A certain debt will be paid in 52 installments from January 1 to Decem

Total = 410(20) + 475(32)
Number of installments = 52
Average = Total/Number of installments = 450.

Answer: B

Total number of installments = 52
Payment per installment for the first 20 installments = 410

Payment per installment for the remaining 32 installments = 410+65=475

Average = (20*410 + 32*475)/52 =(8200+15200)/52=23400/52 = 450

Answer B

Alternatively , we can proceed as -
65 is difference in value between payment per installment in first 20 installments and the payment per installment for the remaining 32 installments.
At this point we can eliminate option A and E .
(Option A= 443 - If the number of installments were in equal proportion , then average would have been = (410+475)/2=442.5
Option E = 475 - It would be true if all the installments were of 475 )

Number of installments are in the ratio = 20/32 = 5/8
Total number of parts = 5+8 = 13
We need to divide 65 into 13 parts , and the parts will be 25 and 40 . And the distances will be in inverse proportion
Therefore 410 +40 = 450

Answer B
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The most efficient way I found was to split the difference into its own entity.

By that,

\(\frac{(20*410 + 32*475)}{52}\) can be rewritten as \(\frac{(20*410 + (32*410 + 32*65))}{52}\)

This can be simplified much more easily compared to the original expression
\(\frac{(52*410 + 32*65)}{52} = 410 + \frac{32*65}{52}\)
\(= 410 + 40 =450\)

Answer is B

\(\frac{(20*410 +32*475)}{52}\) = 450
correct answer option B

I have tried something as below.

32 * 65 = 2080

\(\frac{2080}{52}\) = 40 [ additional amount paid per month]

average = 410 + 40 = 450/ month

Explanation

Let's say $ 410 is the average amount paid for 52 weeks.

We know that for 20 week installment = $ 410.
Next 32 weeks installment exceeded $ 65.
So additional amount paid over $410 = 32 * 65 = $ 2080

=> \(\frac{2080}{52}\)= 40 [additional amount paid over $ 410]
=> 410 + 40 = 450

Ans: Option B

We are given that 52 total payments have to be made and that the first 20 payments are 410 dollars per payment. Thus, 20 x 410 = 8200 dollars was paid in those 20 payments.

Thus, after those 20 payments, there are 52 - 20 = 32 payments left to be made. If each one is 65 dollars more than each of the first 20 payments, each of the 32 remaining payments will be 410 + 65 = 475 dollars, and the total amount of these 32 payments is:

32 x 475 = 10,450

Thus, the average of the 52 payments is:

average = (8,200 + 15,200)/52 = 23,400/52 = 450 dollars

Answer: B
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there is an easy way to calculate:

410 for all 52 installments. So, average = \(\frac{410*52}{52}\) = 410

For the 32 installments, 65 is more for each. So, average= \(\frac{32*65}{52}\) = 40

Therefore, 410+40 = 450

Answer: B

Just another way to look at this - Alligation

Let the new average be a, then

(475 - a) / (a- 410) = 20 / 32

=> 8 x (475 - a) = 5 x (a- 410)

=> 13a = 5850

=> a = 450

Option B