A Contractor Combined x Tons of a Gravel Mixture that Contained 10 percent Gravel G GMAT Data Sufficiency

Question: A contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight. What is the value of x?

1. Y = 10
2. Z = 16

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statement TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“A Contractor Combined x Tons of a Gravel Mixture that Contained 10 percent Gravel G”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

Given: contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight, with y tons of mixture that contained 2 percent of gravel G and produced z tons of a mixture that was 5 percent gravel G, by weight.

Now we are given two values of y and we are asked to tell whether the given data is sufficient to solve the problem or not. If possible find the value of x.

Firstly it is obvious that the amount of gravel G in the mixture before and after mixing will not change.
So, 10% of x + 2% of y = 5% of z
=> (\(\frac{10}{100}\) * x) + (\(\frac{2}{100}\) * y) = (\(\frac{5}{100}\)) * z
=> 0.1x + 0.02y = 0.05z —- (1)
Let this be equation(1)
Now if we observe carefully the total amount of mixtures before and after mixing will also be same.
So it is obvious to say that
x+y = z — (2)
Let this equation be (2)
Now it should be noted that in the cases, we are given the value of y and we have to check whether we can deduce the value of x or not.
So we have to eliminate z from equation (1)
So putting the value of z from equation (2) in (1) we get
0.1x + 0.02y = 0.05(x+y)
0.1x + 0.02y = 0.05x + 0.05y
0.1x - 0.05x = 0.05y - 0.02y
0.05x = 0.03y
Multiplying both sides by 100 we get,
5x = 3y or 5x - 3y = 0 —---(3)
Now for case 1
The value of y is given -> y = 10
Putting the value of y in (3) we get
5x - 3(10) = 0
X = \(\frac{30}{5}\) = 6
This statement is sufficient.
Now for case 2 we are given the value of z,
We know z = x+ y = 16
Y = 16 - x
Also, we know that 5x - 3y = 0
Putting y here we get,
5x - 3(16-x) = 0
5x - 48 +3x = 0
8x = 48
X = \(\frac{48}{8}\) = 6
This statement is also sufficient.

Correct Answer: D

Approach Solution 2:

By using the scale method,

we see that since 10% and 2% we get a weighted average of 5%,

x:y = 3:5
We also know x + y = z.
we have two cases, let us check them one by one:
1. y = 10.
So, if y = 10, then x = 6 as their ratios must be 3:5. (Sufficient.)

2. z = 16
If the sum of x and y is 16, then x must be 6 and y must be 10

Then we get a ratio of 3:5.
So, we may consider:
x : y...... x + y
3 : 5...... 8
Since 8 in ratio terms, then it is actually 16. 3 is actually 6 and 5 is actually 10.

Correct Answer: D

Approach Solution 3:

Let us consider the problem in an equation: 

0.1x+0.02y=0.05(x+y)

this implies 0.1x+0.02y=0.05(x+y),

where x+y=zx+y=z 

this implies 5x=3y5x=3y 

this implies that Q: x=?x=?
we have two cases, let us check them one by one:

(1) y=10y=10

this implies 5x=3y=30
this implies 5x=3y=30 
this implies x=6x=6. Sufficient.

(2) z=x+y=16z

this implies x+y=16 
this implies y=16−xy=16−x
this implies 5x=3y=3(16−x)
this implies 5x=3y=3(16−x) 
this implies x=6x=6. Sufficient.

Correct Answer: D

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