the group of kids who CAN'T sing. 40 - 24 = 16, so !A has 16 members

 

Let !B be the group of kids who CAN'T play drums. 40 - 16 = 24, so !B has 24 members

 

!An!B are kids who do neither. We're told that there exist 10 of them. (n denotes an "intersection", or a logical and operator)

 

So kids who can sing, but don't play drums (An!B) is equal to  24 - 10 = 14 members, and kids who can't sing but do play drums (!AnB) is equal to 16 - 10 = 6

 

That's 14 + 6 = 20 kids total that only do one thing (sing, or play drums) (using logical operators this looks like ((An!B)u(!AnB))). So, 20 kids who only do one thing, 10 kids who do neither things, and x kids who do both, all equaling 40.

 

So, 40 - 10 - 20 = 10 kids who do both.

 

Our solution is d.

 

This problem can be shown using a venn diagram, but yeah, it seems convoluted if you can't get access to the number of kids who do both.

Answer 4

There are four categories:

1) neither

2) both

3) only sing

4) only drums

 

These 4 categories will add up to more than 40. Imagine if in your math class you added up the number of girls and right-handed people. Those would add up to more than the total of the people in your class.

 

So, if 10 can do neither, that means there are 30 students that can do at least one of them.

 

24 + 16 = 40 as you pointed out, and there are only 30 spots left. Therefore, 10 of the kids have to do both.

 

Now, there are 10 neither and 10 both. At this point you can answer the question.

 

To keep going to fully understand it...

 

24 - 10 (both) = 14 that just sing

16 - 10 (both) = 6 that just play the drum

 

14 + 6 + 10 + 10 = 40

 

Good luck!