# Practice GMAT Problem Solving Question

The ratio of a compound, by weight, consisting only of substances x, y, and z is 4:6:10, respectively. Due to a dramatic increase in the surrounding temperature, the composition of the compound is changed such that the ratio of x to y is halved and the ratio of x to z is tripled. In the changed compound, if the total weight is 58 lbs, how much does substance x weigh?
 A) 48 B) 36 C) 24 D) 12 E) 10
1. The old ratio of x to y was 4:6. If this ratio is cut in half, then the new ratio of x to y is 2:6.
2. The old ratio of x to z was 4:10. If this ratio is tripled, then the new ratio of x to z is 12:10.
3. In order to combine these two ratios into a new ratio of x:y:z, we must rewrite them so that the element in common, x, has the same coefficient. With the same x-coefficient, we can compare the ratios of x:y and x:z. Using a multiplier of 6 on the first ratio (x:y = 2:6) yields x:y = 12:36.
4. Since the new ratio of x:z is 12:10, we can combine the new x:y ratio that we multiplied by 6/6 with the new x:z ratio in order to arrive at an x:y:z ratio of 12:36:10. In other words:
x:z = 12:10
x:y = 12:36
With 12 as a common term:
x:y:z = 12:36:10
5. In order to find the weight of substance x in the total changed compound, set up an equation of the combination:
x + y + z = 58 lbs
6. Now substitute an unknown multiplier, m, for each quantity to ensure that the ratios are enforced in the equation (x=12m, y=36m, z=10m):
12m + 36m + 10m = 58 lbs
Note: The unknown multiplier is the ratio by which the 12:36:10 ratio holds true. In other words, if m = 1, the substances will be in the ratio of 12:36:10. If m = 2, the substances will be in the ratio of 12(2):36(2):10(2) = 24:72:20
7. Combining like terms simplifies the equation to 58m=58.
8. Dividing through by 58 shows a multiplier of m=1.
9. Using this multiplier in the original equation we set up, we can see that the weight of x=12(1)=12. Thus, 12lbs or D is correct.