Year 11 General Consumer Arithmetic: Loans And Investments

Rearranging The Simple Interest Formula

Q: How do you rearrange the simple interest formula to find the principal?

Divide the interest by the rate times the time: P equals I divided by r times n.

Q: How do you find the interest rate from a simple interest problem?

Divide the interest by the principal times the time: r equals I divided by P times n. Multiply by 100 to get the percentage.

Q: How do you find the time period for a simple interest problem?

Divide the interest by the principal times the rate: n equals I divided by P times r. The answer is in years.

Q: What if the question gives the final amount instead of the interest?

Find the interest first by subtracting principal from the total: I equals A minus P. Then rearrange as normal.

Q: How do you find the rate from monthly repayments on a flat-rate loan?

Total repaid equals monthly payment times the number of months. Interest equals total repaid minus the principal. Then use r equals I divided by P times n.

Q: How do you express a rate that comes out as a decimal as a percentage?

Multiply by 100. A computed rate of 0.045 means 4.5 percent per year.

9 practice questions 2 video lessons Theory + worked examples

Theory

The simple interest formula $I = P r n$ has four variables. Knowing any three lets you solve for the fourth by rearranging. In every case, divide $I$ by the product of the other two. Watch out for problems that give the final amount $A$ or monthly repayments instead of $I$ directly — find $I$ first, then rearrange.

The simple interest formula $I = P \cdot r \cdot n$ involves four variables: interest $I$ , principal $P$ , rate $r$ , and time $n$ . Knowing any three lets you find the fourth by rearranging.

In all three rearrangements, the unknown sits on the left and the rest of the formula moves to the right. The interest $I$ is always divided by the product of the other two. $P$ , $r$ and $n$ are symmetric in this respect.

If the question gives the final amount $A$ instead of the interest, find $I$ first using $I = A - P$ . If it gives the equal monthly repayment on a flat-rate loan, total repaid is $12 \cdot n \cdot M$ , so $I = 12 \cdot n \cdot M - P$ .

The computed rate is a decimal. To convert to a percentage, multiply by 100. A computed value of r=0.045 means 4.5% per year.

Same pattern every time

Always derive I before solving for P, r, or n

The base formula:

I = P \cdot r \cdot n

Solve for principal:

P = \frac{I}{r \cdot n}

P = \frac{I}{r n}

Solve for rate:

r = \frac{I}{P \cdot n}

r = \frac{I}{P n}

Solve for time:

n = \frac{I}{P \cdot r}

n = \frac{I}{P r}

If the final amount $A$ is given instead of $I$ :

I = A - P

If the equal monthly repayment $M$ on a flat-rate loan over $n$ years is given:

I = 12 \cdot n \cdot M - P

I = 12 n M - P

The computed rate is a decimal. Multiply by 100 for a percentage: r=0.05→5% p.a.

How to solve any rearrangement problem

Identify the unknown. Write down what $P$ , $r$ , $n$ , $I$ , or $A$ you have.
Find $I$ if missing. From $A$ : $I = A - P$ . From monthly repayments: $I = 12 n M - P$ .
Pick the rearrangement matching the unknown ( $P = I / r n$ , $r = I / P n$ , or $n = I / P r$ ).
Substitute and compute. Convert the rate to a decimal and time to years first.
Convert the answer back to the units the question wants — percentage for rate, months for time.

Example 1 — Find the principal

What principal is needed to earn

$ 1,080

interest at

4.5 %

p.a. for

3

years?

Solution

Use $P = I / (r \cdot n)$ with $r = 0.045$ , $n = 3$ .

$P$	$=$	$\frac{1,080}{0.045 \times 3}$
$P$	$=$	$\frac{1,080}{0.135}$
$P$	$=$	$$ 8,000$

P = 8000

A principal of $$8{,}000$ is needed.

Example 2 — Find the rate

$ 5,000

invested for

4

years grew to

$ 6,000

. Find the simple interest rate.

Solution

Find $I$ first, then use $r = I / (P \cdot n)$ .

$I$	$=$	$6,000 - 5,000 = 1,000$
$r$	$=$	$\frac{1,000}{5,000 \times 4}$
$r$	$=$	$0.05 = 5 %$ p.a.

r = 0.05

The rate is $5\%$ p.a.

Example 3 — Find the time

For how many months must

$ 2,500

be invested at

8 %

p.a. to earn

$ 200

interest?

Solution

Use $n = I / (P \cdot r)$ , then convert years to months.

$n$	$=$	$\frac{200}{2,500 \times 0.08}$
$n$	$=$	$\frac{200}{200} = 1$ year
$months$	$=$	$1 \times 12 = 12$

n = 12 months

The investment runs for $12 months$ .

Example 4 — Rate from monthly repayments

Liam borrows

$ 4,800

and repays it over

2

years in equal monthly instalments of

$ 220

. Find the flat rate.

Solution

Total repaid, then interest, then rate.

$total$	$=$	$24 \times 220 = 5,280$
$I$	$=$	$5,280 - 4,800 = 480$
$r$	$=$	$\frac{480}{4,800 \times 2}$
$r$	$=$	$0.05 = 5 %$ p.a.

r = 0.05

The flat rate is $5\%$ p.a.

Common pitfalls

Forgetting to convert $r$ back to a percentage. A computed

r = 0.045

is not "

0.045 %

" — multiply by

100

to get

4.5 %

p.a.

Using $A$ where $I$ is needed. If the question gives the final balance

A

(or "grew to $X"), find

I = A - P

first. Substituting

A

for

I

inflates every answer.

Time stays in years. If the question asks for months but the formula gives years, multiply by

12

at the end. Don't convert inside the formula.

Mixing per-year and per-month rates. Always keep

r

as the annual rate (per annum) and

n

in years. Don't convert the rate to monthly.

Frequently asked questions

How do you rearrange the simple interest formula to find the principal?

$P = \frac{I}{r \cdot n}$ . Divide the interest by the product of the rate (as a decimal) and the time (in years).

How do you find the interest rate from a simple interest problem?

$r = \frac{I}{P \cdot n}$ . The answer is a decimal; multiply by $100$ for a percentage.

How do you find the time period for a simple interest problem?

$n = \frac{I}{P \cdot r}$ . Time is in years. To convert to months, multiply by $12$ .

What if the question gives the final amount instead of the interest?

Find the interest first: $I = A - P$ . Then rearrange as normal.

How do you find the rate from monthly repayments on a flat-rate loan?

Total repaid $= 12 \cdot n \cdot M$ . Interest $= 12 n M - P$ . Then $r = I / (P \cdot n)$ .

How do you express a rate that comes out as a decimal as a percentage?

Multiply by $100$ . $r = 0.045 \to 4.5 %$ p.a.

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Rearranging The Simple Interest Formula

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Theory

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