In a certain group of people, the average (arithmetic mean) weight of

This is a weighted average problem. We know the different partial averages, so we could determine the global average if we knew the weights. In the weighted arithmetic average form the weights would be the number of males and the number of females. However, in any weighted average it's the ratio of the weights that really matters, not their actual values.

The rephrased question: M:F=?

1) We know that M:F=1:2. Thus, the answer to the rephrased question is a unique value. \(\implies\) Sufficient

2) We know that F=M+10. If M=2, then M:F=1:6. However, if M=10, then M:F=1:2. Thus, we can't get a unique value to answer the rephrased question. \(\implies\) Insufficient

Answer: A

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Assume \(m\) and \(f\) are the numbers of males and females, respectively.

The question ask what the value of \(\frac{180m+120f}{m+f}\).

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that A is most likely to be the answer to this question.

Condition 1)

Since \(f = 2m\), we have \(\frac{180m + 120f}{m+f} = \frac{180m+240m}{m+2m} = \frac{420m}{3m} = 140\).

Since condition 1) yields a unique solution, it is sufficient.


Condition 2)
Since \(f = m + 10\), we have \(\frac{180m + 120f}{m+f} = \frac{180m + 120m + 1200}{m + m + 10} = \frac{300m + 1200 }{ 2m+10} = \frac{150m + 600 }{ m+5}\).
If \(m = 5\), then we have the average \(\frac{150*5+600)}{10} = \frac{1350}{10} = 135\).
If \(m = 15\), then we have the average \(\frac{150*15+600}{20} = \frac{2850}{20} = 142.5\).

Since condition 2) does not yield a unique solution, it is not sufficient

Therefore, A is the answer as expected.

though it doesn't matter, I would like to know if it is F=2M or M=2F. Is there any resource for deciphering such statements. I find it a bit difficult to convert the english into Math.

Please find attached the answer to the question.

When questions talks about average weight, height, age, etc. of 2 groups, think in terms of mixtures and alligations. Check the image given below for solution.

Males average weight 180p
Females average weight 120p
1) 2M=F (numbers)
2) F=M+10 (numbers); So, 2M=M+10; M=10 and F=20
(180p+120p)/2=150p
The average weight in the group is 150 pounds ( ~68 kilograms).

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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