UNIT 10 • INTEREST & FINANCE

Compound Interest
and Simple Interest, Year by Year

Compound interest adds each period's interest to the principal, so the next interest is bigger: $A = P\left(1 + \frac{R}{100}\right)^{T}$ . Simple interest is always on the original amount: $SI = \frac{PRT}{100}$ .

THE FORMULASP, R, T

\begin{gathered} SI = \frac{P \times R \times T}{100} \\ A = P\left(1 + \frac{R}{100n}\right)^{nT}, \quad CI = A - P \end{gathered}

1. Rate per PeriodDivide the yearly rate by n: 2 for half-yearly, 4 for quarterly, 12 for monthly.

2. Number of PeriodsMultiply the years by n. 1½ years half-yearly is 3 periods.

3. Grow, then SubtractWork out the amount A, then CI = A − P.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the compound interest on ₹8,000 for 1½ years at 10% a year, compounded half-yearly. Compare it with the simple interest.

STEP 01 • RATE AND PERIODS

Half-yearly means 2 periods a year:

\text{Rate per half-year} = \frac{10}{2} = 5\%, \qquad \text{periods} = 2 \times 1\tfrac{1}{2} = 3

STEP 02 • AMOUNT

A = 8000\left(1 + \frac{5}{100}\right)^{3} = 8000 \times (1.05)^3 = 8000 \times 1.157625 = 9261

STEP 03 • INTEREST, AND THE SIMPLE INTEREST CHECK

CI = A - P = 9261 - 8000 = 1261

SI = \frac{8000 \times 10 \times 1.5}{100} = 1200

Compound interest is ₹61 more, because each half-year's interest also earns interest.

FINAL CONCLUSION

CI = ₹1,261 (amount ₹9,261)

A = 9261, \qquad CI = 1261

The calculator below opens on this question and shows the amount at the end of each year.

LABORATORY

Interactive Interest Calculator

LABORATORY

Interest Engine

Type of interest

Principal PRate R (% a year)Time TTime is inCompoundedCurrency

Common Questions

What is the difference between simple and compound interest?

Simple interest is always worked out on the original principal. Compound interest is worked out on the principal plus the interest already added, so it grows faster. ₹10,000 at 10% for 2 years earns ₹2,000 simple interest but ₹2,100 compound interest.

How does compounding half-yearly or quarterly change the formula?

Divide the yearly rate by the number of times a year and multiply the time by it: half-yearly uses

\frac{R}{2}

per period and

2T

periods, so

A = P\left(1 + \frac{R}{200}\right)^{2T}

What is the effective annual rate?

The yearly rate that, compounded once a year, gives the same result. 12% compounded monthly is the same as 12.68% compounded yearly, because of interest on interest within the year.

What is the rule of 72?

A quick estimate of how long money takes to double: divide 72 by the yearly interest rate. At 8%, about 72 ÷ 8 = 9 years (the exact answer is 9.006 years).

Related Methods

EMI Calculator Percentage Calculator

PreviousPercentage Calculator

Unit 10•46 of 47

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f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 10 • INTEREST & FINANCE

Compound Interest
and Simple Interest, Year by Year

THE FORMULASP, R, T

\begin{gathered} SI = \frac{P \times R \times T}{100} \\ A = P\left(1 + \frac{R}{100n}\right)^{nT}, \quad CI = A - P \end{gathered}

1. Rate per PeriodDivide the yearly rate by n: 2 for half-yearly, 4 for quarterly, 12 for monthly.

2. Number of PeriodsMultiply the years by n. 1½ years half-yearly is 3 periods.

3. Grow, then SubtractWork out the amount A, then CI = A − P.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the compound interest on ₹8,000 for 1½ years at 10% a year, compounded half-yearly. Compare it with the simple interest.

STEP 01 • RATE AND PERIODS

Half-yearly means 2 periods a year:

\text{Rate per half-year} = \frac{10}{2} = 5\%, \qquad \text{periods} = 2 \times 1\tfrac{1}{2} = 3

STEP 02 • AMOUNT

A = 8000\left(1 + \frac{5}{100}\right)^{3} = 8000 \times (1.05)^3 = 8000 \times 1.157625 = 9261

STEP 03 • INTEREST, AND THE SIMPLE INTEREST CHECK

CI = A - P = 9261 - 8000 = 1261

SI = \frac{8000 \times 10 \times 1.5}{100} = 1200

Compound interest is ₹61 more, because each half-year's interest also earns interest.

FINAL CONCLUSION

CI = ₹1,261 (amount ₹9,261)

A = 9261, \qquad CI = 1261

The calculator below opens on this question and shows the amount at the end of each year.

LABORATORY

Interactive Interest Calculator

LABORATORY

Interest Engine

Type of interest

Principal PRate R (% a year)Time TTime is inCompoundedCurrency

Common Questions

What is the difference between simple and compound interest?

How does compounding half-yearly or quarterly change the formula?

Divide the yearly rate by the number of times a year and multiply the time by it: half-yearly uses

\frac{R}{2}

per period and

2T

periods, so

A = P\left(1 + \frac{R}{200}\right)^{2T}

What is the effective annual rate?

The yearly rate that, compounded once a year, gives the same result. 12% compounded monthly is the same as 12.68% compounded yearly, because of interest on interest within the year.

What is the rule of 72?

A quick estimate of how long money takes to double: divide 72 by the yearly interest rate. At 8%, about 72 ÷ 8 = 9 years (the exact answer is 9.006 years).

Related Methods

EMI Calculator Percentage Calculator

PreviousPercentage Calculator

Unit 10•46 of 47

NextEMI Calculator

Compound Interest and Simple Interest, Year by Year

Step-by-Step Worked Example

CI = ₹1,261 (amount ₹9,261)

Interactive Interest Calculator

Interest Engine

Common Questions

NETZ Engine

Compound Interest and Simple Interest, Year by Year

Step-by-Step Worked Example

CI = ₹1,261 (amount ₹9,261)

Interactive Interest Calculator

Interest Engine

Common Questions

Compound Interest
and Simple Interest, Year by Year

Compound Interest
and Simple Interest, Year by Year