GMAT Practice Question: José has a collection of 100 coins, consisting of nickels, dimes, quarters, and half-dollars. If...
Question
José has a collection of 100 coins, consisting of nickels, dimes, quarters, and half-dollars. If he has a total of 35 nickels and dimes, a total of 45 dimes and quarters, and a total of 50 nickels and quarters, how many half-dollars does he have?
- 15
- 20
- 25
- 30
- 35
Topics: word problems, simultaneous equations, translations
Solution
Step 1
We first create variables for each coin type to avoid ambiguity
N = \text{number of nickels} D = \text{number of dimes} Q = \text{number of quarters} H = \text{number of half-dollars}
Step 2
We translate \textbf{\text{"José has a collection of 100 coins"}} into an equation for the total number of coins
N + D + Q + H = 100
Step 3
We translate \textbf{\text{"he has a total of 35 nickels and dimes"}} into an equation
N + D = 35
Step 4
We translate \textbf{\text{"a total of 45 dimes and quarters"}} into an equation
D + Q = 45
Step 5
We translate \textbf{\text{"a total of 50 nickels and quarters"}} into an equation
N + Q = 50
Step 6
We now combine two equations to eliminate N by subtracting N + D = 35 from N + Q = 50
(N + Q) - (N + D) = Q - D = 50 - 35 = 15
Step 7
We then combine two other equations to eliminate D by adding D + Q = 45 to Q - D = 15
(D + Q) + (Q - D) = 2Q = 45 + 15 = 60 Q = \frac{60}{2} = 30
Step 8
We substitute N + D = 35 and Q = 30 into the total equation to solve for H
N + D + Q + H = 100 35 + Q + H = 100 35 + 30 + H = 100 65 + H = 100 H = 100 - 65 = 35
Answer
35
