AFCAT Numerical Ability · Simple and Compound Interest

When a Sum Doubles, Triples or Becomes n Times at Simple Interest

A sum that becomes n times itself has earned n minus 1 times itself as interest, the same amount every year, so rate × time = 100 × (n minus 1).

All Simple and Compound Interest notes

Why this matters

Three of AFCAT's past-year questions are here, from 2015 and 2016, and it is one of the larger pages of the chapter. The one slip is counting the multiple instead of the interest: a sum that triples has earned two principals, not three.

Concept 1 of 3: The multiple and the rate

If a sum becomes 4 times itself, the interest is 3 times the sum, which is 300%. Spread over the years, that is rate × time. A fraction of the principal works the same way: interest of half the sum is 50%.

Definition

Becomes $n$ times in $T$ years: $R \times T = 100(n - 1)$ .
Doubles: $RT = 100$ . Triples: $RT = 200$ .
Interest equal to $\tfrac ab$ of the sum: $RT = 100 \times \tfrac ab$ .
Read the other way: at $R\%$ for $T$ years the sum becomes $1 + \tfrac{RT}{100}$ times itself, and it becomes $n$ times in $\tfrac{100(n - 1)}{R}$ years.
A rate per half-year or per quarter is the yearly rate divided by 2 or 4.

Multiple and rate

R \times T = 100\,(n - 1)

Worked example

A sum becomes 5 times itself in 16 years at simple interest. Find the rate.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 04-206Easy

Example 1 · Simple and Compound Interest · Simple Interest: Sum Becoming n Times

On a certain sum, the simple interest at the end of

6\frac{1}{4}

yr becomes

\frac{3}{8}

th of the sum. The rate per cent is

n times means n minus 1 as interest

A sum that becomes 4 times itself in 12 years has earned 300%, so the rate is 25%. Using 400% gives

33\tfrac13\%

Drill 26 more on the multiple and the rate

Concept 2 of 3: From one multiple to another

At simple interest the sum earns the same amount every year. If it doubles in 6 years it earns one principal every 6 years, so to become 4 times itself (three principals of interest) it needs 18 years.

Definition

Count principals of interest: becoming $n$ times means $n - 1$ principals.
Time is proportional to interest: $\dfrac{T_2}{T_1} = \dfrac{n_2 - 1}{n_1 - 1}$ .
Fractions work the same: $\tfrac54$ times itself means $\tfrac14$ of the sum as interest.
Never multiply the multiples. That is the compound interest rule, which comes later in the chapter.

Time to a new multiple

T_2 = T_1 \times \dfrac{n_2 - 1}{n_1 - 1}

Worked example

A sum doubles in 7 years at simple interest. In how many years will it become 5 times itself?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Simple and Compound Interest · Simple Interest: Sum Becoming n Times

If a sum triples itself in

12

years at simple interest, in how many years will it become

\frac{7}{2}

times?

Count interest, not the multiple

A sum doubling in 3 years at simple interest becomes 4 times itself in 9 years, because 4 times needs three principals of interest. 6 years is the compound interest answer.

Drill 34 more on from one multiple to another

Concept 3 of 3: The multiple, amount or interest after another time

Once one year's interest is known as a share of the sum, any other time follows. A sum that doubles in 20 years earns 5% of itself a year, so after 4 years it is 1.2 times itself.

Definition

The share earned each year is $\dfrac{n - 1}{T}$ .
After $t$ years the sum is $1 + t \times \dfrac{n - 1}{T}$ times itself.
The interest after $t$ years is $t \times \dfrac{n - 1}{T}$ of the sum, which gives the ratio of principal to interest.
With a rupee principal, multiply at the end.

Multiple after t years

1 + t \times \dfrac{n - 1}{T}

Worked example

A sum doubles in 20 years at simple interest. How many times itself is it after 6 years, and what is ₹5,000 worth then?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 04-213Easy

Example 3 · Simple and Compound Interest · Simple Interest: Sum Becoming n Times

If a sum becomes double in 16 yr at the rate of simple interest, how many times will it be in 8 yr?

Add the 1 back

After 10 years at 15% a year a sum has earned 1.5 times itself, so it is 2.5 times itself. Stopping at 1.5 answers 'how much interest', not 'how many times'.

Drill 6 more on the multiple, amount or interest after another time

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)

The multiple and the rate
Multiple and rate
$R \times T = 100\,(n - 1)$
From one multiple to another
Time to a new multiple
$T_2 = T_1 \times \dfrac{n_2 - 1}{n_1 - 1}$
The multiple, amount or interest after another time
Multiple after t years
$1 + t \times \dfrac{n - 1}{T}$

Watch out for (3)

n times means n minus 1 as interest
→ The multiple and the rate
Count interest, not the multiple
→ From one multiple to another
Add the 1 back
→ The multiple, amount or interest after another time

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