SSC CGL Maths · Simple and Compound Interest

Doubling, Part Years and Instalments

Compound growth repeats a factor every period: a sum that doubles keeps doubling, a part year or a shorter period uses its share of the rate, and equal instalments are matched to the loan by present worth.

All Simple and Compound Interest notes

Why this matters

Eleven questions in the bank, eight of them MODERATE and ten from the 2025 shifts: the hardest page of the chapter. A sum that doubles and must become 8 times, compound interest for whole years plus some months, interest compounded more often than yearly, and the equal instalment on a loan. Each one is a rule about the period, so learn the rule and the arithmetic is short.

Concept 1 of 3: A sum that becomes n times itself at compound interest

Under compound interest the sum is multiplied by the same factor every equal stretch of time. If it doubles in 9 years, the next 9 years double the doubled sum, so it is 4 times after 18 years and 8 times after 27.

Definition

If $\left(1 + \frac{r}{100}\right)^t = n$ , then $\left(1 + \frac{r}{100}\right)^{kt} = n^k$ : the multiple is raised to a power as the time is multiplied.
Doubles in $t$ years: 4 times in $2t$ , 8 times in $3t$ , 16 times in $4t$ .
Rate from a multiple: $1 + \frac{r}{100} = n^{1/t}$ . Becoming 4 times in 2 years means a factor of 2 a year, a rate of 100%.
Compare simple interest, where the time grows with $n - 1$ : doubling in 10 years means 4 times in 30 years at simple interest, but in 20 years 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}

nthe multiple reached in $t$ years
khow many such stretches of $t$ years

Worked example

A sum becomes 3 times itself in 4 years at compound interest. In how many years will it become 9 times, and 27 times, itself?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q51Easy

Example 1 · Simple and Compound Interest · Doubling, Part Years and Instalments

A sum of money doubles itself at compound interest in 15 years. In how many years will it become eight times?

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.

Drill 3 more on a sum that becomes n times itself at compound interest

Concept 2 of 3: Part years and compounding more often than yearly

The rate is per year. A shorter period gets its share of the rate, and a leftover part of a year, when interest is compounded yearly, earns its share once on the amount reached so far.

Definition

Compounded yearly for $n$ years and $m$ months: grow for $n$ whole years, then multiply once by $1 + \frac{r m}{1200}$ . 4 months at 9% is 3%.
Compounded half-yearly: rate $\frac{r}{2}$ per period, $2n$ periods. Quarterly: rate $\frac{r}{4}$ , $4n$ periods.
Every $k$ months: rate $\frac{rk}{12}\%$ per period, $\frac{12n}{k}$ periods. Every 4 months for 1 year is 3 periods, each at a third of the yearly rate.
Halve the rate and double the periods together; changing only one of them is the usual slip.

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}

mthe months left over after $n$ whole years
kthe months in one compounding period

Worked example

Find the compound interest on ₹5,000 at 8% per annum for 2 years 3 months, compounded annually.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 1 · Q59Moderate

Example 2 · Simple and Compound Interest · Doubling, Part Years and Instalments

Find the compound interest on ₹10,000 at

10\%

per annum for 2 years 6 months, compounded annually.

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.

Drill 4 more on part years and compounding more often than yearly

Concept 3 of 3: Equal annual instalments

An instalment paid in a year's time is worth less than its face value today. The loan is fair when today's worth of all the instalments together equals the loan.

Definition

Compound interest: the present worth of an instalment $x$ paid after $k$ years is $\frac{x}{(1 + r/100)^k}$ . Add these for every instalment and set the total equal to the loan.
Take off any cash down payment first: only the rest is financed.
Simple interest, a debt due after $n$ years: an instalment paid at the end of year $k$ earns simple interest for the $n - k$ years left. So the debt $= nx + \frac{xr}{100}\left(1 + 2 + \dots + (n - 1)\right)$ .
Check by running the balance: grow what is owed for a year, subtract the instalment, repeat. The last instalment should clear it exactly.

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}

xeach equal instalment
Pthe loan financed today
Athe debt due after $n$ years, under simple interest

Worked example

A loan of ₹2,100 is repaid in 2 equal annual instalments at 10% per annum compound interest. Find each instalment.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 2 · Q58Moderate

Example 3 · Simple and Compound Interest · Doubling, Part Years and Instalments

A TV set is available for ₹19,650 in cash or for ₹3,100 as a cash down payment and three equal annual instalments. If the shopkeeper charges interest at the rate of

10\%

per annum compounded annually, calculate the amount of each instalment.

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.

Drill 1 more on equal annual instalments

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

A sum that becomes n times itself at compound interest
Repeated multiples at compound interest
$\left(1 + \frac{r}{100}\right)^{t} = n \quad \Rightarrow \quad \left(1 + \frac{r}{100}\right)^{kt} = n^{k}$
Part years and compounding more often than yearly
Part years and shorter periods
$A = P\left(1 + \frac{r}{100}\right)^{n}\left(1 + \frac{r m}{1200}\right) \qquad \text{every } k \text{ months: } \frac{rk}{12}\% \text{ per period, } \frac{12n}{k} \text{ periods}$
Equal annual instalments
Equal annual instalments
$\text{CI: } P = \frac{x}{1 + \frac{r}{100}} + \frac{x}{\left(1 + \frac{r}{100}\right)^2} + \dots \qquad \text{SI: } A = nx + \frac{x r}{100} \cdot \frac{n(n - 1)}{2}$

Watch out for (5)

Using the simple interest rule under compound interest
→ A sum that becomes n times itself at compound interest
Compounding the part year at the full rate
→ Part years and compounding more often than yearly
Halving the rate but not doubling the periods
→ Part years and compounding more often than yearly
Charging every instalment interest for the full term
→ Equal annual instalments
Forgetting the down payment
→ Equal annual instalments

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