Problem Solving

AAPL wrote:GMAT Prep

Beth has a collection of 8 boxes of clothing for a charity, and the average (arithmetic mean) number of pieces of clothing per box is c. If she replaces a box in the collections that have 12 pieces of clothing with a box that contains 22 pieces of clothing, what is the average number of pieces of clothing per box for the new collection in terms of c?

A. c - 5/4
B. c + 5/4
C. 8 - 10/c
D. 8 + 10/c
E. 8c - 10

OA B

One box contains 12 items of clothing
Let t, u, v, w, x, y, z = the number of items in each of the other 7 boxes

Beth has a collection of 8 boxes of clothing for a charity, and the average (arithmetic mean) number of pieces of clothing per box is c.
We can write: (t + u + v + w + x + y + z + 12)/8 = c

If she replaces a box in the collections that has 12 pieces of clothing with a box that contains 22 pieces of clothing, what is the average number of pieces of clothing per box for the new collection in terms of c?
If we replace the box with 12 items with a box with 22 items, the NEW average = (t + u + v + w + x + y + z + 22)/8
If we replace 22 with 12 + 10, then the NEW average = (t + u + v + w + x + y + z +12 + 10)/8

USEFUL PROPERTY: (a + b)/c = a/c + b/c
We apply the above property to write: NEW average = (t + u + v + w + x + y + z +12)/8 + 10/8
Replace first fraction with c to get: NEW average = c + 10/8
Simplify remaining fraction to get: NEW average = c + 5/4

Answer: B

Cheers,
Brent

When there are variables in the answer choices in a PS problem, it's often easy to pick our own values for those variables.

With averages, it's often easiest to just treat each member of the group as the same. i.e. let's say that in this case, we have 8 identical boxes with 12 clothing items each.
So the original average c = 12.
If we have 8 boxes with an average of 12 each, the total # of clothing items = 8*12 = 96.

If we change one of those boxes from 12 to 22, we simply have 10 more clothing items total: 96 + 10 = 106 pieces of clothing.
If our new total is 106 but the number of boxes is the same at 8, the new average = 106/8 = 13.25.

To see which answer choice fits, we plug in c = 12 into each answer choice, and see which gives us our target value of 13.25.

A. c - 5/4 --> 12 - 5/4 = 10.75. No.

B. c + 5/4 --> 12 + 5/4 = 13.25. Yes.

C. 8 - 10/c --> 8 - 10/12 7.1666... No.

D. 8 + 10/c --> 8 + 10/12 8.83333.... No.

E. 8c - 10 --> 8*12 - 10 = 86. No.

The answer is B.

AAPL wrote:GMAT Prep

Beth has a collection of 8 boxes of clothing for a charity, and the average (arithmetic mean) number of pieces of clothing per box is c. If she replaces a box in the collections that have 12 pieces of clothing with a box that contains 22 pieces of clothing, what is the average number of pieces of clothing per box for the new collection in terms of c?

A. c-5/4
B. c+5/4
C. 8-10/c
D. 8+10/c
E. 8c-10

OA B

The sum of Beth's original boxes is 8c.

After she replaces a 12-piece box with a 22-piece box, the new sum is 8c - 12 + 22 = 8c + 10, so the new average is:

(8c + 10)/8 = 8c/8 + 10/8 = c + 5/4

Answer: B