SOLUTION: Bethany’s credit card bill in December was 1.5 times her food bill. Her apartment rent was 5 times her utility bill. Her phone bill was $17 less than half of her food bill. Her pho

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Question 1032212: Bethany’s credit card bill in December was 1.5 times her food bill. Her apartment rent was 5 times her utility bill. Her phone bill was $17 less than half of her food bill. Her phone bill was also $18 less than twice her utilities bill. The total of the five bills was $751. What was the amount of each bill?
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your goal is to reduce everything down to one variable which can then be solved.
once you solve for that one variable, you can then find the values of the other variables.

let:

a = apartment rent bill.
c = credit card bill.
p = phone bill.
f = food bill.
u = utility bill.

since the total bill is 751, you get:

a + c + p + f + u = 751.

now you start looking at relationships that will allow you to reduce the number of variables down to 1.

since c = 1.5 * f, and p = f/2 - 17, you can replace c and p with their equivalent values to get:

a + 1.5 * f + f/2 - 17 + f + u = 751.

since a = 5 * u, you can replace a to get:

5 * u + 1.5 * f + f/2 - 17 + f + u = 751.

you're down to 2 variables.

you are given that p = f/2 - 17 and also that p = 2u - 18.

in the second equation, you can solve for u to get u = (p + 18) / 2.

since you also know that p = f/2 - 17, you can replace p in the u = equation to get u = (f/2 - 17 + 18) / 2.

simplify this to get u = (f/2 + 1) / 2.

simplify this further to get u = f/4 + 1/2.

go back to the equation where you only had f and u.

that equation is 5 * u + 1.5 * f + f/2 - 17 + f + u = 751.

replace u with f/4 + 1/2 to get:

5 * (f/4 + 1/2) + 1.5 * f + f/2 - 17 + f + f/4 + 1/2 = 751.

simplify this equation to get:

5f/4 + 5/2 + 1.5f + f/2 - 17 + f + f/4 + 1/2 = 751.

multiply both sides of this equation by 4 to get:

5f + 10 + 6f + 2f - 68 + 4f + f + 2 = 3004.

that got rid of the fractions and makes combining like terms easier.

combine like terms to get 18f - 56 = 3004

add 56 to both sides of the equation to get 18f = 3060.

divide both sides of the equation by 18 to get f = 170.

now that you know f, you can solve for all the other variables.

you will get:

f = 170
c = 255
p = 68
u = 43
a = 215.

they add up to 751 so you're ok there.

the requirements of the problem are also satisfied.

255 / 170 = 1.5 so c = 1.5 * f is satisfied.
5 * 43 = 215 so a = 5 * u is satisfied.
68 = 1/2 * 170 - 17 so p = 1/2 * f - 17 is satisfied.
68 = 2 * 43 - 18 so p = 2 * u - 18 is satisfied.

everything checks out so the answer is good.

the main objective was to reduce all the variables to one variable that could be solved.

once you did that you were able to solve for the rest of the variables.







Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Bethany’s credit card bill in December was 1.5 times her food bill. Her apartment rent was 5 times her utility bill. Her phone bill was $17 less than half of her food bill. Her phone bill was also $18 less than twice her utilities bill. The total of the five bills was $751. What was the amount of each bill?
Let the amount for food, and utilities be F, and U, respectively

Then amount for credit card = 1.5F

Amount for rent = 5U

Amount for phone = 

Also, amount for phone = 2U - 18

We can then say that: 

F - 34 = 4U - 36 -------- Multiplying by LCD, 2

F = 4U - 36 + 34

F = 4U - 2

Since all sum to $751, we then get: U + 1.5(4U - 2) + 5U + 2U - 18 + 4U - 2 = 751

U + 6U - 3 + 5U + 2U - 18 + 4U - 2 = 751

18U - 23 = 751

18U = 751 + 23

18U = 774

U, or amount for utilities = , or 

Amount for food: 4(43) - 2 = 172 - 2, or 

Amount for credit card: 1.5(170) = 

Amount for phone: 2(43) - 18 = 86 - 18 = 

Amount for rent: 5(43) =  

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