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Mathematics

Simple & Compound Interest

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Amari Cross & Matthew Williams
||7 min read
AppreciationConsumer ArithmeticDepreciationInterest

Calculating simple interest, compound interest, appreciation, and depreciation.

Interest questions are about how money changes over time. Simple interest grows by the same amount each period, while compound interest grows on the new total each period.

In CSEC, the hardest part is often choosing the correct model before substituting into a formula. Identify the principal, rate, time, and whether the question involves growth or decrease. Then explain whether you are finding interest only, final amount, appreciation, or depreciation.

Interest is the money a bank PAYS YOU for keeping your money with them (or money YOU PAY the bank for borrowing money).

Understanding Simple Interest

Principal (P): The original amount of money.

Rate (R): The percentage of interest per year (usually written as % per annum, or % p.a.).

Time (T): How long the money is invested or borrowed, in years.

Interest (I): The money earned or paid.

Amount (A): The total after adding interest to principal.

Simple Interest Formula

In simple interest, the interest is calculated only on the ORIGINAL principal, not on accumulated interest.

I=P×R×T100I = \frac{P \times R \times T}{100}

A=P+I=P+P×R×T100A = P + I = P + \frac{P \times R \times T}{100}

Or combined:

A=P(1+R×T100)A = P\left(1 + \frac{R \times T}{100}\right)

Remember

Simple interest = same interest every year, based on the original amount only.

Example

You invest 1000 dollars at 5% per annum for 3 years. How much interest do you earn?

Step 1: Identify values

  • P = 1000 dollars
  • R = 5% per annum
  • T = 3 years

Step 2: Use formula I=P×R×T100=1000×5×3100I = \frac{P \times R \times T}{100} = \frac{1000 \times 5 \times 3}{100}

Step 3: Calculate I=15000100=$150I = \frac{15000}{100} = \$150

Step 4: Find amount A=P+I=1000+150=$1150A = P + I = 1000 + 150 = \$1150

After 3 years, you have 1150 dollars (earned 150 dollars interest).

Each year, you earn exactly 50 dollars interest:

  • Year 1: 1000 + 50 = 1050 dollars
  • Year 2: 1050 + 50 = 1100 dollars
  • Year 3: 1100 + 50 = 1150 dollars

The interest is the SAME every year!

Example

You borrow 2000 dollars at 8% per annum for 2 years. How much total will you owe?

Step 1: Identify values

  • P = 2000 dollars
  • R = 8% per annum
  • T = 2 years

Step 2: Calculate interest I=2000×8×2100=32000100=$320I = \frac{2000 \times 8 \times 2}{100} = \frac{32000}{100} = \$320

Step 3: Calculate total A=2000+320=$2320A = 2000 + 320 = \$2320

You owe 2320 dollars (interest is 320 dollars).

Finding Missing Values in Simple Interest

Sometimes you know the amount, rate, and time — but need to find the principal or rate.

Example

An investment of P at 6% per annum for 4 years earns 240 dollars in interest. Find P.

Step 1: Use the formula I=P×R×T100I = \frac{P \times R \times T}{100}

Step 2: Substitute 240=P×6×4100240 = \frac{P \times 6 \times 4}{100}

Step 3: Simplify 240=24P100240 = \frac{24P}{100}

Step 4: Multiply both sides by 100 24000=24P24000 = 24P

Step 5: Divide by 24 P=2400024=$1000P = \frac{24000}{24} = \$1000

The principal was 1000 dollars.

Check: I=1000×6×4100=24000100=$240I = \frac{1000 \times 6 \times 4}{100} = \frac{24000}{100} = \$240 ✓

Exam Tip

For simple interest problems on CSEC exams:

  1. Always convert time to YEARS (if given in months, divide by 12)
  2. Always use the formula I=P×R×T100I = \frac{P \times R \times T}{100}
  3. Make sure rate is per annum (per year)
  4. Show all steps clearly

Part 4: Compound Interest

Compound interest is when interest is calculated not just on the original principal, but on the interest you've already earned. It's "interest on interest."

Understanding Compound Interest

The key difference: with compound interest, each year you earn interest on a LARGER amount (principal + previous interest).

Compound Interest Formula

A=P(1+R100)TA = P\left(1 + \frac{R}{100}\right)^T

Where:

  • A = Final amount
  • P = Principal
  • R = Rate per annum (%)
  • T = Time in years

Interest earned = A−PA - P

Remember

Compound interest grows faster than simple interest because you earn interest on the interest!

Example

You invest 1000 dollars at 5% per annum for 3 years with compound interest. Find the final amount.

Step 1: Identify values

  • P = 1000 dollars
  • R = 5%
  • T = 3 years

Step 2: Use formula A=P(1+R100)T=1000(1+5100)3A = P\left(1 + \frac{R}{100}\right)^T = 1000\left(1 + \frac{5}{100}\right)^3

Step 3: Simplify the bracket A=1000(1.05)3A = 1000(1.05)^3

Step 4: Calculate (1.05)3(1.05)^3 1.05×1.05=1.10251.05 \times 1.05 = 1.1025 1.1025×1.05=1.1576251.1025 \times 1.05 = 1.157625

Step 5: Multiply by principal A=1000×1.157625=$1157.63A = 1000 \times 1.157625 = \$1157.63

Step 6: Interest earned I=1157.63−1000=$157.63I = 1157.63 - 1000 = \$157.63

After 3 years, you have 1157.63 dollars (earned 157.63 dollars interest).

Compare to simple interest: Only 150 dollars! Compound interest earned 7.63 dollars extra.

Year-by-Year Breakdown

Let's see how compound interest actually works:

YearPrincipal at StartInterest EarnedTotal at End
11000.00 dollars50.00 dollars1050.00 dollars
21050.00 dollars52.50 dollars1102.50 dollars
31102.50 dollars55.13 dollars1157.63 dollars

Notice: Each year, interest is calculated on the PREVIOUS year's total!

Example

Compare simple vs. compound interest:

Investment: 2000 dollars at 8% per annum for 2 years

Simple Interest: I=2000×8×2100=$320I = \frac{2000 \times 8 \times 2}{100} = \$320 A=$2320A = \$2320

Compound Interest: A=2000(1+8100)2=2000(1.08)2A = 2000\left(1 + \frac{8}{100}\right)^2 = 2000(1.08)^2 A=2000×1.1664=$2332.80A = 2000 \times 1.1664 = \$2332.80 I=2332.80−2000=$332.80I = 2332.80 - 2000 = \$332.80

Difference: Compound interest earns 12.80 dollars extra!

Over longer times, this difference grows MUCH larger.

Part 5: Appreciation and Depreciation

Not all money grows through interest. Some assets (things you own) increase or decrease in value over time.

Appreciation

Appreciation: When something INCREASES in value over time.

Example: Property, artwork, classic cars.

Formula (same as compound interest):

V=P(1+R100)TV = P\left(1 + \frac{R}{100}\right)^T

Where V = final value, P = initial value, R = rate of appreciation per annum, T = time in years.

Example

A house is bought for 150,000 dollars. It appreciates at 6% per year. What's it worth after 5 years?

Step 1: Identify values

  • P = 150,000 dollars
  • R = 6%
  • T = 5 years

Step 2: Use formula V=150000(1+6100)5=150000(1.06)5V = 150000\left(1 + \frac{6}{100}\right)^5 = 150000(1.06)^5

Step 3: Calculate (1.06)5(1.06)^5 (1.06)5=1.3382(1.06)^5 = 1.3382

Step 4: Multiply V=150000×1.3382=$200,730V = 150000 \times 1.3382 = \$200,730

The house is worth 200,730 dollars (gained 50,730 dollars value).

Depreciation

Depreciation: When something DECREASES in value over time.

Example: Cars, computers, furniture.

Formula:

V=P(1−R100)TV = P\left(1 - \frac{R}{100}\right)^T

Note the MINUS sign in the bracket!

Example

A car is bought for 30,000 dollars. It depreciates at 15% per year. What's it worth after 3 years?

Step 1: Identify values

  • P = 30,000 dollars
  • R = 15% (depreciation rate)
  • T = 3 years

Step 2: Use formula V=30000(1−15100)3=30000(0.85)3V = 30000\left(1 - \frac{15}{100}\right)^3 = 30000(0.85)^3

Step 3: Calculate (0.85)3(0.85)^3 (0.85)3=0.6141(0.85)^3 = 0.6141

Step 4: Multiply V=30000×0.6141=$18,423V = 30000 \times 0.6141 = \$18,423

The car is worth 18,423 dollars (lost 11,577 dollars value).

Remember
  • Appreciation formula: V=P(1+R100)TV = P(1 + \frac{R}{100})^T
  • Depreciation formula: V=P(1−R100)TV = P(1 - \frac{R}{100})^T

The ONLY difference is + or − in the bracket!