An SAT math question!?

Question

In a group of people, the average age is 16. One leaves and is replaced by someone 12 years older. The new average is 18. How many people are in the group?

I really would love an explanation for this one!

Answer 1

Here's the explanation you seek.

The average age of the group equals the sum of the ages, divided by the number of people in the group, and the answer is 16:

Sum / People = Average = 16

Now, when one of the people is replaced by someone 12 years older, the Sum is increased by 12, People (the number in the group) is unchanged, and the answer is now 18:

(Sum + 12) / People = 18

If you subtract the first equation from the second equation, you get:

12 / People = 18 - 16 = 2

So 12 / People = 2.
Therefore, People must be 6, because 12 / 6 = 2.

If you want the algebra to be easier to follow, try this (using S for Sum, and P for People):

S / P = 16 so S = 16 P
(S + 12) / P = 18 so S + 12 = 18 P
Subtracting the first equation from the second:
12 = 2 P
Dividing both sides of the equation by 2:
6 = P
So there are 6 people in the group.

Answer 2

So (a + b + c + ... + d)/n = 16, where a, b, c, ..., d are the ages, and n is the number of people. Replace a with a+12, and ((a+12) + b + c + ... + d)/n = 18, or (a + b + c + ... + d)/n + 12/n = 18, so 16 + 12/n = 18 by the first equation.

Take it from there.

Answer 3

I would say three.people that the first three average 16 = 6, 18, and 24. If you raise the 6 year olds age by 12 you get 18 and 18+18+24+average of 18

Answer 4

This one looks tough, but if you write out how to go about finding averages, it turns into a pretty easy problem.
I'll explain. Originally, the average age of this group of n people is 16. So, (x_1 + x_2 + ... + x_n)/n = 16. Multiply both sides by n and we get that the sum of everybody in the original group is 16n. Now somebody leaves and somebody 12 years older comes in. We've just added 12 to the total age of the group, 16n + 12. Since the new average is 18, that means that the new sum of the ages should equal 18n. So we have two expressions for the same thing. They have to be equal 16n + 12 = 18n. Solve for n and we get that n = 6. There were 6 people in the group.

Neat, eh. I think I'll give that to my stat class Monday! Thanks!

Answer 5

let t= total age & n=# of people
(1) t/n=16
1 person leaves & is replaced by someone 12 yrs older
(2) t+12)/n=18
from (1) t=16n

substitute into (2)
(16n+12)/n=18
16n+12=18n
2n=12
n=6 there are 6 people in the group

not asked but the total age started as 6*16=96
When increased by 12, it bacame 96+12=108.
108/6=18

Answer 6

SAT questions (at least the hard ones) are designed to have two solution methods: a long one and a quick one. That's so not everyone will finish. You have to look for the quick way to solve it, which is NOT to write out long equations. Just notice that the total went up by 12 but the average went up by 2. 12 divided by 2 is 6. So 6 people were in the group.

Answer 7

Take 6 people whose ages sum to 96 giving average of 16. Now take 6 people whos ages sum to 108 giving average of 18.

Notice that 108 -96 = 12

So you can take any combination of ages for 6 people that add up to 96, and the average will be 16.

Now keep the same 6 ages but replace one of them by a number 12 greater than the one selected. the sum will be 108, and the average age will be 18.

So, the smallest group having this property is 6.

Answer 8

It's easier to work with totals than averages.

If there are x people in the group, total of their ages is
16*x. With the change, the total is
18*x

The replacement means the total goes up by 12, and so
16x + 12 = 18x, which you can solve.

This is basically the same as the solution posted by im2_weird, and since he was first I guess he should get the points.

Answer 9

Say the number of people in the group is n.
Let x1, x2, ..., xn denote the ages of these n individuals.
The average age of the group is: (x1+ x2+ ...xn)/n = 16
So 16n = x1+ x2+ ...xn (Call this equation #1)
Now if the person whose age is x1 leaves, and a person whose 12 years older enters the group, we have:
[(x1+12) + x2 + ...+ xn]/n = 18
So x2 + ...+ xn = 18n - x1 - 12
Now replace x2 +...+ xn by 18n - x1 - 12 in equation #1 to get:
16n = x1 + 18n - x1 - 12
2n = 12 implies n = 6, which is the number of people in the group.

Note: When solving the problem, there is nothing special about removing the person with age x1. We could have removed the person with age xk where k = 1, 2, 3,..., or n. That is, removing the person with age xk for any of the mentioned k will yield the same result for n, namely 6.

Answer 10

x = Ones who leaves
12 is ones who replaced

16(n-1) + x = 16n .................i)
16(n) - x +12 =18n .................ii)

16n - x +12 =16n + 2n

12-2n = x
x = 12 - 2n

2n = 12 -x
n =(12-x)/2

n>2 and n =(12-x)/2

So the answer is
n >= 2
n =(12-x)/2

so the answer is

n>=2 n = {2,3,4,5}

x =8 , n =2
x =6 , n =3
x =4 , n =4
x =2 , n =5

example
if one who leaves is 8 years old the the group is consist of 2 people

24 + x = 16(n)
24 + 12 = 18(n)

24 + 8 = 16(2)
24 + 12 = 18(2)