Simple and Compound Interest — SSC CGL Formulas and Traps

Simple Interest

Simple interest formula and the amount

Simple interest and amount

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

$P$ principal, the sum lent
$R$ rate percent per year
$T$ time in years
$A$ amount, principal plus interest

Simple interest for months and days

Time in years

T = \frac{\text{months}}{12} \qquad T = \frac{\text{days}}{365}

Principal from the amount (present worth)

Principal from amount

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

$A$ the amount, or the sum due later
$P$ the principal, or the present worth

Common traps

Giving the amount when the interest is asked

The options usually print both the interest and the amount. Read the last line of the question again: interest is what is earned, the amount is principal plus interest.

Taking a percentage earning on the interest

A percentage earning or return is always a percent of the sum invested. If ₹50,000 earns a net ₹4,000 after a fee, the earning is

\frac{4000}{50000} = 8\%

, a percent of the sum and not of the interest.

Putting months in place of years

SI = \frac{P \times R \times 8}{100}

for 8 months charges 8 years of interest. Divide the months by 12 first: 8 months is

\frac{2}{3}

year.

Counting both the first and the last day

Money lent on 24 January earns from 25 January. Count the days left in the first month without the start date, then the full months, then the days of the last month up to the repayment date.

Taking the percent off the amount

At 19% for the period, the principal is the amount divided by 1.19, not 81% of the amount. 81% of ₹11,900 is ₹9,639, but the principal is ₹10,000.

Finding the Rate, Time or Principal

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Finding the rate or the time from the interest

Rate, time and principal

R = \frac{100 \times SI}{P \times T} \qquad T = \frac{100 \times SI}{P \times R} \qquad P = \frac{100 \times SI}{R \times T}

Two amounts after two different times

Two amounts at two times

\text{Interest per year} = \frac{A_2 - A_1}{t_2 - t_1} \qquad P = \frac{t_2 A_1 - t_1 A_2}{t_2 - t_1}

$A_1, A_2$ the amounts after $t_1$ and $t_2$ years
$t_1, t_2$ the two times in years, $t_1 < t_2$

Equal simple interest on two sums

Equal simple interest

P_1 R_1 T_1 = P_2 R_2 T_2

Rate tied to the number of years

Rate equal to the years

R^2 = \frac{100 \times SI}{P}

Common traps

Using the amount as the interest

When a question says a sum becomes or amounts to a figure, that figure includes the principal. Subtract the principal before you divide, or the rate comes out far too large.

Rounding the rate too early

SSC prints rate options that share a whole number and differ only in the fraction, such as

10\frac{2}{63}\%

,

10\frac{5}{63}\%

,

10\frac{10}{63}\%

and

10\frac{13}{63}\%

. Keep the fraction exact to the end; a rounded decimal cannot pick between them.

Dividing the rise by the later time

The rise between the two amounts covers only the years between them. From ₹2,240 at 3 years to ₹2,400 at 5 years, one year earns

\frac{160}{2} = 80

, not

\frac{160}{5}

.

Turning the ratio of sums upside down

For equal interest the sums are in the inverse ratio of rate times time. If one loan runs at rate times time 12 and the other at 24, the first sum is twice the second, not half.

Answering the time when the rate is asked

When the rate is a multiple of the years, the options print both numbers. If the time is four times the rate, a 3% rate means 12 years, and both 3 and 12 sit among the options: check which one the question wants.

Interest as a Fraction of the Sum

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Simple interest as a fraction of the principal

Interest as a share of the principal

\frac{SI}{P} = \frac{R \times T}{100} \qquad R = \frac{100}{T} \times \frac{SI}{P}

A sum that becomes n times itself at simple interest

Becoming n times at simple interest

R \times T = 100(n - 1) \qquad T_2 = T_1 \times \frac{n_2 - 1}{n_1 - 1}

$n$ how many times its own value the sum becomes
$T_1, T_2$ the times taken to become $n_1$ and $n_2$ times

Simple interest scales with principal, rate and time

Scaling simple interest

\frac{SI_2}{SI_1} = \frac{P_2 R_2 T_2}{P_1 R_1 T_1} \qquad B^2 = AC

$A, B, C$ a sum, its interest, and the interest on that interest, all at one rate and time

Common traps

Reading the amount's fraction as the interest's fraction

A sum that becomes

\frac{8}{7}

of itself has earned

\frac{1}{7}

of itself, not

\frac{8}{7}

. The amount includes the sum; take 1 away before you find the rate.

Scaling the time by n instead of n minus 1

A sum that doubles in 5 years at simple interest does not become 4 times in 10 years. Doubling is one copy of interest; four times is three copies, so it takes 15 years.

Comparing the rates alone

The higher rate does not always earn more. On the same sum, 9% for 2 years (18%) earns less than 4% for 5 years (20%): compare rate times time.

Sums Lent at Two Rates

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One sum split between two rates

Splitting a sum between two rates

\text{Higher part} = \frac{100 I - r_1 P}{r_2 - r_1} \qquad \frac{\text{Lower part}}{\text{Higher part}} = \frac{r_2 - \bar{r}}{\bar{r} - r_1}

$r_1, r_2$ the lower and higher rates percent
$I$ the total interest for one year
$\bar{r}$ the average rate on the whole sum, $\frac{100I}{P}$

Several loans at one rate

SI_{\text{total}} = \frac{R}{100}\left(P_1 T_1 + P_2 T_2 + \dots\right)

Parts linked by a ratio or by equal interest

Equal interest from every part

a : b : c = \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3}

Common traps

Running alligation on a several-year interest

The average rate is the interest for one year as a percent of the whole sum. If ₹3,000 is earned in 3 years on ₹12,500, the average rate is 8%, not 24%.

Reversing the alligation ratio

The part at the lower rate takes the higher rate minus the average; the part at the higher rate takes the average minus the lower rate. Swapping them gives the other part as the answer, and that is always one of the options.

Charging a later loan for the whole year

Money lent 9 months into the year earns for the last 3 months only. Multiply its rate by

\frac{3}{12}

, not by 1.

Splitting in the ratio of the rates

Equal interest needs the parts in the inverse ratio of the rates. At 4% and 5%, the parts are 5 : 4, not 4 : 5: the lower rate needs more money to earn the same.

Rate Changes and Part Repayments

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Extra interest from a change in rate or time

Extra interest from a change

\Delta SI = \frac{P \times \Delta R \times T}{100} \qquad \Delta SI = \frac{P \times R \times \Delta T}{100}

$\Delta R$ the rise in the rate percent
$\Delta T$ the extra years

Different rates in different years

Rates that change over the years

SI = \frac{P}{100}\left(R_1 T_1 + R_2 T_2 + R_3 T_3\right)

Part repayments and interest added to the principal

Final payment after a part repayment

\text{Final payment} = (P - x) + \frac{R}{100}\left(P + (P - x)\right)

$P$ the sum borrowed
$x$ the principal repaid at the end of year 1

Common traps

Applying a rate rise for one year only

A rate 2% higher on a 5-year loan earns 2% more in each of the 5 years: 10% of the sum in all, not 2%.

Averaging the rates without the years

Rates of 4% for 1 year and 10% for 3 years do not average to 7%. The total percent is

4 + 30 = 34\%

over 4 years, an average of 8.5%.

Charging the second year on the original sum

After a part of the principal is repaid, the second year's interest is on what is still owed. Charging it on the full sum overstates the payment by the rate times the amount repaid.

Leaving out the unpaid first-year interest

If the first year's interest was not paid when it fell due, it is still owed at the end. The final payment is the principal left plus both years' interest.

Compound Interest

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Compound interest amount and principal

Compound interest, yearly

A = P\left(1 + \frac{r}{100}\right)^n \qquad CI = A - P \qquad \text{2 years: } \left(2r + \frac{r^2}{100}\right)\%

$r$ rate percent per year
$n$ number of years, compounded yearly

Rate or time from the growth factor

Growth factor

\left(1 + \frac{r}{100}\right)^n = \frac{A}{P}

Interest year by year and consecutive amounts

Year-by-year compound interest

r = \frac{A_{n+1} - A_n}{A_n} \times 100 \qquad I_k = P\left(1 + \frac{r}{100}\right)^{k-1} \times \frac{r}{100}

$A_n$ the amount after $n$ years
$I_k$ the interest earned in year $k$ alone

Compound and simple interest on the same sum

Gap between compound and simple interest

CI - SI = P\left(\frac{r}{100}\right)^2 \ (2 \text{ years}) \qquad CI - SI = P\left(\frac{r}{100}\right)^2\left(3 + \frac{r}{100}\right) \ (3 \text{ years})

Common traps

Doubling the first year's interest

Compound interest for 2 years is more than twice one year's interest, because year 2 also charges interest on year 1's interest. At 10% on ₹5,000 it is ₹1,050, not ₹1,000.

Giving the amount when compound interest is asked

P(1 + r)^n

is the amount. The compound interest is the amount minus the principal, and both figures usually sit among the options.

Dividing the total growth by the years

A 21% rise over 2 years at compound interest is 10% a year, not 10.5%:

1.1^2 = 1.21

. Splitting the total growth evenly is the simple interest method.

Reading constant growth as a constant addition

A constant growth rate means the same percent each year, so the increase itself grows. From 2,500 to 3,000 is 20%, and the next year rises by 600, not 500.

Using the simple interest method on compound amounts

Under compound interest the gap between two consecutive amounts is the rate on the earlier amount, not on the principal. Stepping back by equal gaps, as for simple interest, gives the wrong principal.

Expecting a gap in the first year

Compounded yearly, compound and simple interest are equal for the first year. The gap starts in year 2, which is why the 2-year gap is just the interest on the first year's interest.

Doubling, Part Years and Instalments

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A sum that becomes n times itself at compound interest

Repeated multiples at compound interest

\left(1 + \frac{r}{100}\right)^{t} = n \quad \Rightarrow \quad \left(1 + \frac{r}{100}\right)^{kt} = n^{k}

$n$ the multiple reached in $t$ years
$k$ how many such stretches of $t$ years

Part years and compounding more often than yearly

Part years and shorter periods

A = P\left(1 + \frac{r}{100}\right)^{n}\left(1 + \frac{r m}{1200}\right) \qquad \text{every } k \text{ months: } \frac{rk}{12}\% \text{ per period, } \frac{12n}{k} \text{ periods}

$m$ the months left over after $n$ whole years
$k$ the months in one compounding period

Equal annual instalments

\text{CI: } P = \frac{x}{1 + \frac{r}{100}} + \frac{x}{\left(1 + \frac{r}{100}\right)^2} + \dots \qquad \text{SI: } A = nx + \frac{x r}{100} \cdot \frac{n(n - 1)}{2}

$x$ each equal instalment
$P$ the loan financed today
$A$ the debt due after $n$ years, under simple interest

Common traps

Using the simple interest rule under compound interest

At compound interest a sum that doubles in 7 years becomes 8 times in 21 years, because

8 = 2^3

. The simple interest rule (seven copies of interest, 49 years) does not apply.

Compounding the part year at the full rate

For 1 year 6 months compounded annually, the last 6 months earn half the rate on the amount after 1 year. Charging the full rate for them treats the half year as a whole year.

Halving the rate but not doubling the periods

Half-yearly compounding for 2 years at 10% means 5% for 4 periods. Using 5% for 2 periods, or 10% for 4 periods, is wrong both ways.

Charging every instalment interest for the full term

Under simple interest the last instalment is paid on the due date and earns no interest. An instalment paid at the end of year 1 of a 4-year debt earns 3 years of interest, not 4.

Forgetting the down payment

When an article is sold for a cash down payment plus instalments, the instalments repay the cash price minus the down payment, not the whole cash price.

Simple and Compound Interest formulas

Simple Interest

Simple interest formula and the amount

Simple interest for months and days

Principal from the amount (present worth)

Common traps

Finding the Rate, Time or Principal

Finding the rate or the time from the interest

Two amounts after two different times

Equal simple interest on two sums

Rate tied to the number of years

Common traps

Interest as a Fraction of the Sum

Simple interest as a fraction of the principal

A sum that becomes n times itself at simple interest

Simple interest scales with principal, rate and time

Common traps

Sums Lent at Two Rates

One sum split between two rates

Parts linked by a ratio or by equal interest

Common traps

Rate Changes and Part Repayments

Extra interest from a change in rate or time

Different rates in different years

Part repayments and interest added to the principal

Common traps

Compound Interest

Compound interest amount and principal

Rate or time from the growth factor

Interest year by year and consecutive amounts

Compound and simple interest on the same sum

Common traps

Doubling, Part Years and Instalments

A sum that becomes n times itself at compound interest

Part years and compounding more often than yearly

Common traps

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