AFCAT Numerical Ability · Simple and Compound Interest

Simple Interest: the Formula, Both Ways

Simple interest is the same amount every year, P × R × T ÷ 100; the amount is the principal plus that interest, and the same formula read backwards gives the principal, the rate or the time.

All Simple and Compound Interest notes

Why this matters

This is the base of the whole chapter, and most of the page is EASY. The marks go on time: months and days have to become years first, and a rate quoted per quarter or per half-year has to become a yearly rate before it goes into the formula.

Concept 1 of 3: Simple interest and amount

At simple interest only the original sum earns. ₹2,000 at 9% earns ₹180 every year, so 4 years earn ₹720 and the lender gets back ₹2,720.

Definition

$SI = \dfrac{P \times R \times T}{100}$ , with $P$ the principal, $R$ the rate per cent a year and $T$ the time in years.
The amount is $A = P + SI$ : what is paid back at the end.
Months: divide by 12. 9 months is $\tfrac34$ of a year; 2 years 4 months is $2\tfrac13$ years.
Days: divide by 365. Leave out the day the money is lent and count the day it is repaid. From 10 March to 22 May is $21 + 30 + 22 = 73$ days, which is $\tfrac{73}{365} = \tfrac15$ of a year.

Simple interest

SI = \dfrac{P \times R \times T}{100}, \quad A = P + SI

Worked example

Find the simple interest and the amount on ₹4,500 at 8% a year for 2 years 4 months.

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The same idea in a practice question:

Practice questionModerate

Example 1 · Simple and Compound Interest · Simple Interest: Interest and Amount

What will be the simple interest on ₹6400 at the rate of 6% per annum from 11th Feb to 7th July (of a non-leap year)?

Turn months into years

At 12% a year, 8 months is

\tfrac{8}{12}

of a year, so the interest on ₹5,000 is ₹400. Putting

T = 8

gives ₹4,800.

Interest or amount

₹6,000 at 5% for 2 years earns ₹600 and amounts to ₹6,600. Read which one is asked: the options usually carry both.

Drill 13 more on simple interest and amount

Concept 2 of 3: Finding the principal, rate or time

The formula links four quantities. Know any three and the fourth follows: the same formula is simply read backwards.

Definition

$P = \dfrac{100 \times SI}{R \times T}$ , $R = \dfrac{100 \times SI}{P \times T}$ , $T = \dfrac{100 \times SI}{P \times R}$ .
Given the amount and the principal, take the interest out first: $SI = A - P$ .
Given the amount but not the principal: the amount is $P\left(1 + \tfrac{RT}{100}\right)$ . At 6% for 5 years that is $1.3P$ , so an amount of ₹5,200 means $P = 4000$ .
'The interest is ₹x less than the principal' is the equation $P - SI = x$ . Write the interest as a share of $P$ first.
A rate that changes over the years: add rate × years for each stretch and treat the total as one percentage of the sum.

Finding the rate or time

R = \dfrac{100 \times SI}{P \times T}, \quad T = \dfrac{100 \times SI}{P \times R}

Worked example

₹6,000 amounts to ₹7,440 at simple interest in 4 years. Find the rate.

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The same idea in a real exam question:

AFCAT · 2015 · AFCAT (II) 2015 — Numerical Ability · 04-212Easy

Example 2 · Simple and Compound Interest · Simple Interest: Interest and Amount

A man lent a sum of money at the rate of simple interest of 4%. If the interest for 8 yr is ₹340 less than the principal, the principal is

Take the interest out of the amount first

₹6,000 amounting to ₹7,440 has earned ₹1,440. Dividing the whole ₹7,440 by

6000 \times 4

as if it were the interest gives a rate of 31%.

Drill 28 more on finding the principal, rate or time

Concept 3 of 3: Rates per month, quarter or half-year

Simple interest never earns on itself, so how often it is 'calculated' changes nothing. A rate of 2% a quarter is 8% a year, and 1% a month is 12% a year. Make the rate yearly and the time years, then use the formula.

Definition

Yearly rate = rate per quarter × 4 = rate per half-year × 2 = rate per month × 12.
Or keep both in the same unit: 2% a quarter for 6 quarters is 12% of the sum.
'Calculated half-yearly' or 'quarterly' at simple interest gives the same interest as yearly, so the difference between them is zero.
Asked for a rate per quarter: find the yearly rate first, then divide by 4.

Matching the units

SI = \dfrac{P \times (\text{rate per period}) \times (\text{number of periods})}{100}

Worked example

Find the simple interest on ₹15,000 at 1.25% a month for 8 months.

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The same idea in a practice question:

Practice questionHard

Example 3 · Simple and Compound Interest · Simple Interest: Interest and Amount

A man borrows Rs.

18,000

4.5\%

per half year. What interest will he pay for

10

months?

A rate per period is not a yearly rate

At 3% a quarter the yearly rate is 12%. Using 3% as the yearly rate makes the interest a quarter of what it should be.

Compounding words at simple interest

'Calculated half-yearly' changes nothing at simple interest. Only compound interest grows faster when it is added more often; that comes later in the chapter.

Drill 23 more on rates per month, quarter or half-year

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)

Simple interest and amount
Simple interest
$SI = \dfrac{P \times R \times T}{100}, \quad A = P + SI$
Finding the principal, rate or time
Finding the rate or time
$R = \dfrac{100 \times SI}{P \times T}, \quad T = \dfrac{100 \times SI}{P \times R}$
Rates per month, quarter or half-year
Matching the units
$SI = \dfrac{P \times (\text{rate per period}) \times (\text{number of periods})}{100}$

Watch out for (5)

Turn months into years
→ Simple interest and amount
Interest or amount
→ Simple interest and amount
Take the interest out of the amount first
→ Finding the principal, rate or time
A rate per period is not a yearly rate
→ Rates per month, quarter or half-year
Compounding words at simple interest
→ Rates per month, quarter or half-year

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