On Monday, a person mailed 8 packages weighing an average (arithmetic

SOLUTION

On Monday, a person mailed 8 packages weighing an average (arithmetic mean) of \(12\frac{3}{8}\) pounds, and on Tuesday, 4 packages weighing an average of \(15\frac{1}{4}\) pounds. What was the average weight, in pounds, of all the packages the person mailed on both days?

(A) \(13\frac{1}{3}\)

(B) \(13\frac{13}{16}\)

(C) \(15\frac{1}{2}\)

(D) \(15\frac{15}{16}\)

(E) \(16\frac{1}{2}\)

The total weight of 8 packages is \(8*12\frac{3}{8}=99\) pounds;

The total weight of 4 packages is \(4*15\frac{1}{4}=61\) pounds;

The average weight of all 12 packages is \(\frac{total \ weight}{# \ of \ packages}=\frac{99+61}{12}=13\frac{1}{3}\).

Answer: A.
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Since the final average cannot be greater than \(15\frac{1}{4}\), answers C, D and E are out.

We can use the property of weighted averages.
\(15\frac{1}{4}=15\frac{2}{8}\), the distance between the two initial averages is almost 3.
Since the number of packages are in a ratio of 8:4 = 2:1, the differences between the final average and the initial averages are in a ratio 1:2.
So, the distance between \(12\frac{3}{8}\) and the final average is almost 1, close to \(12\frac{3}{8}+1\approx{13}\frac{1}{4}\).
The final answer should be close to \(13\frac{1}{4}\).

Answer A.
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There must be something easy that i just don't get for me :
8*12(3/8) = 36 as you simplify the 8 between them. Therefore how do you manage to arrive at 99?

I guess it must be something different of spelling or something?

Thanks a lot for your help !

It's not 12 multiplied by 3/8. it's \(12\frac{3}{8}=\frac{12*8+3}{8}=\frac{99}{8}\) (the same way as \(1\frac{1}{2}=\frac{3}{2}\)).
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I used ratio of packages, which is 2:1.
converted both to the same fractions, so
15 1/4 = 15 2/8

2x(12 3/8) + 1x( 15 2/8) = 24+15+ 6/8+2/8 = 39 and 8/8, 8/8 is also obviously 1. Could also be together 40 but that's not easily divisible with three and you know you're left with a remainder. Instead just:
39/3 + 1/3 = 13 and 1/3

Total weight on Monday \(= 8*12 + 8 * \frac{3}{8} = 96 + 3\)

Total weight on Tuesday \(= 4*15 + 4 * \frac{1}{4} = 60 + 1\)

Average of all days \(= \frac{96 + 60 + 4}{12} = 8 + 5 + \frac{4}{12} = 13\frac{1}{3}\)

Answer = A

Solution:

To solve this question we can use the weighted average equation.

Weighted Average = (Sum of Weighted Terms) / (Total Number of Items)

We'll first determine the sum (numerator). We see that on the first day we had 8 items that averaged 12 3/8 pounds. We don't know the weights of the individual packages, but we can determine that the sum of all 8 packages is:

Sum of first day's packages = 8 x 12 3/8 = 99 pounds

Similarly, the sum of the second day's packages is:

Sum of second day's packages = 4 x 15 ¼ = 61

We now can use the weighted average equation to find the average weight of the 12 packages:

Weighted Average = (99 + 61) / 12

Weighted Average = 160 /12

Weighted Average = 13 1/3

Answer: A
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Hey,

You need to know the common fractions in and out. Haven’t found the original post but pls see attached a picture from my notes. A lot of question will translate easy when you know the fractions and their decimal equivalent and reversely as well.

We're going to make weighted average of \(12\frac{3}{8}\) (with 8 packages) and \(15\frac{1}{4}\) (4 packages). So, the correct answer must be between these two numbers. Choice C,D, and E are beyond of these two numbers, so they are out.
We're going to make the weighted average, but choice B is simply the ''regular'' average (\(12\frac{3}{8}\) + \(15\frac{1}{4}\))/2= \(13\frac{13}{16}\), which indicates choice B. So, B is out. Correct choice is A.
Hope it helps.
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