Joe has 24 quarters and half dollars. If he had twice as many half dollars and half as many quarters, he would have 2 dollars more. How much money does he have?

Question

Joe has 24 quarters and half dollars. If he had twice as many half dollars and half as many quarters, he would have 2 dollars more. How much money does he have?

Answer 1

We have 3 unknowns

(1) the number of half-dollars, H

(2) the number of quarters, Q

(3) the value of the 24 coins consisting of quarters and half dollars, V

We need 3 equations!!!

Q + h = 24 <--- there are 24 coins

25Q + 50H = V <---- value in pennies

25(Q/2) + 50(2H) = V + 200 <---- per the 3rd statement, half the quarters and twice the half dollars

yields a $2 gain or 200 pennies

we have 3 equations

simplifying the 3rd equation:

(25/2) Q + 100H = V + 200

substitutes the 2nd equation into that:

(25/2)Q + 100 H = 25Q + 50H + 200

0 = (25/2)Q - 50H + 200 <--- subtracts (25/2)Q and 100H from both sides

-200 = (25/2)Q - 50H <--- now, it is an equation containing Q and H only

200 = 50H - (25/2)Q <-- multiplies everything by -1 to clear away negatives

The 2 x 2 system is:

200 = 50H - (25/2)Q

Q + h = 24 <-- original first equation

h = 24 - q <--- solves it for h

200 = 50(24 - q) - (25/2)q <--- substitution method

200 = 1200 - 50q - (25/2)q <--- distributes the 50

200 = 1200 - (100/2)q - (25/2)q <--- common denominator

200 = 1200 - (125/2)q <--- combines like terms

-1000 = (-125/2) q <--- subtracts 1200 from both sides

2000 = 125q <---- multiplies both sides by 2

q =2000/125 = 16

h =24-q = 24-16 = 8

16 quarters and 8 half dollars

In pennies, that's 16*25 + 8*50 = 400 + 400 = 800 = $8

half the quarters, and double the half dollars:

8 quarters and 16 half dollars

8 * 25 + 16 * 50 = 200 + 800 = 1000 = $10 , which is $2 more

16 quarters and 8 half dollars

Proven to be correct.