Last Updated on May 9, 2023
GMAT OFFICIAL GUIDE DS –
Solution:
We are given that a store sold gift certificates of either $10 or $50. Let’s define some variables for the numberof $10 and $50 gift certificates sold.
x = the number of $10 gift certificates sold
y = the number of $50 gift certificates sold
We are also given that the store sold more than 5 $50 gift certificates. Thus, y > 5.
We need to determine the total number of gift certificates sold, that is, the sum of x and y.
Statement One Alone:
Yesterday the bookstore sold fewer than 10 gift certificates that cost $10 each.
With the information in statement one we know that x < 10. However, we do not have enough information to determine the value of x + y. We can eliminate answer choices A and D.
Statement Two Alone:
The total cost of the gift certificates sold yesterday by the bookstore was $460.
From the information in statement two we can create the following equation:
10x + 50y = 460
Divide both sides by 10:
x + 5y = 46
We were given that y is greater than 5. Let’s substitute 6 in for y in the equation x + 5y = 46.
x + 5(6) = 46
x + 30 = 46
x = 16
We see that if y = 6, then x = 16 and x + y = 22.
Let’s now substitute 7 (the next higher integer value) for y in the equation x + 5y = 46.
x + 5(7) = 46
x + 35 = 46
x = 11
We see that if y = 7, then x = 11 and x + y = 18.
Becausewe have two different values for x + y, statement two is not sufficient to answer the question. We can eliminate answer choice B.
Statements One and Two Together:
From statement one, we know that x < 10. From statement two, we know that x + 5y = 46 and in particular, x + y can be either 22 or 18 (or even some other values). However, by knowing that x < 10, we know that x + y can’t be 22 because, in order for x + y to be 22, y = 6 and x = 16, but we know that x < 10.
Similarly, x + y can’t be 18. In order for x + y to be 18, y = 7 and x = 11, but we know that x < 10.
The question becomes: are there any values for x and y such that x + 5y = 46 and x < 10 and y > 5?
To find out, let’s substitute 8 (the next integer after 7) for y:
x + 5(8) = 46
x + 40 = 46
x = 6
We see that if y = 8, then x = 6 (which is less than 10) and x + y = 14. This solution is plausible.
Now let’s substitute 9 (the next integer after 8) in for y:
x + 5(9) = 46
x + 45 = 46
x = 1
We see that if y = 9, then x = 1 (which is less than 10) and x + y = 10. This solution is also plausible.
However, we still have two different values for x + y; thus, both statements together are still not sufficient to determine a unique value for x + y, the total number of gift certificates sold.
Answer: E
