A wholesaler of nuts sells two mixtures of peanuts and cashews

A wholesaler of nuts sells two mixtures of peanuts and cashews. The first mixture is 3/4 peanuts by weight. The second mixture has the same number of pounds of cashews as the first mixture but 8 fewer pounds of peanuts. If 7/10 of the weight of the second mixture is peanuts, how many pounds of nuts are in the first mixture?

One thing that jumps out about this question is that the quantity of cashews is the same in both cases.

A second thing that jumps out is that, if we know the proportion of peanuts in each mixture, we also know the proportion of cashews and the relative proportions of peanuts and cashews in each mixtures.

So, given those two facts, we can base all of our work on the quantity of cashews.

Let \(C\) be the quantity of cashews in both mixtures.

Then, since the first mixture is \(3/4\) peanuts by weight, \(C = 1/4\) of the first mixture. So, the weight of the first mixture is \(4C\).

In the second mixture, peanuts represent \(7/10\) and \(C\) represents \(3/10\). So, the total weight of the second mixture is the following:

\(\frac{7}{3} C + C = \frac{10}{3} C\)

Since the difference between the weights of the mixture is 8 pounds, we have the following:

\(4C - 8 = \frac{10}{3} C\)

\(12C - 24 = 10C\)

\(2C = 24\)

\(C = 12\)

So, the total weight of the first mixture is \(4C = 48\).

Correct Answer

What are your thoughts on solving with the method? (pure ratio reasoning)

The ratio of the first mixture of Peanuts:Cashews:Total is 3x:1x:4x (given)

The ratio of the second mixture is 7x:3x:10x (given)

1 ) The common ratios between the first and the second mixtures are 9x:3x:12x

2)The difference between the total of the 2 ratios is 8 pounds (given) and 2 ratio units (12x-10x), so:

3)
2x = 8
x = 4

12x = 48

Is not easy to spot right on this reasoning but I think with practice it gets easier.

I have considered that in 1st mixture, Peanut is P pounds, Cashew is C pounds. So the total weight of 1st mixture is P+C pounds.
As per the question, in 1st mixture, (3/4)*(P+C)=P or, P=3C. In 2nd mixture, (P-8+C)*(7/10)=P-8. Substituting P=3C, we can achieve C=12.
Therefore, P=3*12=36. So, 36+12=48 (C).

I solved it a bit differently than seen above and I'm wondering if anyone can say if how I did it is repeatable.

3x-8 / 1 = 7x/3

9x - 24 = 7x

2x = 24

x = 12 then multiply 12 by the denominator of nuts in the first mixture. 12 * 4 = 48

(3x-8)/(4x-8) = 7/10
30x - 80 = 28x - 56
2x = 24
x = 12

4x = 12*4 = 48
Answer C

My way:

Say X is total weight of first mixture and Y is total weight of second mixture
Peanuts in X are equal to peanuts in Y plus 8 pounds --> 3X/4 = 7Y/10 + 8
We also know that Y = X-8
Then, 3X/4 = 7*(X-8)/10 + 8 --> x= 48

Let wt. of mix I be X,
In mix I, peanuts = 3/4*X, cashew = 1/4*X

Let wt. of mix II be Y,
In mix II, peanuts = 7/10*Y, cashew = 3/10*Y

As per the question in mix II wt of cashew is same as mix I
1/4*X = 3/10*Y
Y = 5/6*X

Also, peanuts in mix II = peanuts in mix I - 8
7/10*Y = 3/4*X - 8
Solvoing this,
X = 48 lbs

Correct Option: C

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Harsha

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Thanks for your method. It really helps me.

Whenever I see this kind of question my mind just go straightforward literally interpret the all wordings into equations and just use sheer force to solve them in the most straightforward way possible, not trying to simplify or steamline at all. I don't know why but I think I would always lose more time/brain power to think of simpler equations to solve it.

My abomination method of trying to solve it with sheer force, no simplication and still got it wrong:­
Spot my mistake and got it right at last.... Too many variables makes me highly risks solving equations wrong... really.­

It sounds like Mix II has 8 fewer Peanuts is that incorrect? Thats how I made my mistake

The second mixture has 8 fewer pounds of peanuts than the first one.

KarishmaB

I tried to solve it as a weighted average but failed. Any thoughts?

First Mix: Peanuts : Cashews = 3 : 1 = 9: 3

Second Mix: Peanuts : Cashews = 7 : 3

Cashews are the same weight in both. On the ratio scale, there are 2 fewer units of peanuts in second mix. In actual values, there are 8 fewer units of peanuts.
Hence multiplier is 4.
Hence actual weight of total first mix = (9 + 3) * 4 = 48 pounds

Answer (C)

Ratios discussed here: https://youtu.be/5ODENGG5dvc
Ratios Post: https://anaprep.com/arithmetic-ratios-the-one-where-it-all-starts/The two mixes are not mixed together hence the question is not a weighted average question.

I solved it by 3/10W2 = 1/4W1, which is because the weight of cashews is the same.
i got the ratio W1/W2 = 12/10. We know that W2 = W1 - 8.
Equating the ratios we get 12(W1-8) = 10W1 ---> 2W1 = 12*8 ---> W1 = 6*8 = 48 (C)