AFCAT Numerical Ability · Simple and Compound Interest

The Difference Between Compound and Simple Interest

For the same sum, rate and time, compound interest is more than simple interest by the interest earned on interest: P(R/100)² over 2 years and P(R/100)²(3 + R/100) over 3 years.

All Simple and Compound Interest notes

Why this matters

Three of AFCAT's past-year questions are on this page, from 2016, 2018 and 2019, and about half the page is EASY. For two years the whole question is one line. Marks go on three years, where the gap is not three times the two-year gap, and on rows that give one interest and ask for the other.

Concept 1 of 5: Two years: P(R/100)²

Over two years both kinds of interest give R% of the principal each year. Compound interest also gives R% on the first year's interest during the second year. That extra is

P \times \tfrac{R}{100} \times \tfrac{R}{100}

Definition

$CI - SI = P\left(\dfrac{R}{100}\right)^2$ for 2 years compounded annually.
The sum from the gap: $P = \dfrac{\text{gap}}{(R/100)^2}$ . At 10% divide by 0.01, at 5% by 0.0025.
'SI and CI differ by' and 'CI exceeds SI by' mean the same positive gap.

Two-year gap

CI - SI = P\left(\dfrac{R}{100}\right)^2

Worked example

Find the difference between the compound and simple interest on ₹12,000 for 2 years at 5% a year.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 04-202Easy

Example 1 · Simple and Compound Interest · Difference Between CI and SI

The difference between compound and simple interest on a sum of money at 4% per annum for 2 yr is ₹8. The sum is

Square the rate as a fraction

At 5% the factor is

0.05^2 = 0.0025

, not 0.025. Using 0.025 makes the gap ten times too big.

Drill 33 more on two years: p(r/100)²

Concept 2 of 5: The rate from the two-year gap

Turn the same formula round:

(R/100)^2 = \text{gap} \div P

. In every such question the right side is a perfect square, and its square root is the rate as a fraction.

Definition

$\left(\dfrac{R}{100}\right)^2 = \dfrac{\text{gap}}{P}$ .
Recognise the square: 0.0025 is $0.05^2$ , 0.0049 is $0.07^2$ , 0.0081 is $0.09^2$ , 0.0225 is $0.15^2$ .
Or work in whole numbers: $R^2 = \dfrac{10000 \times \text{gap}}{P}$ .

Rate from the gap

R = 100\sqrt{\dfrac{CI - SI}{P}}

Worked example

On ₹20,000 for 2 years, compound interest is more than simple interest by ₹98. Find the rate.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Simple and Compound Interest · Difference Between CI and SI

Rate p.a. if CI−SI on Rs.

\;6250

for 2 yrs is Rs.

\;10

R squared is not R

R^2 = 49

, the rate is 7%, not 49% and not 4.9%.

Drill 9 more on the rate from the two-year gap

Concept 3 of 5: Three years

Over three years the interest on interest piles up: the first year's interest earns in years 2 and 3, the second year's interest earns in year 3, and the first year's interest on interest earns once more. Together that is

P(R/100)^2(3 + R/100)

Definition

$CI - SI = P\left(\dfrac{R}{100}\right)^2\left(3 + \dfrac{R}{100}\right)$ .
At 5% that is $0.007625P$ ; at 20% it is $0.128P$ .
Or work it out in full: compound interest from $(1 + R/100)^3 - 1$ , simple interest as $3R\%$ , and subtract.

Three-year gap

CI - SI = P\left(\dfrac{R}{100}\right)^2\left(3 + \dfrac{R}{100}\right)

Worked example

Find CI minus SI on ₹16,000 for 3 years at 5% a year.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 04-204Moderate

Example 3 · Simple and Compound Interest · Difference Between CI and SI

The difference between compound interest and simple interest for 3 yr at a rate of 10% per annum is ₹77.5. What is the principal?

Three years is not three times two

At 20% on ₹1,000 the two-year gap is ₹40, but the three-year gap is ₹128. Tripling ₹40 misses the interest earned on interest on interest.

Drill 8 more on three years

Concept 4 of 5: From one interest to the other

A question that gives the compound interest and asks for the simple interest (or the other way round) takes two steps: find the sum from the interest you have, then work out the other one.

Definition

From compound interest: $P = \dfrac{CI}{(1 + R/100)^n - 1}$ , then $SI = \dfrac{P R n}{100}$ .
From simple interest: $P = \dfrac{100\,SI}{R n}$ , then the compound interest or the gap.
Comparing both as percentages of the sum skips $P$ : at 8% for 2 years compound interest is 16.64% of the sum and simple interest 16%, so $SI = CI \times \tfrac{16}{16.64}$ .
For 2 years, $CI : SI = (2 + R/100) : 2$ .

Sum from compound interest

P = \dfrac{CI}{\left(1 + \frac{R}{100}\right)^n - 1}

Worked example

The compound interest on a sum for 2 years at 5% is ₹1,230. Find the simple interest for the same time and rate.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Simple and Compound Interest · Difference Between CI and SI

The compound interest on a sum for 2 years at 10% per annum is ₹2310. Find the simple interest on the same sum for 2 years at the same rate.

Divide by the compound share

At 5% for 2 years compound interest is 10.25% of the sum, so ₹1,230 of it means a sum of ₹12,000. Dividing by 10% instead treats it as simple interest and overstates the sum.

Drill 17 more on from one interest to the other

Concept 5 of 5: Both interests given: interest on interest

For 2 years, half the simple interest is one year's interest. The extra in the compound interest is that one year's interest earning for a year, so extra ÷ one year's interest is the rate.

Definition

One year's interest is $\dfrac{SI}{2}$ .
Rate: $\dfrac{R}{100} = \dfrac{CI - SI}{SI/2}$ .
Then the sum is $\dfrac{SI/2}{R/100}$ .

Rate from both interests

\dfrac{R}{100} = \dfrac{CI - SI}{SI / 2}

Worked example

The simple and compound interest on a sum for 2 years are ₹1,000 and ₹1,050. Find the rate and the sum.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 5 · Simple and Compound Interest · Difference Between CI and SI

If simple interest and compound interest on a certain principal at a rate of interest for 2 years are Rs. 1,600 and Rs. 1,760 respectively, then the principal is:

Halve the simple interest first

With ₹1,000 and ₹1,050 for 2 years, the rate is

\tfrac{50}{500} = 10\%

. Dividing by the whole ₹1,000 gives 5%.

Drill 3 more on both interests given: interest on interest

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 (5)

Two years: P(R/100)²
Two-year gap
$CI - SI = P\left(\dfrac{R}{100}\right)^2$
The rate from the two-year gap
Rate from the gap
$R = 100\sqrt{\dfrac{CI - SI}{P}}$
Three years
Three-year gap
$CI - SI = P\left(\dfrac{R}{100}\right)^2\left(3 + \dfrac{R}{100}\right)$
From one interest to the other
Sum from compound interest
$P = \dfrac{CI}{\left(1 + \frac{R}{100}\right)^n - 1}$
Both interests given: interest on interest
Rate from both interests
$\dfrac{R}{100} = \dfrac{CI - SI}{SI / 2}$

Watch out for (5)

Square the rate as a fraction
→ Two years: P(R/100)²
R squared is not R
→ The rate from the two-year gap
Three years is not three times two
→ Three years
Divide by the compound share
→ From one interest to the other
Halve the simple interest first
→ Both interests given: interest on interest

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