Arithmetic Statistics Problem

[andreanasrallah]

Hello,

I have come accross this problem and I seem to have adifficulty understanding what is being done to solve this, although it is all explained in the book. Can someone please explain? I quote:

"Problem:

List S consists of 10 consecutive odd integers, and list T consists of 5 consecutive even integers. If the least integer in S is 7 more than the least integer in T, how much greater is the average (arithmetic mean) of the integers in S than the average of the integers in T?

(A) 2
(B) 7
(C) 8
(D) 12
(E) 22

Solution:

Let the integers in S be s, s+2, s+4....,s+18 where s is odd. Let the integers in T be t,t+2, t+4, t+6. t+8, where t is even. Given that s = t+7, it follows that s-t = 7. The average of the integers in s is (10s+90)/10 = s+9, and similarly the average of the integers in T is (5t+20)/5 = t+4.

The difference in these averages is (s+9) - (t+4) = (s-t) + (9-4) = 7+5 = 12.

Correct answer (D)

Can someone explain why we are adding 2 then 4 then 6 in both cases to each consecutive term? Isn't the pattern supposed to be plus 2 in each case since they are consecutive integers? Or does the word consecutive signify something else?

Also how come we add 10s to 90? And 5t to 20?

I would really appreciate your help. Thanks.

[jnelson0612]

What is the original source for this problem? Is it the Official Guide?

[krishnan.anju1987]

Hi,

Regarding your first question

Can someone explain why we are adding 2 then 4 then 6 in both cases to each consecutive term? Isn't the pattern supposed to be plus 2 in each case since they are consecutive integers? Or does the word consecutive signify something else?

As mentioned T has 5 consecutive even integers and S has 10 consecutive odd integers

Now take examples,

Lets take T={2,4,6,8,10}
This would constitute even consecutive integers. Now if I consider 2 to be x, then 4=x+2, 6=x+4, 8=x+6 and 10=x+8.

Take other examples and check.

Similar case exists for odd consecutive integers. Hence, the increment by 2 is done when consecutive even/odd integers is mentioned with reference to one base value.

Also how come we add 10s to 90? And 5t to 20?

2) Regarding your second question

Also how come we add 10s to 90? And 5t to 20?

Now we know T can be written as {x, x+2, x+4, x+6, x+8}
and S can be written as {s, s+2, s+4, ...., s+18}

add all the values of T and S.

Then sum(t)=5x+20
and sum(s)=10s+90

where s=x+7

so now, divide t by 5 and s by 10 to find the average and subtract to get your answer.

Hope this clarifies.

[jnelson0612]

Hey all,
Per forum guidelines, I am still looking for the original source.