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Archie, Betty and Coach bought a radio. Archie paid the leas

This topic has 4 expert replies and 2 member replies

Archie, Betty and Coach bought a radio. Archie paid the leas

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Archie, Betty and Coach bought a radio. Archie paid the least amount of money, Coach paid twice as much as Archie, and Betty paid the most. Archie paid seventy dollars less than the difference between what Betty and Coach paid. If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less. Which of the following equals the amount Betty paid?

A. 40
B. 90
C. 130
D. 140
E. 150

OA C

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vinni.k wrote:
Archie, Betty and Coach bought a radio. Archie paid the least amount of money, Coach paid twice as much as Archie, and Betty paid the most. Archie paid seventy dollars less than the difference between what Betty and Coach paid. If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less. Which of the following equals the amount Betty paid?

A. 40
B. 90
C. 130
D. 140
E. 150

OA C
If Coach paid twice as much as Archie, and we designate Archie as 'A' then Coach is '2A.' We know that Archie paid 70 less than the difference between Betty and Coach, so we know that
A = Betty - 2A - 70
Or Betty = 3A + 70

Now we have everyone in terms of 'A.'

Archie: A
Betty: 3A + 70
Coach: 2A
Total: 6A + 70

If Archie paid twice as much, or 2A, then Betty and Coach would have paid $10 less. (Presumably, the total would remain the same.) This yields the following

Archie: 2A
Betty: 3A + 60
Coach: 2A - 10
Total: 6A + 70

Now we have: 2A + 3A + 60 + 2A - 10 = 6A + 70
7A + 50 = 6A + 70
A = 20

We want the amount Betty paid, or 3A + 70. So we get 3*20 + 70 = 60 + 70 = 130. The answer is C

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Thanks David,

I think the total should be 7A + 50 and not 6A + 70 as given below :
Quote:
Archie: 2A
Betty: 3A + 60
Coach: 2A - 10
Total: 6A + 70
Then we can equate since we have two totals satisfying the two conditions

Quote:
7A + 50 = 6A + 70
Am i correct ?

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vinni.k wrote:
Thanks David,

I think the total should be 7A + 50 and not 6A + 70 as given below :
Quote:
Archie: 2A
Betty: 3A + 60
Coach: 2A - 10
Total: 6A + 70
Then we can equate since we have two totals satisfying the two conditions

Quote:
7A + 50 = 6A + 70
Am i correct ?
You can think of it like that. Because we know that the total hasn't changed, we can either rewrite "6A + 70" for the second total, as I did, or you can sum up the expressions and set the new total of 7A + 50 equal to the original 6A + 70. Ultimately, you're doing the same thing - acknowledging that those expressions are identical.

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vinni.k wrote:
Archie, Betty and Coach bought a radio. Archie paid the least amount of money, Coach paid twice as much as Archie, and Betty paid the most. Archie paid seventy dollars less than the difference between what Betty and Coach paid. If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less. Which of the following equals the amount Betty paid?

A. 40
B. 90
C. 130
D. 140
E. 150
Since A, B and C each contribute to the total cost, we get:
T = A + B + C.

If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less.
Thus:
T = 2A + (B-10) + (C-10)
T = 2A + B + C - 20.

Since the blue expressions above both represent the total cost, they must be equal:
2A + B + C - 20 = A + B + C
A = 20.

Coach paid twice as much as Archie.
Thus, C = 2A = 2*20 = 40.

Archie paid seventy dollars less than the difference between what Betty and Coach paid.
Thus:
A = (B-C) - 70.
Substituting A=20 and C=40 into A = (B-C) - 70, we get:
20 = B - 40 - 70
130 = B.

The correct answer is C.

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Mitch and David,
Thank you for your solutions... really appreciate it Very Happy

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vinni.k wrote:
Archie, Betty and Coach bought a radio. Archie paid the least amount of money, Coach paid twice as much as Archie, and Betty paid the most. Archie paid seventy dollars less than the difference between what Betty and Coach paid. If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less. Which of the following equals the amount Betty paid?

A. 40
B. 90
C. 130
D. 140
E. 150
The key to solving this problem is this sentence: “If Archie had paid twice as much as he did pay, Betty and Coach would have each paid ten dollars less.” This means that Betty and Coach together would have paid 20 dollars less, which means that Archie would have paid 20 dollars more than he paid. Since the extra 20 dollars would have resulted in Archie’s paying twice as much, he must have paid 20 dollars originally (so that the doubled amount would be 40 dollars).

Since Archie paid 20 dollars and Coach paid twice as much, Coach paid 40 dollars.

Since Betty paid the most and Archie paid seventy dollars less than the difference between what Betty and Coach paid, if we let x = the amount Betty paid, we have:

20 = x - 40 - 70

130 = x

Answer: C

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