Hire purchase maths problem!?

So this is my maths problem, and I have the answer to the question and I'm not sure where I'm going wrong.





"A bicycle has a marked price of $300. It can be bought through hire-purchase with a deposit of $60, and 10% interest on the outstanding balance, to be repaid in 10 monthly installments".





Calculate: The amount of monthly installment ; and the total cost of buying the bike by hire purchase





Ok, so the total cost is $324. This is the correct answer since the amount of monthly installment is 26.40 and you times that by 10 months, along with 60 on top of all that. So, 60 + (10X26.40) = 324 (which is the correct answer).





My question is, how do you find the monthly installment amount. 10% of the "outstanding balance"...isn't that 10 / 100 * 240 (since 300-60 deposit)...that gives me $24 bucks which as you can see, is wrong.





Help is much appreciated. Thank you.|||P = A*[1 - 1/(1 + i)^n]/i





240 = A*[1 - 1/(1 + 0.1/12)^10]/(0.1/12)


A = $25.11





Using Excel:


=PMT(0.1/12,10,-240) = $25.11








60 + 10*25.11 = 311.10|||so, it goes like this:





10% out of the remaining balance is:


240+240*10%


which means:


240+24=264


so, the monthly payment is 26.4.


:)|||$300 - $60 = $240 has to be paid in 10 months.





Let X denote the monthly payment.





1st month outstanding balance


$240 + 10% of interest - X





2nd month outstanding balance


($240 + 10% of interest of $240 - X) + 10% of interest - X





... (3rd ~ 9th month outstanding balances omitted)





10th month outstanding balance


X - X = 0





Under these conditons, I got X = $37.97 (rounded) so you have to pay $379.70 total.





1) Using Differential equation (Recommended)


P: outstanding balance


X: monthly payment


t: time in month {1, 2, 3, ... ,9, 10}





dP/dt = -X + 0.1P


1/(-X + 0.1P) dP = dt





If we integrate both sides,


P = Ce^(0.1t) + 10X





Using the conditions P(0) = 240 and P(10) = 0, C and X can be calculated.


C = 240/(1-e) = -139.67 (rounded)


X = (240 - C)/10 = 37.97 (rounded)


So X = $37.97 and since we are payin 10X dollars, the total amount to be paid is $379.7.





2) Use sequences.


P_n = The outstanding balance in n_th month





P_0 = $240


P_1 = P_0 + P_0 * 0.1 - X = 1.1 P_0 - X


P_2 = P_1 + P_1 * 0.1 - X = 1.1 P_1 -X





so, P_n = 1.1 P_(n-1) - X





P_10 = 1.1 P_9 - X = 0 -%26gt; P_9 = X / 1.1


P_9 = 1.1 P_8 - X = X / 1.1 -%26gt; P_8 = 2.1 X / (1.1)^2