AFCAT Numerical Ability · Simple and Compound Interest

Finding the Principal or Rate from Two Facts

When a sum is given at two different times, or a rate has to be found in one case and used in another, the first step is one year's interest.

All Simple and Compound Interest notes

Why this matters

Two of AFCAT's past-year questions are here, from 2016 and 2018, and most of the page is MODERATE. Two amounts give one year's interest at once; the slip is taking off the wrong number of years of it to reach the principal.

Concept 1 of 2: A sum given at two different times

Between the two dates the principal does not change; only interest is added. So the difference between the two amounts is the interest for the years in between, and dividing gives one year's interest.

Definition

One year's interest: $I_1 = \dfrac{A_2 - A_1}{t_2 - t_1}$ .
Principal: $P = A_1 - t_1 \times I_1$ . Use the earlier amount with its own number of years.
Rate: $R = \dfrac{I_1}{P} \times 100$ .
When will it reach a third amount? (target minus principal) ÷ $I_1$ , counted from the start.
'After a further 4 years' means the second date is the first date plus 4.

One year's interest

I_1 = \dfrac{A_2 - A_1}{t_2 - t_1}, \quad P = A_1 - t_1 I_1

Worked example

A sum amounts to ₹1,610 in 3 years and to ₹1,750 in 5 years at simple interest. Find the principal and the rate.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 04-211Easy

Example 1 · Simple and Compound Interest · Simple Interest: Finding Principal, Rate or Time

A sum of money lent out at simple interest amounts to ₹720 after 2 yr and to ₹1020 after a further period of 5 yr. The sum is

Take off the right number of years

With ₹1,610 after 3 years and ₹70 a year, the principal is

1610 - 3 \times 70 = 1400

. Taking off only one year's interest, or the years of the later date, gives a wrong principal that is still among the options.

'A further' period

'₹2,200 after 2 years and ₹2,600 after a further 4 years' puts the second date at year 6. The ₹400 is four years' interest, not six.

Drill 25 more on a sum given at two different times

Concept 2 of 2: Find the rate in one case, use it in another

Many questions give one complete case (a sum, its amount and the time) and then ask about another sum or another time 'at the same rate'. Find the rate from the complete case; the second case is then the plain formula.

Definition

From the complete case: $R = \dfrac{100\,(A - P)}{P \times T}$ .
Use that rate with the new principal and time.
'The rate is doubled' or 'raised by 2%': change the rate you found before using it.
Two interests said to be equal: work out the one that is fully given, then solve the other for its missing rate, time or principal.

Rate from a complete case

R = \dfrac{100\,(A - P)}{P \times T}

Worked example

₹2,500 amounts to ₹2,950 in 3 years at simple interest. What will ₹3,500 amount to in 6 years at the same rate?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (I) 2018 — Numerical Ability · 04-205Moderate

Example 2 · Simple and Compound Interest · Simple Interest: Finding Principal, Rate or Time

₹6000 becomes ₹7200 in 4 yr at a certain rate of simple interest. If the rate becomes 1.5 times itself, the amount of the same principal in 5 yr will be

Change the rate, not the amount

If the rate is doubled, a rate of 4% becomes 8% and the interest doubles. Doubling the old amount, principal included, is a different and wrong number.

Drill 16 more on find the rate in one case, use it in another

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)

A sum given at two different times
One year's interest
$I_1 = \dfrac{A_2 - A_1}{t_2 - t_1}, \quad P = A_1 - t_1 I_1$
Find the rate in one case, use it in another
Rate from a complete case
$R = \dfrac{100\,(A - P)}{P \times T}$

Watch out for (3)

Take off the right number of years
→ A sum given at two different times
'A further' period
→ A sum given at two different times
Change the rate, not the amount
→ Find the rate in one case, use it in another

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