Candles
You have $100 and need 100 candles for your great-grandmother’s hundredth birthday. Type A candles cost 50¢ each. Type B candles cost $5.50 each, and Type C cost $9.50. If you must buy at least one of each candle for a total of 100 candles, how many candles of each type would you purchase?
One way to solve the problem
Let A = the number of type A candles I will buy.
B = the number of type B candles I will buy.
C = the number of type C candles I will buy.
A + B + C = 100 (1)
.5A + 5.5B + 9.5C = 100 (2)
Multiplying (2) by 2,
A + 11B +19C = 200 (3)
Subtracting (1) from (3),
10B + 18C = 100
18C = 100 - 10B
9C = 50 - 5B
9C = 5(10 - B)
C = 5/9*(10 - B) (4)
B and C must be whole numbers (no parts of candles are allowed!).
So the right side of (4) must be a whole number. So 10 - B must be divisible by 9. B cannot be 0 or negative. So B = 1.
Thus
C = 5/9*(10 - 1) = 5
So A = 100 - B - C = 100 - 1 - 5 = 94.
Let’s check the prices to see if they sum to $100.
.5(94) + 5.5(1) + 9.5(5) = 47 + 5.5 + 47.5 = $100.