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AFCAT Numerical Ability · Simple and Compound Interest

Changing the Rate, Time or Sum, and Splitting a Sum

Simple interest is proportional to the principal, the rate and the time, so a change in any of them scales the interest; a difference of two interests gives the sum, and a sum split between two rates is one equation.

All Simple and Compound Interest notes

Why this matters

Two of AFCAT's past-year questions are here, from 2016 and 2019: a sum that would earn more at a higher rate, and equal sums lent for different times. Each is one percentage of the sum, so nothing needs computing twice. Most of the HARD rows split one sum between two rates.

Concept 1 of 3: Interest scales with the sum, the rate and the time

Double the rate and the interest doubles; lend for 3 years instead of 2 and it grows by half. There is no need to find the principal: scale the interest you know.

Definition

  • Same principal: new interest = old interest × R2T2R1T1\dfrac{R_2 T_2}{R_1 T_1}.
  • A different principal too: multiply by P2R2T2P1R1T1\dfrac{P_2 R_2 T_2}{P_1 R_1 T_1}.
  • On a known sum, a change of rate adds P×(R2−R1) T100P \times \dfrac{(R_2 - R_1)\,T}{100}; a change of time works the same way.

Scaling the interest

SI2=SI1×P2R2T2P1R1T1SI_2 = SI_1 \times \dfrac{P_2 R_2 T_2}{P_1 R_1 T_1}

Worked example

The simple interest on a sum at 5% a year for 4 years is ₹900. Find the interest on the same sum at 7% for 6 years.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Simple and Compound Interest · Simple Interest: Changing the Rate, Time or Sum

The simple interest on a sum for 55 years at 6%6\% per annum is Rs. 600600. What will be the simple interest on the same sum for 88 years at 9%9\% per annum?

Scale, do not add

Raising the rate from 5% to 7% multiplies the interest by 75\tfrac75. Adding ₹2 to the interest, or 2% of the interest, is not the same thing.
Drill 23 more on interest scales with the sum, the rate and the time

Concept 2 of 3: A difference of two interests gives the sum

Two interests on the same sum differ by a percentage of the sum: the difference of rate × time. Divide the rupee difference by that percentage and the sum appears.

Definition

  • Same sum and time, rates R1R_1 and R2R_2: the difference is P×(R2−R1) T100P \times \dfrac{(R_2 - R_1)\,T}{100}.
  • Same sum and rate, times T1T_1 and T2T_2: the difference is P×R (T2−T1)100P \times \dfrac{R\,(T_2 - T_1)}{100}.
  • Different rates and different times: write each interest as a percentage of the sum (8% for 2 years is 16%, 12% for 3 years is 36%) and subtract.
  • An unknown difference of rates: R2−R1=100×differenceP×TR_2 - R_1 = \dfrac{100 \times \text{difference}}{P \times T}.
  • Two sums in a ratio: write them as 2x2x and 3x3x and do the same.

Sum from a difference

P=100×difference(R2−R1) TP = \dfrac{100 \times \text{difference}}{(R_2 - R_1)\,T}

Worked example

The simple interest on a sum at 9% a year for 4 years is ₹360 more than at 7% for 4 years. Find the sum.
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 04-209Easy

Example 2 · Simple and Compound Interest · Simple Interest: Changing the Rate, Time or Sum

A sum was invested for 3 yr at simple interest at a certain rate. Had it been invested at a 4% higher rate of interest, it would have fetched ₹600 more. The sum is

Multiply the rate difference by the time

Rates of 5% and 8% for 2 years differ by 6% of the sum, not 3%. Forgetting the time doubles the sum you find.
Drill 17 more on a difference of two interests gives the sum

Concept 3 of 3: One sum split between two rates

Call one part xx; the other is the total minus xx. The two interests add up to the total given, or are equal, and either way that is one equation in xx.

Definition

  • Total interest given: x⋅R1T1100+(S−x)⋅R2T2100=Ix \cdot \dfrac{R_1 T_1}{100} + (S - x) \cdot \dfrac{R_2 T_2}{100} = I.
  • Equal interests: xR1T1=(S−x)R2T2x R_1 T_1 = (S - x) R_2 T_2, so the parts are in the ratio R2T2:R1T1R_2 T_2 : R_1 T_1.
  • A quicker way for a total: put the whole sum at the lower rate. The interest falls short, and the shortfall is earned by the part at the higher rate, at the difference of the rates.
  • Fractions of an income at different rates: write each part's interest as a share of the whole and add.
  • A rate per quarter is made yearly first, as on the first page.

Splitting a sum

x⋅R1T1100+(S−x)⋅R2T2100=Ix \cdot \dfrac{R_1 T_1}{100} + (S - x) \cdot \dfrac{R_2 T_2}{100} = I

Worked example

₹9,000 is lent partly at 6% and partly at 9% simple interest. The total interest for one year is ₹660. Find the part at 6%.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Simple and Compound Interest · Simple Interest: Changing the Rate, Time or Sum

A sum of Rs. 40004000 is lent out in two parts, one at 8%8\% and the other at 10%10\% simple interest per annum. If the annual interest is Rs. 350350, find the sum lent at 8%8\%.

Answer the part that is asked

If xx is the part at the lower rate, the question may ask for the part at the higher rate, S−xS - x. Both are among the options.

Equal interest means the inverse ratio

Equal interest at 12% and at 4% puts more money at the lower rate: the parts at 12% and at 4% are in the ratio 4:12=1:34 : 12 = 1 : 3, not 3:13 : 1.
Drill 9 more on one sum split between two rates

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)

Watch out for (4)

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