SOLUTION: Rama was getting some items for the new school year. First, he bought some textbooks with $8 more than 1/3 of his money. Next, Next, he bought his stationery with $12.20 less than

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Rama was getting some items for the new school year. First, he bought some textbooks with $8 more than 1/3 of his money. Next, Next, he bought his stationery with $12.20 less than       Log On


   

Question 1209350: Rama was getting some items for the new school year. First, he bought some textbooks with $8 more than 1/3 of his money. Next, Next, he bought his stationery with $12.20 less than 1/2 of his remaining money. Lastly, he bought some school socks with $2.80 more than 1/2 of the money left. Then, he had $15.40 with him. How much money did he have at first?
Found 4 solutions by proyaop, mccravyedwin, greenestamps, MathTherapy:
Answer by proyaop(69) About Me  (Show Source):
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**1. Let's represent:**
* The initial amount of money Rama had as 'x'
**2. Formulate the equation:**
* **Textbooks:** Rama spent (x/3) + 8 dollars on textbooks.
* **Remaining money after textbooks:** x - (x/3) - 8 = (2/3)x - 8
* **Stationery:** Rama spent 1/2 * ((2/3)x - 8) - 12.20 dollars on stationery.
* **Remaining money after stationery:** (2/3)x - 8 - 1/2 * ((2/3)x - 8) + 12.20 = (2/3)x - 8 - (1/3)x + 4 + 12.20 = (1/3)x + 8.20
* **School Socks:** Rama spent 1/2 * ((1/3)x + 8.20) + 2.80 dollars on socks.
* **Remaining money:** (1/3)x + 8.20 - 1/2 * ((1/3)x + 8.20) - 2.80 = 15.40
**3. Simplify and solve the equation:**
* (1/3)x + 8.20 - (1/6)x - 4.10 - 2.80 = 15.40
* (1/6)x + 1.30 = 15.40
* (1/6)x = 14.10
* x = 14.10 * 6
* x = 84.60
**Therefore, Rama had $84.60 at first.**

Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!
Looks like AI got this one correct.  AI will get better and better, but you
can't trust it with everything yet.  There is still some use for us humans.
So I'll do it the human way.
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Rama was getting some items for the new school year.

Let x = the amount at first.

First, he bought some textbooks with $8 more than 1/3 of his money. 

Let y = how much he had left after buying textbooks.
y=x+-+%288+%2B+expr%281%2F3%29x%29
y=x+-+8+-+expr%281%2F3%29x
y=expr%282%2F3%29x-8 

Next, he bought his stationery with $12.20 less than 1/2 of his remaining money.

Let z = how much he had left after buying stationery.
z=y-%28expr%281%2F2%29y-12.20%29 
z=y-expr%281%2F2%29y%2B12.20
z=expr%281%2F2%29y%2B12.20

Lastly, he bought some school socks with $2.80 more than 1/2 of the money left.

Let w = how much he had left after buying socks.
w=z-%28expr%281%2F2%29z%2B2.80%29
w=z-expr%281%2F2%29z-2.80
w=expr%281%2F2%29z-2.80

Then, he had $15.40 with him. 
w=15.40

 

How much money did he have at first?

Substitute the 4th equation in the 3rd equation

15.40=expr%281%2F2%29z-2.80
18.20=expr%281%2F2%29z
36.40=z

Substitute that in the 2nd equation

36.40=expr%281%2F2%29y%2B12.20
24.20=expr%281%2F2%29y
48.40=y

Substitute that in the 1st equation

48.40=expr%282%2F3%29x-8
56.40=expr%282%2F3%29x
169.20=2x
84.60=x

Answer: $84.60   <--SOLUTION!

Checking:

He had $84.60 
First, he bought some textbooks with $8 more than 1/3 of his money. 

1/3 of his 84.60 or 28.20, plus 8 is 36.20, which was what he spent,
so he had 84.60-36.20 or 48.40 left.

Next, he bought his stationery with $12.20 less than 1/2 of his remaining money. 

1/2 of his remaining money was 1/2 of 48.40, or 24.20, minus 12.20, or 12, 
which was what he spent, so he had 48.40-12 or 36.40 left.

Lastly, he bought some school socks with $2.80 more than 1/2 of the money left.

1/2 of his remaining money was 1/2 of 36.40, or 18.20, plus 2.80, or 21, 
which was what he spent, so he had 36.40-21 or 15.40 left.

Then, he had $15.40 with him.

So now we know we have the right answer, $84.60.

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Working the problem "forwards", as shown in the other responses, requires doing some back-substitution to get the final answer.

You can get the answer directly by working the problem backwards.

He finished with $15.40 after spending $2.80 more than half of his money on school socks.

let x = amount he had before buying the socks
x-%28%281%2F2%29x%2B2.80%29=15.40
%281%2F2%29x-2.80=15.40
%281%2F2%29x=18.20
x=36.40

Before buying the socks, the amount he had was $36.40. That was what he had after spending $12.20 less than half his money on stationery.

let y = amount he had before buying the stationery
y-%28%281%2F2%29y-12.20%29=36.40
%281%2F2%29y%2B12.20=36.40
%281%2F2%29y=24.20
y=48.40

The amount he had before buying the stationery was $48.40. That was what he had after spending $8 more than 1/3 of his money on textbooks.

let z = amount he had before buying the textbooks (i.e., the amount he had at first)
z-%28%281%2F3%29z%2B8%29=48.40
%282%2F3%29z-8=48.40
%282%2F3%29z=56.40
z=84.60

ANSWER: The amount he started with was $86.40

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Rama was getting some items for the new school year. First, he bought some textbooks with $8 more than 1/3 of his money. Next, Next, he bought his stationery with $12.20 less than 1/2 of his remaining money. Lastly, he bought some school socks with $2.80 more than 1/2 of the money left. Then, he had $15.40 with him. How much money did he have at first?

Let original amount he had be M
Remainder after spending $8 more than 1%2F3 of his money: 1+-+%281%2F3%29M - 8 = 2M%2F3+-+8
Remainder after spending $12.20 less than 1%2F2 of his 1st REMAINDER: %281%2F2%29%282M%2F3++-++8%29+%2B+12.20 = M%2F3++-++4+%2B+12.20 = M%2F3+%2B+8.2
Remainder after spending $2.80 less than 1%2F2 of his 2nd REMAINDER: %281%2F2%29%28M%2F3+%2B+8.2%29+-+2.80 = M%2F6+%2B+4.1++-++2.8 = M%2F6+%2B+1.3
Since his last (3rd) REMAINING amount was M%2F6+%2B+1.3, which was given as $15.4, we get: 
                     Cross-multiplying, we get original amount he had, or M as: 6(14.1) = $84.60