Percentages

Hong Kong

Stage 1 - Stage 3

Lesson

A deferred payment plan is an arrangement that essentially allows customers to "buy now and pay later." It is a bit like lay-by in that you pay and deposit and do not have to pay the balance until sometime in the future. However, the period of time to pay off the balance is normally specified in a contract, so payment must be made by a certain date or multiple repayments can be made until the full amount is paid off. Customers are also usually charged interest on the balance on deferred payment plans. Depending on the specific arrangement, interest might be added to the amount due starting immediately or after a certain amount of time.

Some items are pretty expensive and you may not just have the cash lying around to buy a new flat screen TV! So when a customer can't pay the full amount of a purchase right away but is confident that they will be able to by a certain date (e.g. because they will continue to receive income from their job), a deferred payment plan makes sense for both the buyer and the seller.

A lot of people get excited about the "buy now" and forget about the "pay later," meaning they may not actually be able to afford the item, even with the deferred payment plan. Remember, most of the time, deferred payment plans incur interest, and more specifically, compound interest. If you can't make the necessary payments, the amount owed goes up exponentially, meaning your purchase will become very expensive very quickly! Some companies run credit checks to see whether your record suggests that you will be able to pay off the loan. Other companies let everyone purchase items on deferred payment plans, so just because you get approved for a deferred payment plan doesn't necessarily make it a good idea.

So how do we calculate how much a deferred payment plan will actually cost us? Let's run through the different components of a deferred payment plan. Some terms you may already be familiar with and other may be new.

**Deposit:** an initial, partial payment for the cost of an item.

**Balance:** the amount owing after a deposit has been paid, i.e. $\cos t-deposit=balance$`c``o``s``t`−`d``e``p``o``s``i``t`=`b``a``l``a``n``c``e`.

**Installment:** a fixed amount paid at regular time periods to repay the balance and any interest incurred, i.e. $\frac{\text{balance + interest}}{\text{number of time periods}}=installment$balance + interestnumber of time periods=`i``n``s``t``a``l``l``m``e``n``t`.

**Interest:** an amount charged for loaning you the amount of the balance. Usually calculated as a percentage of the balancing owning using the compound interest formula.

**Fees & charges:** any other costs that are incurred on top of the balance and interest e.g. a service fee.

**Annual:** yearly. e.g. the annual interest rate means the yearly interest rate.

James purchased a car, valued at $\$36653$$36653, on a deferred payment plan. He paid a $\$1714$$1714 deposit, followed by nothing for the first $6$6 months and then $31$31 monthly installments of $\$1266$$1266.

**A)** What was the balance owing after he paid the deposit?

**Think:** The balance is the deposit subtracted from the cost of the car.

**Do:** $36653-1714=\$34939$36653−1714=$34939

**B)** What was the total amount of installments paid?

**Think:** How many installments did James have to pay? How much was each installment?

**Do:** $31\times1266=\$39246$31×1266=$39246

**C)** How much did he pay for the car altogether?

**Think:** He had to pay the deposit plus all the installments.

**Do:** $39246+1714=\$40960$39246+1714=$40960

**D)** How much interest did he pay?

**Think:** The interest is the *additional* amount you have to pay on top of the actual cost of the item.

**Do:** $40960-36653=\$4307$40960−36653=$4307

**E)** What was the interest charged as a percentage of the cost of the car? Write your answer as a percentage to $2$2 decimal places.

**Think:** We want to express the interest as a fraction of the cost, then convert it to a percentage.

**Do:** $\frac{4307}{36653}\times100%=11.750$430736653×100%=11.750...

= $11.75%$11.75% (to 2 d.p.)

**F)** What was the annual rate of interest charged for buying on a deferred payment plan? Write your answer as a percentage to $2$2 decimal places.

**Think:** How can we use the simple interest formula to help us solve this question?

**Do:**

$I$I |
$=$= | $PRT$PRT |
(substitute the values we know in) |

$4307$4307 | $=$= | $36653\times R\times\frac{37}{12}$36653×R×3712 |
(simplify the expression) |

$4307$4307 | $=$= | $R\times113013.4$R×113013.4 ... |
(rearrange the equation) |

$R$R |
$=$= | $4307\div113013.4$4307÷113013.4 ... | |

$R$R |
$=$= | $0.03811$0.03811 ... | (convert to a percentage) |

$R$R |
$=$= | $3.81%$3.81% |

A store offered interest free terms for $20$20 months on all purchases. Valentina purchased a $\$220$$220 treadmill by paying an initial $\$37$$37 deposit followed by $5$5 monthly instalments. If she was also charged an account keeping fee of $\$5$$5 per month, what is:

The size of each instalment?

Write your answer to the nearest cent.

The total amount paid for the treadmill?

Beth purchased a $\$3200$$3200 second hand car on terms, and was given the choice of two payment plans.

Plan 1: $7%$7% deposit with monthly instalments of $\$104$$104 over $3$3 years.

Plan 2: $\$362$$362 deposit with $77$77 weekly instalments of $\$63$$63.

Find the total cost of plan 1.

Find the total cost of plan 2.

Which of the 2 plans is better for Beth?

Plan 1

APlan 2

B