PSAT Math Multiple-Choice Question 784: Answer and Explanation

Question: 784

David has two quarters for every five dimes in his change dish, with no other coins present. If he has a total of $2 in coins in the dish, how many total coins does he have?

• A. 12
• B. 14
• C. 16
• D. 18

Correct Answer: B

Explanation:

(B) For this problem, you need to create a system of equations. First, having two quarters for every five dimes means that the ratio of quarters to dimes is:

Next, we need to come up with an expression that represents the value of the coins. Because quarters are worth 25 cents, the number of cents David has from q quarters will be 25q. Similarly, the number of cents he has from d dimes will be 10d. Because these expressions are in cents, we need the amount of money he has also to be in cents. There are 100 cents in 1 dollar, so David has 200 cents. Therefore:
25q + 10d = 200
Since , we have:

Next, plug this fraction in for q in the second equation:

Combine the d-terms:
20d = 200 → d = 10
Thus, there are 10 dimes. Plug 10 in for d in the equation that expresses q in terms of d:

Thus, there are four quarters and ten dimes, giving David 14 coins total, which is choice (B).