The shop sells the TV at $3999.

The down payment made is 10% of $3999, or $399.90.

The unpaid sum at this moment is $3599.10. This value will attract 4% interest until the sum is paid. The question is very ambiguous, in the sense that they did not mention specifically whether simple or compound interest is charged.

Assume compound interest at a rate of 4% per annum for 2 years (24 months), compounded annually.

Amount in 2 years
= P (1 + r%)^n
= 3599.10 x (1 + 4%)^2
= 3892.78656

This is the sum to be paid in 24 monthly installments.

Value of each installment
= 3892.78656 / 24
= $162.19944
= $162.20

If the interest charged on the $3599.10 balance is a simple interest format, however,

Interest charged on the $3599.10
= PRT%
= 3599.10 (4%) (2)
= $287.928

Total sum paid in installments
= 3599.10 + 287.928
= $3887.028

This is to be divided into 24 monthly installments.

Value of each installment
= 3887.028 / 24
= $161.9595
= $161.96

For compound interest, since its over 24 monthly instalments, the interest is compounded monthly. So you'll have to take 4% ÷ 12 = ⅓%. Then apply the formula, for 24 periods.

And the phrasing in the question usually implies that it's compound interest. It's actually quite common for this syllabus (and other syllabi as well)

Edit :

It appears that the question wants annual compounding, monthly payments, which is less common but still occurs in real life. The majority of loans compound monthly though.

Interesting. Means that if the payment frequency is a per month basis, the compounding is also on a per month basis.

Assume compound interest at a rate of 4% per annum for 2 years (24 months), compounded monthly in line with the frequency of payments.

Amount in 2 years
= P (1 + r%)^n
= 3599.10 x (1 + 4%/12)^(2 x 12)
= 3898.339824

This is the sum to be paid in 24 monthly installments.

Value of each installment
= 3898.339824 / 24
= $162.430826
= $162.43

The answer states that she has to pay $311.92 for each month though

It works the same way with investment plans/bank deposits that pay in periods.

E.g. about 0.005% interest is compounded and paid monthly for most savings accounts.

The bank is effectively the borrower/debtor in this case and pays the account holder monthly, similar to how
the purchaser pays the monthly installment. Exception is that the bank doesn't pay the principal deposited amoun

Your answer key is probably erroneous. $311.92 x 24 = $7486.08, which is way too high.

That's $3886.98 of interest on $3599.10 of principal amount, which is 1.08 times. Crazy

A case of the “answer key gone wrong”.

The answer key appears valid for “12 monthly installments”.

The answer key is based on the compounding annually for only 1 year, then paid over 12 months. This is a case of annual compounding, monthly payments.

total to be paid
= P(1 + r/100)ⁿ

= $3599.10(1 + 4/100)¹
= $3743.064

Monthly installments
= $3743.064 ÷ 12
= $311.922