Word Problem

Question

Bill and Jill sold candy bars to raise money for their band trip. Bill generated $176 in revenue. Jill thought she could get more revenue by lowering the price of her candy bars. She lowered her price by 15 cents, sold 10 more bars than Bill, but in the end generated $2 less in revenue than Bill. Determine the price per candy bar that Bill and Jill each charged, as well as the number of candy bars they each sold.

Answer 1

First, let's set some variables:

 

Let b = # candy bars Bill sold

Let c_b = price Bill charged

 

Let j = # candy bars Jill sold

Let c_j = price Jill charged

 

Bill generated $176 revenue... # bars times price per bar:

 

b(c_b) = 176  __________________ eqn 1

 

Jill charged 15 cents or $0.15 less than Bill per candy bar:

 

c_j = c_b - 0.15 _______________ eqn 2

 

Jill generated $2 less revenue than Bill or $174:

 

j(c_j) = 174 __________________ eqn 3

 

Jill sold 10 more candy bars than Bill:

 

j = b + 10 ___________________ eqn 4

 

Combining for total revenue:

 

b(c_b) + j(c_j) = 174 + 176

b(c_b) + j(c_j) = 350

 

substituting from eqn 4, j = b+10

and from eqn 2, c_j = c_b - 0.15

 

b(c_b) + (b+10)(c_b-0.15) = 350

bc_b + bc_b - 0.15b + 10c_b - 1.5 = 350

combine like terms and add 1.5 to both sides

2bc_b + 10c_b - 0.15b = 351.5

 

From eqn 1: bc_b = 176

 

2(176) + 10c_b - 0.15b = 351.5

352 + 10c_b - 0.15b = 351.5

10c_b - 0.15b = -0.5

 

Also from eqn 1 we can get c_b = 176/b

 

10(176/b) - 0.15b = -0.5

Multiply through by b

 

1760 - 0.15b² = -0.5b

Move all to one side of equal sign

add (0.15b² -1760) to both sides

 

0.15b² -0.5b - 1760 = 0

To clear decimals multiply through by 100

15b² - 50b - 176000 = 0

 

From quadratic formula:

b = [50 ± √((-50)²-4(15)(-176000))]/[2(15)]

 

b = [50 ± √10562500]/30

b = (50 ± 3250)/30

Disregarding the negative since Bill did not sell a negative

number of candy bars

b = 3300/30 = 110 candy bars

 

bill sold 110 candy bars

jill sold 110 + 10 = 120 candy bars

 

bc_b = 176

110c_b = 176

c_b = 176/110 = $1.60

 

Bill charged $1.60 per candy bar

 

Jill charged 0.15 less per candy bar

Jill charged $1.45 per candy bar