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Question 1187279: Please support with explanation , thk
Bill spent 1/6 of his money and an additional $11 on food. He then spent 2/3 of the remaining money and an additional $7 on books. Given that he was left with $16, how much money did Bill have at first?
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i get the following:
let s = the original amount of money.
he spent 1/6 * x + 11 on food.
he was left with 5/6 * x - 11.
he then spent 2/3 of that plus an additional 7 on books.
he was left with 1/3 * (5/6 * x - 11) - 7.
that was equal to 16 dollars.
you get:
1/3 * (5/6 * x - 11) - 7 = 16
add 7 to both sides to get:
1/3 * (5/6 * x - 11) = 23
simplify to get:
5/18 * x - 11/3 = 23
multiply both sides by 18 to get:
5 * x - 11 * 6 = 23 * 18
simplify to get:
5 * x - 66 = 414
add 66 to both sides to get:
5 * x = 414 + 66
simplify to get:
5 * x = 480
divide both sides by 5 to get:
x = 96
that's the original amount.
to confirm, follow the steps with 96 as the original amount.
he spent 1/6 * 96 + 11.
he had 5/6 * 96 - 11 left.
that was equal to 69
he then spent 2/3 of that plus 7.
he had 1/3 * 69 - 7 left.
that was equal to 16.
solution is that the original amount he started with was 96 dollars.
this was confirmed to be good.
Answer by greenestamps(13215)  (Show Source):
You can put this solution on YOUR website!
I will copy the setup of the problem from the other tutor...
let x = the original amount of money.
He spent 1/6 * x + 11 on food.
He was left with 5/6 * x - 11.
He then spent 2/3 of that plus an additional 7 on books.
He was left with 1/3 * (5/6 * x - 11) - 7.
That was equal to 16 dollars.
You get:
1/3 * (5/6 * x - 11) - 7 = 16 [1]
... and then I will take a slightly different path from there, for reasons you will see later.
add 7 to both sides:
1/3 * (5/6 * x - 11) = 23
Multiply by 3:
(5/6 * x - 11 = 69
Add 11:
5/6 * x = 80
Multiply by 6/5:
x = 96
ANSWER: He started with $96
That's a good algebraic method for solving the problem. However, often a problem like this is solved more easily by working backwards, like this:
He finished with $16, and the last thing he did was spend $7 on books, so before buying the books he had $16+$7=$23.
Before that, he spent 2/3 of his money, so the $23 he had was 1/3 of what he had previously. So the amount he had before spending 1/3 of it was 3*$23 = $69.
Before that, the last thing he did was spend $11 on food, so before buying the food he had $69+$11=$80.
And before that, he spent 1/6 of his original amount, so the $80 he had left was 5/6 of his original amount. So his original amount was $80*(6/5) = $96.
ANSWER: (again, of course) He started with $96.
The reason I used a different path than the other tutor in my algebraic solution to the problem is that my two solutions -- working forward and backward -- use exactly the same sequence of calculations.
And the reason for showing how to solve the problem by working backwards is that it will be a much easier method if the problem has more steps.
Suppose, for example, that there were one more step in this problem, in which he spent 1/4 of what he had left and $3 more, ending with $9. Then the equation for solving the problem "forwards" would be
3/4 (1/3 * (5/6 * x - 11) - 7) - 3 = 9
Writing that whole equation correctly is difficult; whereas working the problem backwards, one step at a time, will make finding the answer much easier.
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