Unsure about how to do specific questions regarding loans. (Consumer Arithmetic) (1 Viewer)

[randomuser123]

Does anyone know how to do any of these? Please provide working out
Questions were taken from Mathscape 10 extension.
1) In each case, the loan amount is $100000 and the interest rate is 9.25% p.a. monthly reducible. Chris makes one payment of $822 per month, while his sister Rita makes separate payments of $411 each fortnight. Rita finishes paying off her loan within 21 years and 6 months. Chris finishes paying his loan within 30 years. How much interest does Rita save over the term of the loan. ($83844 is the answer at the back of the textbook)
2) Wendy borrows $300000 at 9% p.a. monthly reducible interest over 25 years. At the end of each month, the interest on the outstanding amount is added, then Wendy makes a payment of $2500. How much of the loan will she have paid off after 6 months? ($1528.41 is the answer)

Help is appreciated. Thanks in advance!

 

[snowflakeptical]

I can't remember consumer but hopefully someone who does, sees this

 

[InteGrand]

Here's how to do Question 2. (These questions seem like HSC 2U maths consumer arithmetic Q's)

Let the amount owing after k months be $A_k. So A₀ = 300 000. Let P = 300 000.

The monthly payment is M = 2500 (in dollars). The monthly interest rate is . Hence, the factor by which the amount owing increases from the previous month is . Let R = 1.0075.

Now, A₀ = 300 000.

A₁ = 300,000R – 2500 = PR – M (because the initial amount owing, P, is increased by the factor of R due to interest, and then M is repaid, so PR – M is the amount left owing.

Similarly, A₂ = A₁R – M = (PR – M)R – M = PR² – MR – M.

Maybe you can see the pattern now for A_k in terms of k (if not, continue by using A₃ = A₂R – M and expanding everything. In general, A_k = A_k-1R – M).

The pattern is, A_k = PR^k – M(1 + R + R² + ... + R^k-1).

This is where 2U knowledge can be helpful. That part in brackets is what's called a geometric series; it's a series (a sum of terms), where each term is a fixed factor times the previous term (in this case, that factor is R: each term is R times the previous term).

There's a 2U formula for the sum of a geometric series, and it tells us that .

So, .

After 6 months (so k = 6), amount remaining is .

If you calculate this by plugging in the values for P, R and M, you get some value. Then, the amount paid off is the initial amount owing (300 000) minus this value, i.e. $300,000 – $A₆. This comes out to be $1528.41 to the nearest cent (calculated here: http://www.wolframalpha.com/input/?...0075)^6+-+2500*((1.0075)^6+-+1)/(1.0075+-+1)) ), hence the answer given.

Question 1 will use a similar method.

Note. You didn't need that 2U formula to calculate that geometric series, since the number of terms in the series was just 6, so you could calculate it by typing each term into the calculator (it'd be a little tedious, but doable). However, if the number of months in the question was something big, like 50, you'd need to 2U formula, since it'd be impractical to type each term of the series into a calculator.

 

[randomuser123]

thanks so much for your reply/reasoning, it really helped!