Compound Interest: Methods & Shortcuts — SSC CGL Maths: Methods & Shortcuts Notes (English Medium)

In simple interest (SI) the interest each year is charged only on the original sum (the principal, $P$ ). In compound interest (CI) the interest is added to the principal at the end of each year, and next year’s interest is charged on the bigger sum. So the interest itself earns interest. The total you get at the end is the amount ( $A$ ): $A = P + C I$ .

∑ Formula

A = P (1 + \frac{R}{100})^{n}, C I = A - P

The basic formula for annual compounding

$P$: principal
$R$: rate per year (%)
$n$: number of years
$A$: amount

The formula is correct, but multiplying big powers wastes time in the exam. So in this chapter we learn methods: the ratio method (write the rate as a fraction), the interest tree (shows each year’s interest separately), and the short formulas that come from them. After each method there are solved examples and then practice questions — solve them yourself first, then open the solution.

RememberOne key opens the whole chapter: every year the money is multiplied by the same factor $(1 + \frac{R}{100})$ . Multiply, do not add.

1

Foundations: multiply, don’t add

Successive % · ratio · effective %

Seeing CI as successive percentage change, the ratio method and the effective percent.

The whole chapter stands on these three methods. Learn them well first.

Successive percentage change

x + y + \frac{x y}{100}

One percentage change after another — the root of compound interest.

The idea

Suppose a price first rises 20% and then another 10%. The total rise is not 30%, because the second rise acts on the new price, not the old one. A price of 100 first becomes 120; now 10% means 10% of 120 = 12, giving 132. The total rise is 32%.

Where did the extra 2% come from? The second rise (10%) also acted on the first rise (20): 10% of 20 = 2. Compound interest is exactly this — each year’s growth also acts on the growth of earlier years.

∑ Formula

Net % = x + y + \frac{x y}{100}

Two successive changes (put a minus sign on a fall or a discount)

$x, y$: the two percent changes
$\frac{x y}{100}$: the effect of one change on the other

The fraction way: 10% $= \frac{1}{10}$ , so a rise is 10 → 11 and a fall is 10 → 9. Write each change as a ratio and multiply them all; (last − first) ÷ first × 100 = the overall percent.

Percent	Fraction	Rise ratio	Fall ratio
5%	$\frac{1}{20}$	20 : 21	20 : 19
10%	$\frac{1}{10}$	10 : 11	10 : 9
$12 \frac{1}{2} %$	$\frac{1}{8}$	8 : 9	8 : 7
$16 \frac{2}{3} %$	$\frac{1}{6}$	6 : 7	6 : 5
20%	$\frac{1}{5}$	5 : 6	5 : 4
25%	$\frac{1}{4}$	4 : 5	4 : 3
$33 \frac{1}{3} %$	$\frac{1}{3}$	3 : 4	3 : 2
15%	$\frac{3}{20}$	20 : 23	20 : 17

Spot itUse this method when the question says “first… then…”, “successive discounts”, “two years in a row”.

Steps

Write each change with its sign: $+$ for a rise, $-$ for a fall or discount.
Combine the first two with $x + y + \frac{x y}{100}$ .
If there is a third change, combine it with this result by the same formula.
A positive answer is an overall rise, a negative one an overall fall. Check by starting from 100.

Example 1 · Basic

The price of a cooler rises by 20% and then by another 10%. What is the total percentage rise?

Solution, step by step

1Percent questions are easiest on 100, so take the old price as 100. After the 20% rise the price is $100 + 20 = 120$ .
2Now the 10% rise is on the new price 120, not on 100: 10% of 120 = 12, so the price becomes $120 + 12 = 132$ .
3Overall 100 → 132, i.e. a rise of 32%.
4The formula gives the same: $20 + 10 + \frac{20 \times 10}{100} = 30 + 2 = 32%$ . The extra 2 is the 10% rise acting on the first rise of 20.

Answer 32% rise

Not 30%! Successive percents never simply add.

Example 2 · Standard

A shop gives three successive discounts of 25%, 20% and 10% on the marked price. What single discount equals them?

Solution, step by step

1A discount lowers the price, so write all three with a minus sign: $- 25, - 20, - 10$ .
2Combine the first two: $- 25 - 20 + \frac{( - 25 ) ( - 20 )}{100} = - 45 + 5 = - 40$ . So 25% and 20% together equal a 40% discount (not 45%).
3Now combine $- 40$ with the third discount $- 10$ : $- 40 - 10 + \frac{( - 40 ) ( - 10 )}{100} = - 50 + 4 = - 46$ .
4Check from 100: 100 → 75 → 60 → 54. The buyer pays 54 instead of 100, i.e. a 46% discount.

Answer 46%

With three changes, combine two at a time; the order does not change the answer.

Example 3 · Standard

An employee’s salary rises first by 25% and then by 20%. Find the total rise by the fraction method.

Solution, step by step

125% $= \frac{1}{4}$ : every 4 rupees become 5, ratio $4 : 5$ .
220% $= \frac{1}{5}$ : every 5 rupees become 6, ratio $5 : 6$ .
3The changes come one after the other, so multiply the ratios: before $4 \times 5 = 20$ , after $5 \times 6 = 30$ .
4A rise of 10 on 20 ⟹ $\frac{10}{20} \times 100 = 50%$ . By the formula: $25 + 20 + \frac{25 \times 20}{100} = 50%$ ✓.

Answer 50% rise

The fraction method is exactly what becomes the ratio method for CI.

Example 4 · Advanced

A sum grows by 25% every year. By what percent does it grow in 3 years? Use this to find the compound interest on ₹6,400 at 25% per annum for 3 years.

Solution, step by step

1A 25% rise each year = three successive rises $+ 25, + 25, + 25$ .
2Fraction method: 25% $= \frac{1}{4}$ ⟹ $4 : 5$ each year. In three years $4^{3} : 5^{3} = 64 : 125$ .
3A rise of 61 on 64 ⟹ $\frac{61}{64} \times 100 = 95.3125%$ . (By the formula: $25 + 25 + 6.25 = 56.25$ , then $56.25 + 25 + 14.0625 = 95.3125$ .)
4₹6,400 = 64 units, so 1 unit = ₹100; interest = 61 units = ₹6,100.

Answer 95.3125%; interest ₹6,100

This thinking is the base of compound interest: each year is one more successive rise.

BasicTry it yourself first

The price of a TV changes by

+ 10%

,

+ 15%

, one after another. What is the overall percentage change?

BasicTry it yourself first

A shopkeeper gives successive discounts of

30%

,

20%

on the marked price. What single discount is equal to all of them?

BasicTry it yourself first

Radha’s salary changes by

+ 20%

,

- 10%

, one after another. What is the overall percentage change?

StandardTry it yourself first

The number of visitors to a museum changes by

- 20%

,

- 25%

, one after another. What is the overall percentage change?

StandardTry it yourself first

The rent of a shop changes by

+ 30%

,

- 30%

, one after another. What is the overall percentage change?

StandardTry it yourself first

A shopkeeper gives successive discounts of

10%

,

20%

,

30%

on the marked price. What single discount is equal to all of them?

AdvancedTry it yourself first

The output of a factory changes by

+ 25%

,

+ 20%

,

- 20%

, one after another. What is the overall percentage change?

AdvancedTry it yourself first

The number of trees in a village changes by

+ 10%

,

+ 10%

,

+ 10%

, one after another. What is the overall percentage change?

The ratio (unit) method

Rate = fraction ⟹

P : A = b^{n} : (b + a)^{n}

Writing the rate as a fraction and seeing principal, amount and interest in units.

The idea

Suppose the rate is 10%, i.e. $\frac{1}{10}$ . Then every 10 rupees become 11 rupees in a year: for one year principal : amount = 10 : 11. In year two the same multiplication happens again, so in 2 years it is $1 0^{2} : 1 1^{2} = 100 : 121$ and in 3 years $1000 : 1331$ . Interest = the difference: 21 units in 2 years, 331 units in 3 years.

A “unit” here is an unknown amount of money. If the question gives any one of principal, amount or interest, that gives the value of 1 unit, and then everything else. This is just the formula $A = P (1 + \frac{R}{100})^{n}$ — written in fractions, so the work can be done in your head.

∑ Formula

R % = \frac{a}{b} \Rightarrow P : A = b^{n} : (b + a)^{n}, C I = (b + a)^{n} - b^{n} units

The ratio for

n

years

$\frac{a}{b}$: the rate as a fraction in lowest terms

Rate	Fraction	1 year P : A	2 years P : A	3 years P : A
5%	$\frac{1}{20}$	20 : 21	400 : 441	8000 : 9261
10%	$\frac{1}{10}$	10 : 11	100 : 121	1000 : 1331
$12 \frac{1}{2} %$	$\frac{1}{8}$	8 : 9	64 : 81	512 : 729
$16 \frac{2}{3} %$	$\frac{1}{6}$	6 : 7	36 : 49	216 : 343
20%	$\frac{1}{5}$	5 : 6	25 : 36	125 : 216
25%	$\frac{1}{4}$	4 : 5	16 : 25	64 : 125
$14 \frac{2}{7} %$	$\frac{1}{7}$	7 : 8	49 : 64	343 : 512
15%	$\frac{3}{20}$	20 : 23	400 : 529	8000 : 12167

NoteThis method is only for questions about principal, amount and total interest. If one particular year’s interest, the simple interest or CI−SI is asked, use the interest tree of the next branch.

Steps

Write the rate as a fraction $\frac{a}{b}$ in lowest terms.
Write one year’s ratio $P : A = b : (b + a)$ .
Raise it to the number of years; interest = the difference of the two parts.
Find 1 unit from the part whose money value is given.
Multiply the part asked for by 1 unit.

Example 5 · Basic

Find the compound interest on ₹6,250 at 20% per annum for 2 years.

Solution, step by step

120% $= \frac{1}{5}$ : every 5 rupees become 6 in a year, $P : A = 5 : 6$ .
2In 2 years this happens twice: $5^{2} : 6^{2} = 25 : 36$ . Interest $= 36 - 25 = 11$ units.
3The principal 25 units = ₹6,250 ⟹ 1 unit $= \frac{6250}{25} = 250$ .
4Interest $= 11 \times 250 = ₹2, 750$ . Check by the formula: $6250 \times 1.2 = 7500$ , $7500 \times 1.2 = 9000$ , interest $= 9000 - 6250 = 2, 750$ ✓.

Answer ₹2,750

Example 6 · Standard

A sum earned ₹3,255 as compound interest in 3 years at

12.5%

per annum. Find the sum.

Solution, step by step

1 $12 \frac{1}{2} % = \frac{1}{8}$ , so in one year $P : A = 8 : 9$ .
2For 3 years take cubes: $8^{3} : 9^{3} = 512 : 729$ . Interest $= 729 - 512 = 217$ units.
3217 units $= 3, 255$ ⟹ 1 unit $= \frac{3 , 255}{217} = 15$ .
4Principal $= 512 \times 15 = ₹7, 680$ .

Answer ₹7,680

Knowing cubes saves time:

8^{3} = 512, 9^{3} = 729

.

Example 7 · Standard

A sum becomes ₹12,800 in 2 years at

14 \frac{2}{7} %

per annum compound interest. Find the principal and the interest.

Solution, step by step

1 $14 \frac{2}{7} % = \frac{100}{7} % = \frac{1}{7}$ , so in one year $P : A = 7 : 8$ .
22 years: $49 : 64$ . The amount is given, so we start from the 64-unit part.
364 units $= 12, 800$ ⟹ 1 unit $= 200$ .
4Principal $= 49 \times 200 = ₹9, 800$ ; interest $= 15 \times 200 = ₹3, 000$ .

Answer ₹9,800, ₹3,000

Rates like

14 \frac{2}{7} %, 11 \frac{1}{9} %, 9 \frac{1}{11} %

are a hint to work with the fraction (

\frac{1}{7}, \frac{1}{9}, \frac{1}{11}

).

Example 8 · Advanced

At 15% per annum compound interest a sum earns ₹3,870 in 2 years. Find the principal and the amount.

Solution, step by step

115% $= \frac{15}{100} = \frac{3}{20}$ . Here the numerator is 3, not 1: every 20 rupees become 23 in a year, $P : A = 20 : 23$ .
22 years: $2 0^{2} : 2 3^{2} = 400 : 529$ ; interest $= 129$ units.
3129 units $= 3, 870$ ⟹ 1 unit $= 30$ .
4Principal $= 400 \times 30 = ₹12, 000$ , amount $= 529 \times 30 = ₹15, 870$ .

Answer ₹12,000, ₹15,870

The method is the same when the numerator is not 1; only the numbers get bigger (

2 3^{2} = 529

).

BasicTry it yourself first

₹4,000 is kept in a bank for 3 years at

10%

per annum, compounded annually. How much compound interest will it earn?

BasicTry it yourself first

₹4,800 is kept in a post-office scheme for 2 years at

25%

per annum, compounded annually. How much compound interest will it earn?

BasicTry it yourself first

A sum becomes ₹2,025 in 2 years at

12.5%

per annum compound interest. What is the principal?

StandardTry it yourself first

A sum kept in a company bond at

16 \frac{2}{3} %

per annum compound interest earned ₹585 interest in 2 years. What was the sum?

StandardTry it yourself first

At

25%

per annum compound interest a sum earns ₹1,220 in 3 years. What will the amount be?

StandardTry it yourself first

₹12,000 is kept in a mutual fund for 2 years at

5%

per annum, compounded annually. How much compound interest will it earn?

AdvancedTry it yourself first

A sum kept in a bank at

11 \frac{1}{9} %

per annum compound interest earned ₹950 interest in 2 years. What was the sum?

AdvancedTry it yourself first

A sum becomes ₹8,000 in 3 years at

33 \frac{1}{3} %

per annum compound interest. What is the principal?

AdvancedTry it yourself first

What will ₹5,000 amount to after 2 years at

12%

per annum compound interest?

Effective percent (2 and 3 years)

2 r + \frac{r ^{2}}{100}

·

3 r + \frac{3 r ^{2}}{100} + \frac{r ^{3}}{1 0 ^{4}}

When the rate is not a handy fraction (6%, 7%, 9%…), find the interest as a percent of the principal.

The idea

The fraction for 7% is $\frac{7}{100}$ — its powers give big numbers. Then use the successive percent formula directly: 2 years are two successive rises of $r %$ and $r %$ , so the total rise $= r + r + \frac{r \cdot r}{100} = 2 r + \frac{r ^{2}}{100}$ percent. This is the effective percent: the 2-year interest is this percent of the principal.

For 3 years, combining once more with $r %$ gives $3 r + \frac{3 r ^{2}}{100} + \frac{r ^{3}}{10000}$ . The three terms mean: $3 r$ = three years of simple interest, $\frac{3 r ^{2}}{100}$ = three pieces of interest on interest, $\frac{r ^{3}}{10000}$ = interest on the interest-on-interest.

∑ Formula

C I_{2} = P \cdot \frac{2 r + \frac{r ^{2}}{100}}{100}, C I_{3} = P \cdot \frac{3 r + \frac{3 r ^{2}}{100} + \frac{r ^{3}}{10000}}{100}

Interest for 2 and 3 years by the effective percent

Rate	2-year CI (% of P)	3-year CI (% of P)
4%	8.16%	12.4864%
5%	10.25%	15.7625%
6%	12.36%	19.1016%
8%	16.64%	25.9712%
10%	21%	33.1%
12%	25.44%	40.4928%
15%	32.25%	52.0875%
20%	44%	72.8%

NoteWrite the 3-year percent by place value: at 4%,

12 + 0.48 + 0.0064 = 12.4864

. A decimal in the wrong place is the commonest slip here.

Steps

Find the effective percent for the number of years (2-year or 3-year formula).
Interest = principal × effective % ÷ 100.
If the principal is asked, reverse it: principal = interest × 100 ÷ effective %.

Example 9 · Basic

Find the compound interest on ₹7,500 at 6% per annum for 2 years.

Solution, step by step

1The fraction for 6% is $\frac{3}{50}$ , whose powers are big; so use the effective percent.
22-year effective % $= 2 \times 6 + \frac{6 ^{2}}{100} = 12 + 0.36 = 12.36%$ .
3Interest $= 7500 \times \frac{12.36}{100} = ₹927$ .

Answer ₹927

The 0.36% part is the interest earned in year two on year one’s interest (6%).

Example 10 · Standard

Find the compound interest on ₹10,000 at 7% per annum for 3 years.

Solution, step by step

13-year effective % $= 3 r + \frac{3 r ^{2}}{100} + \frac{r ^{3}}{10000}$ , with $r = 7$ .
2 $= 21 + \frac{3 \times 49}{100} + \frac{343}{10000} = 21 + 1.47 + 0.0343 = 22.5043%$ .
3Interest $= 10000 \times \frac{22.5043}{100} = ₹2, 250.43$ .

Answer ₹2,250.43

Example 11 · Standard

The 2-year compound interest on a sum at 9% per annum is ₹2,821.50. What is the sum?

Solution, step by step

12-year effective % $= 18 + \frac{81}{100} = 18.81%$ , so the interest is 18.81% of the principal.
2Principal × $\frac{18.81}{100} = 2, 821.50$ ⟹ principal $= \frac{2 , 821.50 \times 100}{18.81} = ₹15, 000$ .

Answer ₹15,000

With options, checking is even faster: 18.81% of ₹15,000 = ₹2,821.50 ✓.

BasicTry it yourself first

Meena deposits ₹12,500 for 2 years at

8%

per annum compound interest. How much interest does Meena get?

BasicTry it yourself first

What will ₹62,500 amount to in 2 years at

4%

per annum compound interest?

StandardTry it yourself first

Salma deposits ₹20,000 for 2 years at

11%

per annum compound interest. How much interest does Salma get?

StandardTry it yourself first

The compound interest on a sum for 2 years at

7%

per annum is ₹4,347. Find the sum.

StandardTry it yourself first

Anil deposits ₹50,000 for 3 years at

6%

per annum compound interest. How much interest does Anil get?

AdvancedTry it yourself first

The compound interest on a sum for 3 years at

9%

per annum is ₹5,900.58. Find the sum.

AdvancedTry it yourself first

What will ₹40,000 amount to in 3 years at

3%

per annum compound interest?

2

The year-wise tree and CI − SI

Seeing each year’s interest

Breaking the interest into “interest on principal” and “interest on interest” to get each year’s interest, CI−SI and the ratios built on them.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

The interest tree: 2 years

Principal

= b^{2}

units

Splitting two years of interest into interest on the principal and interest on interest.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

The interest tree: 3 years

Principal

= b^{3}

· gap

3 x + 1

Each year’s interest, total CI and CI−SI over three years.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

CI − SI formulas

P (\frac{R}{100})^{2}

·

\times (3 + \frac{R}{100})

The 2- and 3-year difference formulas, and finding the principal or the rate from them.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Ratio questions on CI and SI

Take SI as the base

Finding the rate from ratios like CI : SI, CI₂ : SI₁, D₃ : D₂.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Link between consecutive years

Next = previous ×

(1 + \frac{R}{100})

One year’s interest (or amount) grows R% by the next year — finding the rate and principal from this.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

3

When the rate or the period changes

Varying rates · part years · half-yearly, quarterly

A different rate each year, some months or days beyond whole years, and interest added more than once a year.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

A different rate each year

P (1 + \frac{R _{1}}{100}) (1 + \frac{R _{2}}{100}) \dots

Different rates in years one, two, three — separate factors, then multiply them all.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Whole years + some months or days

Simple interest on the leftover

Times like 2 years 4 months or 1 year 73 days with annual compounding.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Half-yearly and quarterly compounding

Rate ÷ 2 or 4, periods × 2 or 4

When interest joins the principal twice or four times a year.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

4

Working backwards: rate, time, principal

Roots · times · two amounts

Finding the rate, the time or the principal when amounts are known.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Rate or time by taking roots

n A / P

= one year’s factor

From principal and amount: the rate (take a root) or the time (spot the power).

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

“Becomes x times” questions

x

times in

T

⟹

x^{k}

times in

k T

A sum doubles or triples in some years — then the time for another multiple, or the rate.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Two amounts at two times

Equal gaps:

P = \frac{A _{1}^{2}}{A _{2}}

The rate and the principal from the amounts at two different times.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

5

Loans: instalments, repayments, shares

Value today = later money ÷ factor

Equal annual instalments, paying back part each year, and splitting a sum so that everyone gets the same later.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Equal annual instalments

Add each instalment’s value today

Repaying a loan in equal yearly instalments — the loan from the instalment or the instalment from the loan.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Paying back part each year

× factor, then − payment

Some money is paid back at each year end — how much is left at the end, or how much was paid earlier.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Splitting a sum so the amounts end up equal

Shares ∝ 1 ÷ growth

Parts that grow for different times but reach equal amounts.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

6

Growth and decay

Population · depreciation

The compound formula beyond money: population, trees, bacteria, and the falling value of a machine or vehicle.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Population growth

P (1 \pm \frac{R}{100})^{n}

Population rising and falling, a different rate each year, birth and death rates, and earlier values.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Depreciation (falling value)

Factor

1 - \frac{R}{100}

A machine’s, vehicle’s or computer’s value falling by a percent every year.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

7

Summary, mixed practice, revision

All methods in one place

One table of all methods, exam-style mixed questions and revision cards.

Summary of all methods

Which method when

One page to pick the method as soon as you read a question.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Mixed practice — set 1

Basic to standard

All kinds mixed together; spot the method first, then solve.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Mixed practice — set 2

Standard to advanced

Mixed questions from the hard end of the exam.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)

Revision — cards and traps

Flip and recall

Read the front, think, then flip.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Also in these passes: RAS Pre 2026 — 40 Day Study Plan + Complete Notes (₹300.00) · Competitive Maths (English Medium) (₹90.00) · RRB NTPC Graduation Level 2026 (₹300.00) · RRB NTPC Under Graduate Level 2026 (₹300.00)