Simple interest

When you bor­row money, you pay back more than you bor­rowed. When you keep money in a bank, the bank pays you extra. That extra money is called inter­est. In this les­son you will meet the words used for bor­row­ing and lend­ing, see the dif­fer­ence between sim­ple and com­pound inter­est, and learn the for­mula for sim­ple inter­est. Then you will use it to find the inter­est, the amount, the time, the sum and the rate.

The words#

If a per­son bor­rows money from some­one for a cer­tain time, the bor­rower has to pay back some extra money. This extra money is called inter­est.

• The money bor­rowed is called the prin­ci­pal ($P$). It is also called the sum.
• The prin­ci­pal and the inter­est together are called the amount ($A$).
• The inter­est is usu­ally given as a per­cent­age of the prin­ci­pal for each year. This is the rate of inter­est ($R$). "Per annum" (p.a.) means "for each year".
• The length of time the money is kept is the time ($T$), usu­ally in years.

$$\text{amount} = \text{principal} + \text{interest}$$

For exam­ple, sup­pose the prin­ci­pal is ₹100 and the rate is 10% per annum. After one year the inter­est is $\displaystyle 100 \times \frac{10}{100} = 10$ rupees, and the amount is $100 + 10 = 110$ rupees.

Sim­ple inter­est and com­pound inter­est#

Find the inter­est on ₹12,600 for 2 years at 10% per annum. There are two ways inter­est can be worked out.

First way. Inter­est for the first year $\displaystyle = 12600 \times \frac{10}{100} = 1260$ rupees. In the sec­ond year, inter­est is again worked out on the same ₹12,600, so it is ₹1260 again.

Total inter­est for 2 years $= 1260 + 1260 = 2520$ rupees. This is called sim­ple inter­est (S.I.). The inter­est is always worked out on the first prin­ci­pal.

Sec­ond way. Inter­est for the first year is again ₹1260. But now this inter­est is added to the prin­ci­pal. The new prin­ci­pal for the sec­ond year is $12600 + 1260 = 13860$ rupees.

Inter­est for the sec­ond year $\displaystyle = 13860 \times \frac{10}{100} = 1386$ rupees.

Total inter­est for 2 years $= 1260 + 1386 = 2646$ rupees. This is called com­pound inter­est (C.I.). The inter­est earns inter­est too.

In this les­son we study sim­ple inter­est. Com­pound inter­est comes in the next les­son.

The for­mula for sim­ple inter­est#

$$\displaystyle \text{S.I.} = \frac{P \times R \times T}{100}$$

where $P$ is the prin­ci­pal, $R$ is the rate per cent per annum and $T$ is the time in years. Then

$$A = P + \text{S.I.}$$

Check with the first way above: $\displaystyle \frac{12600 \times 10 \times 2}{100} = 2520$. It agrees.

Exam­ple 1#

Find the sim­ple inter­est on ₹625 at $\displaystyle 6\frac{1}{2}\%$ per annum for $\displaystyle 2\frac{1}{2}$ years.

Write the mixed num­bers as frac­tions: $\displaystyle R = 6\frac{1}{2} = \frac{13}{2}$ and $\displaystyle T = 2\frac{1}{2} = \frac{5}{2}$.

$$\displaystyle \begin{aligned} \text{S.I.} &= \frac{625 \times \frac{5}{2} \times \frac{13}{2}}{100} \\ &= \frac{625 \times 5 \times 13}{2 \times 2 \times 100} \\ &= \frac{1625}{16} = 101.5625 \end{aligned}$$

The sim­ple inter­est is about ₹101.56.

Exam­ple 2: Time in months#

Find the amount on ₹585 at $\displaystyle 4\frac{3}{4}\%$ per annum after 6 months.

$P = 585$, $\displaystyle R = 4\frac{3}{4} = \frac{19}{4}$, and $T = 6$ months $\displaystyle = \frac{6}{12} = \frac{1}{2}$ year.

$$\displaystyle \text{S.I.} = \frac{585 \times \frac{1}{2} \times \frac{19}{4}}{100} = \frac{2223}{160} \approx 13.89$$

Amount $= 585 + 13.89 = 598.89$, so the amount is about ₹598.89.

Exam­ple 3: Time in days#

Find the sim­ple inter­est on ₹700 at 9% per annum from 5th Feb­ru­ary 1984 to 18th April 1984.

When we count days for inter­est, we do not count the day the money is given, but we do count the day it is paid back. 1984 was a leap year, so Feb­ru­ary had 29 days and the year had 366 days.

So $\displaystyle T = \frac{73}{366}$ year.

$$\displaystyle \text{S.I.} = \frac{700 \times 9 \times 73}{100 \times 366} = \frac{1533}{122} \approx 12.57$$

The sim­ple inter­est is about ₹12.57.

Find­ing the time, the sum or the rate#

The for­mula has four let­ters. If you know three of them, you can find the fourth.

Exam­ple 4: Find­ing the time#

The sim­ple inter­est is ₹108 and the amount is ₹540. The rate is 5% per annum. Find the time.

First find the prin­ci­pal: $P = 540 - 108 = 432$ rupees.

$$\displaystyle \begin{aligned} 108 &= \frac{432 \times T \times 5}{100} \\ T &= \frac{108 \times 100}{432 \times 5} = \frac{10800}{2160} = 5 \end{aligned}$$

The time is 5 years.

Exam­ple 5: Find­ing the sum#

A sum of money at sim­ple inter­est of $\displaystyle 3\frac{1}{2}\%$ per annum amounts to ₹364.80 in 4 years. Find the sum.

"Sum" means the prin­ci­pal, and "reck­oned" means worked out.

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

Here $T = 4$ and $\displaystyle R = \frac{7}{2}$, so $\displaystyle \frac{T \times R}{100} = \frac{4 \times 7}{2 \times 100} = \frac{28}{200}$.

$$\displaystyle \begin{aligned} 364.80 &= P\left(1 + \frac{28}{200}\right) = P \times \frac{228}{200} \\ P &= 364.80 \times \frac{200}{228} = \frac{36480}{114} = 320 \end{aligned}$$

The sum is ₹320.

Exam­ple 6: Find­ing the sum and the rate#

A sum of money amounts to ₹678 in 2 years and to ₹736.50 in $\displaystyle 3\frac{1}{2}$ years at sim­ple inter­est. Find the sum and the rate.

Write both facts and sub­tract:

The prin­ci­pal can­cels when we sub­tract. So the inter­est for $\displaystyle 1\frac{1}{2}$ years is ₹58.50.

Sim­ple inter­est grows at the same rate every year, so

$$\displaystyle \text{S.I. for 2 years} = \frac{2}{1\frac{1}{2}} \times 58.50 = 2 \times \frac{2}{3} \times 58.50 = 78$$

From the sec­ond fact, $P + 78 = 678$, so $P = 600$.

Now find the rate:

$$\displaystyle \begin{aligned} 78 &= \frac{600 \times 2 \times R}{100} \\ R &= \frac{78 \times 100}{600 \times 2} = \frac{78}{12} = 6\frac{1}{2} \end{aligned}$$

The sum is ₹600 and the rate is $\displaystyle 6\frac{1}{2}\%$ per annum.

Prac­tice#

1. Find the sim­ple inter­est on ₹4000 at 8% per annum for 3 years.
2. Find the sim­ple inter­est on ₹2500 at 6% per annum for 9 months.
3. Find the amount on ₹1800 at 5% per annum for 4 years.
4. What sum gives ₹1050 as sim­ple inter­est in 5 years at 7% per annum?
5. ₹1600 earns ₹384 as sim­ple inter­est in 3 years. Find the rate.
6. In how many years will ₹2000 amount to ₹2600 at 10% per annum sim­ple inter­est?
7. A sum amounts to ₹1380 in 3 years and to ₹1500 in 5 years at sim­ple inter­est. Find the sum and the rate.
8. A sum of money dou­bles itself in 8 years at sim­ple inter­est. Find the rate.
9. Find the sim­ple inter­est on ₹1460 at 10% per annum from 10th March to 22nd May of a year that is not a leap year.

Answers#