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Stage 5.1-2

# 7.04 Interest rates

Lesson

While interest rates are usually presented and calculated using yearly rates, we can also use different time periods like days, weeks, or months when accumulating interest.

When an interest rate is presented using one time period but is calculated using another, these conversions can help with the simple interest calculations.

### Converting interest rate periods

Converting interest rate periods works the same way that most conversions do.

When converting from one time period to another, we calculate how many of the new time period can fit into one of the old time periods. We can then divide the interest percentage by this number to fully convert the interest rate.

#### Exploration

Suppose that an investment has a simple interest rate of $6%$6%p.a., accumulating monthly. What is the monthly interest rate?

We are given that the interest rate per annum is $6%$6%. This is a yearly interest rate and we want to find the monthly one.

Since there are $12$12 months in a year, we can divide the yearly interest rate of $6%$6% by $12$12 to find the monthly interest rate. This conversion can presented as the working out:

Monthly interest rate $=$= $6%$6% per $12$12 months

Which can be simplified to give us:

Monthly interest rate $=$= $0.5%$0.5% per month

As expected, the monthly interest rate was $\frac{1}{12}$112 of the yearly interest rate, since a month is equal to $\frac{1}{12}$112 of a year.

We can use this same conversion method to convert between any two interest rate periods.

For our conversions, we can assume that there will be:

• $12$12 months in a year
• $52$52 weeks in a year
• $365$365 days in a year
• $7$7 days in a week
• $2$2 weeks in a fortnight

Other time periods like half-years and quarters carry their conversion factors in their names, being $\frac{1}{2}$12 and $\frac{1}{4}$14 of a year respectively.

Caution

Notice that there are no direct conversions between weeks and months or days and months. For these conversions, we need to convert from one period into years, then from years into the other period.

For example: since a week is $\frac{1}{52}$152 of a year and a year is $12$12 months, we can combine these two conversions to find that a week is $\frac{12}{52}$1252 of a month.

### Appreciation and depreciation

Appreciation occurs when an item increases in value by some percentage. This is equivalent to accruing interest on an investment, where the value of the item is equivalent to the investment total.

Depreciation occurs when an item decreases in value by some percentage. This can be calculated the same way as appreciation is calculated, except that the percentage amount is subtracted from the original value.

#### Practice questions

##### Question 1

Peter takes out a loan which accrues interest at a flat rate of $0.18%$0.18% per week.

What is the equivalent yearly simple interest rate? Assume that there are $52$52 weeks in a year.

Calculate the simple interest earned on an investment of $\$6050$$6050 at 0.7%0.7% per quarter for 33 years. Give your answer to the nearest cent. ##### Question 3 A camera valued at \400$$400 depreciates at a rate of $\$3232 per year.

How much will the camera be worth after:

1. One year?

2. Two years?

3. Ten years?

### Outcomes

#### MA5.1-4NA

solves financial problems involving earning, spending and investing money