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?

GMAT Panda · Accepted Answer

35 We first create variables for each coin type to avoid ambiguity N = number of nickels D = number of dimes Q = number of quarters H = number of half-dollars We translate "José has a collection of 100 coins" into an equation for the total number of coins N + D + Q + H = 100 We translate "he has a total of 35 nickels and dimes" into an equation N + D = 35 We translate "a total of 45 dimes and quarters" into an equation D + Q = 45 We translate "a total of 50 nickels and quarters" into an equation N + Q = 50 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 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 = 60/2 = 30 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

Explanation	Calculations	Help
$We first create variables for each coin type to avoid ambiguity$	$N = number of nickels D = number of dimes Q = number of quarters H = number of half-dollars$
$We translate "Jos \overset{ˊ}{e} has a collection of 100 coins" into an equation for the total number of coins$	$N + D + Q + H = 100$
$We translate "he has a total of 35 nickels and dimes" into an equation$	$N + D = 35$
$We translate "a total of 45 dimes and quarters" into an equation$	$D + Q = 45$
$We translate "a total of 50 nickels and quarters" into an equation$	$N + Q = 50$
$We now combine two equations to eliminate N by subtracting N + D = 35 from N + Q = 50$	$(N + Q) - (N + D) = Q - D = 50 - 35 = 15$
$We then combine two other equations to eliminate D by adding D + Q = 45 to Q - D = 15$	$(D + Q) + (Q - D) = 2 Q = 45 + 15 = 60 Q = \frac{60}{2} = 30$
$We substitute N + D = 35 and Q = 30 into the total equation to solve for H$	$N + D + Q + H = 100 35 + Q + H = 100 35 + 30 + H = 100 65 + H = 100 H = 100 - 65 = 35$

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