sondenso wrote:
Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75 % fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X?
A. 10%
B. 33 1/3%
C. 40%
D. 50%
E. 66 2/3%
This is a great problem because we can solve it using equations and substitution, or we can employ some estimation because the equation we create is a weighted average. Let’s first do the equation-substitution method.
Equation-Substitution MethodWe are given information about ryegrass, bluegrass, and fescue. However, we are told the mixture of X and Y is 30 percent ryegrass. Thus, we only care about the ryegrass, so we can ignore fescue or bluegrass.
We are given that seed mixture X is 40%
ryegrass and that seed mixture Y is 25% ryegrass. We are also given that the weight of the combined mixture is 30% ryegrass. With x representing the total weight of mixture X, and y representing the total weight of mixture Y, we can create the following equation:
0.4x + 0.25y = 0.3(x + y)
We can multiply the entire equation by 100, and we have:
40x + 25y = 30x + 30y
10x = 5y
2x = y
The question asks what percent of the weight of the mixture is x.
We can create an expression for this:
x/(x+y) * 100 = ?
Since we know that 2x = y, we can substitute in 2x for y in our expression. So we have:
x/(x+2x) * 100 = x/(3x) * 100 = 1/3 * 100 = 33.33%
The answer is B.
Estimation MethodThis is a method I would suggest using only if you are short on time.
We are given that seed mixture X is 40% ryegrass and that seed mixture Y is 25% ryegrass. We are also given that the overall weight of the mixture is 30% ryegrass.
Since 30% is closer to 25% than it is to 40%, we know that there is
more of mixture Y than there is of mixture X, but not by a large amount. So let’s analyze our answer choices.
A) 10%
This tells us that the mixture is made up of 10% x and 90% y. That discrepancy is much too large to be correct, based on the overall weight of the mixture.
B) 33 1/3%
This tells us that the mixture is made up of 33 1/3% x and 66 2/3% y. This could be the answer.
C) 40%
This tells us that the mixture is made up of 40% x and 60% y. This could be the answer.
D) 50%
This tells us that the mixture is made up of 50% x and 50% y. Since we know that y is weighted more heavily in the mixture, we know this cannot be the answer.
E) 66.67%
This tells us that the mixture is made up of 66 2/3% x and 33 1/3% y. Since we know that y is weighted more heavily in the mixture, we know this cannot be the answer.
So now we have our answer choices narrowed down to B and C. However, there is something interesting about answer choice B. Notice that if we were to add together answer choice B (33 1/3%) and answer choice E (66 2/3%), our sum would be 100%. These are strategic answer choices, because the GMAT is hoping that if we make a mistake in our calculations we will determine that x represents 66 2/3% of the mixture rather than 33 1/3% of the mixture. Since answer choice C (40%) does not have a corresponding trap answer there is a higher likelihood that the answer is B (33 1/3%) than it is C (40%).
As mentioned at the beginning, the surest solution technique is to create an equation to solve it and actually complete the math. However, if you are short on time or have trouble setting up the equation, estimation is a method you can employ.
The answer is B.
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