Compound Interest

When you borrow money, you pay the lender for the use of it; when you save in a bank, the bank pays you. That payment is interest. The words from Grade 8:

Word	Meaning	Letter
Principal	the money borrowed or deposited	P
Rate	the interest on Rs 100 for one year, as a percentage (“per annum”, p.a.)	R
Time	how long, in years	T
Simple interest	interest worked out on the principal alone, the same every year	SI = P × T × R/100
Amount	principal + interest, what is paid back at the end	A = P + SI

2.1.1 Compound interest compounded annually

a. First year: $I_1 = \frac{500000 \times 1 \times 12}{100} = 60000$, so Rs 60,000.

b. The unpaid Rs 60,000 joins the loan: the second year’s principal is Rs 5,00,000 + Rs 60,000 = Rs 5,60,000.

c. Second year: $I_2 = \frac{560000 \times 1 \times 12}{100} = 67200$, so Rs 67,200.

In all he pays Rs 60,000 + Rs 67,200 = Rs 1,27,200 in interest, not the Rs 1,20,000 of simple interest. Why is the second year’s interest bigger? The extra Rs 7,200 is exactly 12% of Rs 60,000: interest on the first year’s interest.

When the interest at the end of each period (a year, a half-year or a quarter) is added to the principal, and the next period’s interest is worked out on that bigger principal, the total interest is called compound interest (CI). The principal and compound interest together are the compound amount (CA). Each period at whose end interest is added is a compounding period: you say the interest is compounded yearly, half-yearly or quarterly.

In rupees:

Year	Bishal’s interest	Badri’s principal	Badri’s interest
1	3,000	30,000	3,000
2	3,000	33,000	3,300
3	3,000	36,300	3,630
Total	9,000		9,930

Bishal’s principal stays Rs 30,000, so his interest is 30000 × 10 ÷ 100 = 3000 every year. Badri’s interest is 10% of each year’s principal: his second-year principal is Rs 30,000 + Rs 3,000 = Rs 33,000, and his third-year principal Rs 33,000 + Rs 3,300 = Rs 36,300.

Badri pays Rs 930 more at the same rate. With simple interest the principal stays Rs 30,000 all three years. With compound interest each year’s principal is the last year’s amount: principal plus that year’s interest. So the interest is never smaller than the year before, and after the first year it is always bigger.

Fill in Badri’s ladder yourself, one number at a time. This is the method “without using the formula” that exam questions ask for.

The formula, and why it works

Writing the ladder out year by year works, but for 10 years, or 40 quarters, you need a shortcut. Watch what happens to a principal P at R% a year (Activity 3).

At the end of year 1 the interest is PR/100, so the amount is

$$P + \frac{PR}{100} = P\left(1 + \frac{R}{100}\right)$$

Adding R% is the same as multiplying by (1 + R/100). At 10%, adding 10% is multiplying by 1.10; at 4.55% it is multiplying by 1.0455.

That amount is the principal for year 2, and year 2 adds R% again: one more multiplication by the same number.

$$\begin{aligned} \text{Amount after 1 year} &= P\left(1 + \frac{R}{100}\right) \\ \text{Amount after 2 years} &= P\left(1 + \frac{R}{100}\right)\left(1 + \frac{R}{100}\right) = P\left(1 + \frac{R}{100}\right)^2 \\ \text{Amount after 3 years} &= P\left(1 + \frac{R}{100}\right)^3 \end{aligned}$$

Every year is one more factor of (1 + R/100). So after T years:

$$\text{CA} = P\left(1 + \frac{R}{100}\right)^T \qquad \text{CI} = \text{CA} - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]$$

Check it on Badri: $30000 \times 1.1^3 = 30000 \times 1.331 = 39930$, and 39930 − 30000 = 9930. ✓

This is the difference in one line. Simple interest adds the same amount every year; compound interest multiplies by the same number every year. Adding gives a straight line; multiplying gives a curve that bends upwards, slowly at first and then faster and faster.

Real deposits and loans don’t always run for whole years at one rate. Activity 4 asks how the formula changes when:

• a.the rate is different every year,
• b.the time is T years and M months,
• c.interest is added every six months (half-yearly),
• d.interest is added every three months (quarterly).

The answer to all four is the same idea: one factor of (1 + rate for that period/100) for every period.

a. A different rate every year

If the rate is R₁% in the first year, R₂% in the second and R₃% in the third, each year still multiplies by its own factor:

$$\text{CA} = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right) \qquad \text{CI} = \text{CA} - P$$

Since multiplication can be done in any order, 8% then 12% gives the same amount as 12% then 8%.

b. T years and M months

For the whole years, the yearly factor T times. The extra M months are less than a year, so the interest on them is not compounded: it is simple interest for M/12 of a year, at the rate R%, on the amount after T years. Simple interest for M/12 of a year is M/12 × R% = MR/1200%, so

$$\text{CA} = P\left(1 + \frac{R}{100}\right)^T\left(1 + \frac{MR}{1200}\right) \qquad \text{CI} = \text{CA} - P$$

c. Half-yearly compound interest

Some banks add the interest twice a year, for example on the first of Magh and the first of Shrawan. Then each half-year earns half the yearly rate, and the next half-year’s interest is on the principal plus the last half-year’s interest. This is half-yearly (or semi-annual) compound interest.

A year has two half-years, so in T years there are 2T periods, each at R/2%:

$$\text{CA} = P\left(1 + \frac{R}{2 \times 100}\right)^{2T} = P\left(1 + \frac{R}{200}\right)^{2T} \qquad \text{CI} = P\left[\left(1 + \frac{R}{200}\right)^{2T} - 1\right]$$

How is this different from yearly? The interest of the first half-year starts earning interest six months sooner. Yearly compounding waits until the end of the year to do that.

d. Quarterly compound interest

Added every three months (Kartik 1, Magh 1, Baisakh 1 and Shrawan 1), the rate per quarter is R/4%, and T years have 4T quarters:

$$\text{CA} = P\left(1 + \frac{R}{400}\right)^{4T} \qquad \text{CI} = P\left[\left(1 + \frac{R}{400}\right)^{4T} - 1\right]$$

Quarterly compound interest is worked out exactly as yearly and half-yearly: only the rate per period and the number of periods change. Some books and exam papers call it terminal compound interest (interest added “terminally”, every three-month term): it means the same, quarterly.

With P = 100000, R = 15 and T = 1:

• Yearly: $\text{CI} = 100000\left[\left(1 + \frac{15}{100}\right)^1 - 1\right] = 15000$: Rs 15,000.
• Half-yearly: $\text{CI} = 100000\left[\left(1 + \frac{15}{200}\right)^2 - 1\right] = 15562.50$: Rs 15,562.50.
• Quarterly: $\text{CI} = 100000\left[\left(1 + \frac{15}{400}\right)^4 - 1\right] \approx 15865.04$: Rs 15,865.04.

Choose quarterly. It pays Rs 865.04 more than yearly and Rs 302.54 more than half-yearly, at the same 15%, because its interest starts earning interest soonest. For a loan it is the other way round: the more often interest is added, the more the borrower pays.

Back to the remittance

Rs 1,00,000 at 4.55% a year for 10 years becomes about Rs 1,56,000 with interest added yearly, and about Rs 1,57,200 with it added every quarter: some Rs 10,500 to Rs 11,700 more than simple interest’s Rs 1,45,500.

$$\begin{aligned} \text{Yearly: } 100000 \times 1.0455^{10} &\approx 156042 \\ \text{Quarterly: } 100000 \times \left(1 + \frac{4.55}{400}\right)^{40} &\approx 157213 \end{aligned}$$

Two things make the real figure smaller, and one makes the waiting longer:

• Tax. The bank keeps back a tax on the interest each time it is added (6% of it for a person’s deposit, at the 2024/25 rates). So the deposit really grows at $4.55 \times 0.94 \approx 4.28\%$ a year: about Rs 1,53,000 after 10 years with quarterly interest.
• The rate changes. Bank rates in Nepal are revised every month, and a fixed deposit’s rate holds only for its term. The 4.55% of July 2026 is not a promise for ten years.
• Prices rise too. Rs 1,53,000 in ten years buys less than it does today. That is Unit 3’s kind of question: a price that grows by a percentage every year grows by the same formula.

And the loan from the opening: at 12% a year, $1.12^6 \approx 1.97$, so in 6 years an unpaid loan has almost doubled. Interest added monthly makes it a little worse.

In each one, read the question, try it yourself, then open the solution one step at a time. Write down P, R and T first, choose the formula, then put the numbers in.

Year by year, and by the formula

Simple against compound

Finding the rate

A different rate each year

Two amounts, two unknowns

Another way to see part a: the third year’s interest is 15972 − 14520 = 1452, earned on Rs 14,520, and $\frac{1452}{14520} \times 100\% = 10\%$.

Interest after tax, and a change of plan

Splitting a sum

A quicker start: the two amounts are equal, and Jibanlal’s has one more year, so Babulal’s loan is exactly 1.1 times Jibanlal’s: 1.1y + y = 52500, y = 25000.

Finding the time

There is one kind of question the formula can’t be turned round for with Grade 10 algebra: how many years? With small numbers, multiply by the factor year after year until you reach the amount.

In the SEE’s Compulsory Mathematics paper, Arithmetic carries 13 marks in three questions (compound interest; growth and depreciation; currency and exchange), with knowledge, understanding, application and higher-ability parts spread among them (CDC’s specification grid). In CDC’s model paper the compound-interest question looks like this:

Part	Level	Marks	What it asks
a.	knowledge	1	which option gives more interest, or a formula
b.	understanding	2	an amount or interest with the formula
c.	application	2	a twist: the plan changes halfway, a tax, a comparison

A paper may add a higher-ability part of 1 mark: a “why” or a “check this claim”. Here is a question in that shape, and one with a higher-ability part.

Question 1

Define:

Question 2

Question 3

Without using the formula, find the compound interest and compound amount:

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Questions to discuss