# Compound Interest - Compounding Variations

Lesson

In Earning Interest on Interest, we were introduced to the compound interest, where new interest was calculated based on the amount of the principal and any previously earned interest. This was done using the formula:

Compound interest formula

$A=P\left(1+r\right)^n$A=P(1+r)n

However, the questions we have looked at already all compounded annually. How would we work out the total amount of an investment that was calculated weekly, monthly or semiannually (twice a year)?

Well, we still use the same formula but we just need to tweak a couple of things based on how the interest is calculated in each individual question. Basically, we need to find equivalent values so everything is expressed in the correct measurement unit. We need to change:

• the number of time periods, $n$n, e.g. if our investment compounded monthly for $3$3 years, our number of time periods would be $3\times12$3×12 or $36$36 months.
• the interest rate, $r$r, e.g. if our investment compounded weekly at a rate of $5.2%$5.2% p.a. for one year, our equivalent interest rate would be $5.2\div52$5.2÷​52 or $0.1%$0.1% per week.

Once everything is expressed in the correct units of measurement, we can use these values and apply the compound interest formula as normal. You'll notice that because the interest is being calculated more frequently, the total amount of the investment will increase faster than if interest is only calculated annually.

Alternatively, if you can easily work out the number of time periods, we can solve these kinds of questions using a slight variation of the compound interest formula:

Variation of the compound interest formula

$A=P\left(1+\frac{r}{k}\right)^{nk}$A=P(1+rk)nk

where:

$P$P is the principal amount

$r$r is the interest rate (this is usually expressed as an annual rate)

$n$n is the total duration of the investment (usually expressed in years)

$k$k is the number of times the interest accrues per time interval

#### Practice questions

##### Question 1

Valentina's investment of $£8000$£8000 earns interest at $2%$2% p.a., compounded semiannually over $2$2 years. Answer the following questions by repeated multiplication.

1. What is the value of the investment after $2$2 years, to the nearest cent?

2. What is the amount of interest earned to the nearest cent?

##### Question 2

A $£70000$£70000 investment earns interest at $4%$4% p.a., compounded semiannually over $20$20 years. Using the compound interest table, calculate:

Total amount from a $£1000$£1000 investment using compound interest.
Periods
Interest rate per period $35$35 $40$40 $45$45 $50$50 $55$55
$1%$1% $1417.83$1417.83 $1490.34$1490.34 $1566.55$1566.55 $1646.67$1646.67 $1730.88$1730.88
$2%$2% $2006.76$2006.76 $2216.72$2216.72 $2448.63$2448.63 $2704.81$2704.81 $2987.80$2987.80
$3%$3% $2835.46$2835.46 $3290.66$3290.66 $3818.95$3818.95 $4432.05$4432.05 $5143.57$5143.57
$4%$4% $3999.56$3999.56 $4875.44$4875.44 $5943.13$5943.13 $7244.65$7244.65 $8831.18$8831.18
$5%$5% $5632.10$5632.10 $7209.57$7209.57 $9228.86$9228.86 $11813.72$11813.72 $15122.56$15122.56

1. the value of this investment, correct to the nearest penny.

2. the amount of interest earned, correct to the nearest penny.

##### Question 3

The price of goods is rising at $0.75%$0.75% per quarter. If $1$1kg of chicken costs $£12.10$£12.10 today, what is the price expected to be in $2$2 years to the nearest penny?