CDS Mathematics · Simple and Compound Interest

Compound Interest

Compound interest adds each period's interest to the sum, so the amount is multiplied by (1 + r) every period: A = P(1 + r)ⁿ.

All Simple and Compound Interest notes CDS Mathematics strategy

Why this matters

Thirteen PYQs, three of them HARD. Two moves: the amount formula with the rate and number of periods adjusted for half-yearly or quarterly compounding, and growth questions (doubles in 5 years, so four times in 10; more than 100 times needs logarithms).

Concept 1 of 2: The amount formula and compounding periods

Each period multiplies the amount by the same factor. Compounding quarterly at

12\%

a year uses

3\%

a quarter over four times as many periods.

Definition

$A = P\left(1 + \dfrac{r}{100}\right)^n$ ; CI $= A - P$ .
Half-yearly: rate $\dfrac r2$ , periods $2n$ . Quarterly: rate $\dfrac r4$ , periods $4n$ .
Interest earned in the second year alone: $P \cdot \dfrac{r}{100}\left(1 + \dfrac r{100}\right)$ .
More frequent compounding at the same yearly rate gives slightly more.

Compound amount

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

Worked example

Find the compound interest on Rs.

8000

for

1

year at

10\%

a year, compounded half-yearly.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q33Moderate

Example 1 · Simple and Compound Interest · Compound Interest

A merchant commences with a certain capital and gains annually at the rate of 25%. At the end of 3 years he has Rs. 10,000. What is the original amount that the merchant invested?

Adjust both the rate and the periods

Quarterly compounding at

12\%

3\%

for FOUR periods a year. Using

12\%

with four periods, or

3\%

for one, gives a wrong amount.

Drill 6 more on the amount formula and compounding periods

Concept 2 of 2: Doubling times and growth factors

Under compound interest the sum is multiplied by the same factor over any equal stretch of time. So if it doubles in

5

years, it doubles again in the next

5

Definition

Doubles in $T$ years: becomes $2^k$ times in $kT$ years.
'More than $k$ times': the least $n$ with $\left(1 + \tfrac r{100}\right)^n > k$ ; try powers or use logarithms.
$n \log(1 + r) > \log k$ with $\log 1.2 = 2\log 2 + \log 3 - 1$ .
If $y$ is the CI on $x$ and $z$ the CI on $y$ (same rate and time), then $y^2 = xz$ .

Growth condition

\left(1 + \dfrac{r}{100}\right)^n > k

Worked example

A sum trebles in

4

years at compound interest. In how many years does it become

27

times?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q37Easy

Example 2 · Simple and Compound Interest · Compound Interest

A sum of money compounded annually doubles itself in 5 years. In how many years will it become four times of itself ?

Four times is two doublings, not four

A sum that doubles in

5

years is four times itself after

10

, not

20

. Compound growth multiplies; it does not add.

Drill 5 more on doubling times and growth factors

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 (2)

The amount formula and compounding periods
Compound amount
$A = P\left(1 + \dfrac{r}{100}\right)^n$
Doubling times and growth factors
Growth condition
$\left(1 + \dfrac{r}{100}\right)^n > k$

Watch out for (2)

Adjust both the rate and the periods
→ The amount formula and compounding periods
Four times is two doublings, not four
→ Doubling times and growth factors

Test yourself on Simple and Compound Interest

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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