SSC CGL Maths · Simple and Compound Interest

Sums Lent at Two Rates

When money is spread over two or more rates, add the interest from each part; a split of one total is fastest by putting it all at the lower rate first.

All Simple and Compound Interest notes

Why this matters

Twenty-three questions in the bank, fourteen of them from the 2025 shifts and ten MODERATE: one sum split between two rates, several loans at one rate, and parts tied by a ratio or by equal interest. The equation always works, but the lower-rate shortcut and alligation save most of the time.

Concept 1 of 3: One sum split between two rates

If the whole sum sat at the lower rate it would earn a little less than it does. The shortfall is made up by the part at the higher rate, which earns the gap between the rates on every rupee.

Definition

Equation: with $x$ at the higher rate $r_2$ and $P - x$ at the lower rate $r_1$ , one year's interest is $\frac{r_2 x + r_1(P - x)}{100} = I$ .
Fast route: put all of $P$ at $r_1$ . The extra interest $I - \frac{r_1 P}{100}$ comes from the gap $r_2 - r_1$ on the higher part, so higher part $= \frac{100I - r_1 P}{r_2 - r_1}$ .
Alligation: the average rate is $\bar{r} = \frac{100 I}{P}$ . Lower part : higher part $= (r_2 - \bar{r}) : (\bar{r} - r_1)$ .
Interest for several years: divide by the years first to get $I$ for one year.
Difference of the two interests given: write the higher part's interest minus the lower part's and solve the same way.

Splitting a sum between two rates

\text{Higher part} = \frac{100 I - r_1 P}{r_2 - r_1} \qquad \frac{\text{Lower part}}{\text{Higher part}} = \frac{r_2 - \bar{r}}{\bar{r} - r_1}

$r_1, r_2$ the lower and higher rates percent
Ithe total interest for one year
$\bar{r}$ the average rate on the whole sum, $\frac{100I}{P}$

Worked example

₹12,000 is lent in two parts at 6% and 10% per annum simple interest. The total interest for one year is ₹1,000. How much is lent at 10%?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 1 · Q57Easy

Example 1 · Simple and Compound Interest · Sums Lent at Two Rates

A sum of ₹10000 is lent out in two parts, one at

7\%

simple interest and the other at

12\%

simple interest. If the annual interest is ₹900, the sum lent at

12\%

is:

Running alligation on a several-year interest

The average rate is the interest for one year as a percent of the whole sum. If ₹3,000 is earned in 3 years on ₹12,500, the average rate is 8%, not 24%.

Reversing the alligation ratio

The part at the lower rate takes the higher rate minus the average; the part at the higher rate takes the average minus the lower rate. Swapping them gives the other part as the answer, and that is always one of the options.

Drill 10 more on one sum split between two rates

Concept 2 of 3: Several loans at one rate

When every loan runs at the same rate, the rate can be taken out. What is left to add is principal times years for each loan, as if one big loan had run for one year.

Definition

Same rate on every loan: total $SI = \frac{R}{100} \times (P_1 T_1 + P_2 T_2 + \dots)$ . So $R = \frac{100 \times SI}{\sum PT}$ .
Same time, different rates: one year's total interest is the sum of each part's interest; time $=$ total interest $\div$ one year's total.
A loan made later in the year earns only for the months it is out.
One loan known, one unknown: take the known loan's interest out of the total, then find the other.

Several loans at one rate

SI_{\text{total}} = \frac{R}{100}\left(P_1 T_1 + P_2 T_2 + \dots\right)

Worked example

₹2,000 is lent for 2 years and ₹1,500 for 4 years, both at the same rate of simple interest. The total interest is ₹1,000. Find the rate.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 3 · Q70Easy

Example 2 · Simple and Compound Interest · Sums Lent at Two Rates

A lent Rs. 5000 to B for a duration of 2 years and Rs. 3000 to C for a period of 4 years both at the same simple interest rate. In total, he received Rs. 2200 as interest from both borrowers. What is the rate of interest per annum?

Charging a later loan for the whole year

Money lent 9 months into the year earns for the last 3 months only. Multiply its rate by

\frac{3}{12}

, not by 1.

Drill 6 more on several loans at one rate

Concept 3 of 3: Parts linked by a ratio or by equal interest

When the parts are tied to each other, there is only one unknown. Name the smallest part, write every other part through it, and one equation finishes the job.

Definition

Parts tied by percents (scheme C is 180% of A): write each part through one letter, add the interests, solve.
Equal interest from every part at rates $r_1, r_2, r_3$ : $r_1 a = r_2 b = r_3 c$ , so $a : b : c = \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3}$ .
Clear the fractions by multiplying by the LCM of the rates, then share the total in that ratio.
The lowest rate holds the most money.

Equal interest from every part

a : b : c = \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3}

Worked example

₹4,200 is invested in three parts at 3%, 5% and 6% per annum simple interest. Each part earns the same interest in a year. Find the three parts.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 1 · Q59Moderate

Example 3 · Simple and Compound Interest · Sums Lent at Two Rates

A person invested a total sum of ₹2200 in three different schemes of simple interest at

4\%

5\%

, and

10\%

per annum. At the end of one year, he got the same interest from all three schemes. What was the amount (in ₹) invested at

5\%

Splitting in the ratio of the rates

Equal interest needs the parts in the inverse ratio of the rates. At 4% and 5%, the parts are 5 : 4, not 4 : 5: the lower rate needs more money to earn the same.

Drill 4 more on parts linked by a ratio or by equal interest

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)

One sum split between two rates
Splitting a sum between two rates
$\text{Higher part} = \frac{100 I - r_1 P}{r_2 - r_1} \qquad \frac{\text{Lower part}}{\text{Higher part}} = \frac{r_2 - \bar{r}}{\bar{r} - r_1}$
Several loans at one rate
Several loans at one rate
$SI_{\text{total}} = \frac{R}{100}\left(P_1 T_1 + P_2 T_2 + \dots\right)$
Parts linked by a ratio or by equal interest
Equal interest from every part
$a : b : c = \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3}$

Watch out for (4)

Running alligation on a several-year interest
→ One sum split between two rates
Reversing the alligation ratio
→ One sum split between two rates
Charging a later loan for the whole year
→ Several loans at one rate
Splitting in the ratio of the rates
→ Parts linked by a ratio or by equal interest

Test yourself on Simple and Compound Interest

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