Part A
x = amount of money she starts with
(4/9)x = cost of the dress
(5/9)x = amount of money leftover after buying the dress
note that 4/9 and 5/9 add to 1 to represent 100% of her money
(3/4)*(5/9)x = (15/36)x = (5/12)x = amount spent on shoes
We see that she spent 5/12 of her original amount of money on shoes.
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We can look at a more concrete example if the previous section does not make sense.
Let's multiply the denominators 9 and 4 to get 9*4 = 36. Doing this trick will mean that the later steps won't involve decimal values.
Let's say she starts off with $36
Multiply that with 4/9 to get (4/9)*36 = 16. So in this hypothetical situation, the dress costs $16 and she's left with 36-16 = 20 dollars.
Multiply that 20 by 3/4 to get (3/4)*20 = 15
In this situation, the shoes cost $15
We'll then divide the cost of the shoes (15) over the original amount of money she started with (36)
15/36 = (3*5)/(3*12) = 5/12
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Answer: 5/12
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Part B
From the first section in part A, we found that:
- (4/9)x = amount of money spent on the dress
- (5/12)x = amount of money spent on shoes
Add those two expressions
(4/9)x + (5/12)x = (16/36)x + (15/36)x = (31/36)x
This indicates that buying the dress and shoes will cost (31/36)x dollars, where x is the original amount of money Jing has to start with.
The amount left over is (5/36)x because 5/36 adds with 31/36 to get 1.
Take 1/3 of this to get:
(1/3)*(5/36)x = (5/108)x
Then set this equal to 18 and solve for x
(5/108)x = 18
5x = 18*108
5x = 1944
x = 1944/5
x = 388.80
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Let's check the answer
If she starts with 388.80 dollars, then the dress costs (4/9)*388.80 = 172.80
She's left with 388.80 - 172.80 = 216 dollars after buying the dress.
She then spends the remaining 3/4 of that on shoes. So (3/4)*216 = 162 dollars is spent on the shoes.
Jing now has 216 - 162 = 54 dollars left over.
1/3 of which is spent on the shoe bag, meaning it costs (1/3)*54 = 18 dollars
This confirms our answer.
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Answer: $388.80