In this section, we will continue the Simple and Compound Interest topic.

Today we will discuss some questions which involve the Concepts of both SI and CI (SI – CI Mix) and see some problems which come in Exam under this topic.

SI – CI Mix:

There are certain problems which involve concepts of both Simple Interest and Compound Interest. It is important to understand the concept involved and then we have to proceed accordingly.

Let’s start by an example from RRB NTPC Quant Questions which contain both the concepts.

E.g. 1: On a certain principal, simple interest amounts to Rs. 1,000 in 1 year at the rate of 10% p.a. What will be the effective rate of interest if the same is compounded on half yearly basis?

Now here we see that we have both the concepts involved in a question. Let’s try to solve it.

Step 1:- Take note of given values

Here, Simple Interest (SI) = Rs. 1,000

         Time (T) = 1 year

         Rate of Simple Interest (RSI) = 10% p.a.

Step 2:- See what we can find using given values

Here, we can find the Principal using formula of SI

SI = (P × R × T)/100

⇒ 1000 = (P × 10 ×1)/100                               [Cancelling 100 and 10 by 10]

⇒ 1000 = (P × 1 × 1)/10

⇒ 1000 = (P)/10

⇒ 1000 × 10 = P

⇒ P = Rs. 10000

Step 3:- Change the values according to what we have to find

Now for CI compounded half yearly we have,

P = Rs. 10,000

RCI = R/2 % = 10/2 % = 5%     [From the table we discussed in the last blog]

TCI = T × 2 = 1 × 2 = 2 half years

Step 4: Find the other unknown values

Here, we can found amount

A = P × {1 + R/100}T

⇒ A = P × {1 + R/100}T

          = 10,000 × {1+ 5/100}2                                               [Cancelling 100 by 5]

          = 10,000 × {1+ 1/20}2

          = 10,000 × {(1 × 20 + 1)/20}2

          = 10,000 × {(1 × 20 + 1)/20}2

          = 10,000 × {21/20}2

         = 10,000 × 441/400

          = 100 × 441/4

          = 25 × 441

          = 11, 025

Therefore, net effective rate of interest for 1 year = (11025 – 10000)/10000 × 100 %

                                                                                = 1025/100 %

                                                                                = 10.25% (Ans).

So these are the steps through which we solve questions that involve both the concepts. However, this process is too long and let’s try to solve it with any other method.

This question is all about two successive percentage changes of 5%. So, we could easily solve these types of questions by a b principle. We will discuss this method in Percentage and Profit and loss in depth so here let’s just understand it very quickly.

a b Principle

a b principle is used to get the result of two successive changes. If a% and b% are two successive percentage changes then, the equivalent percentage change = a + b + ab/100.

For example, In this case initially we have 10% rate of simple interest. But the rate has to be compounded half yearly, so we will have to divide it by 2. So new rate of interest will be 5% per half year and time becomes 2 half years so that means, there is need of two successive changes of 5%.

Using a b principle,

The equivalent percentage change = a + b + ab/100

                                                        = 5 + 5 + 5 × 5/100

                                                        = 10 + 25/100

                                                        = 10 + 0.25

                                                        = 10.25 (Ans).

Let’s solve some other examples from RRB NTPC Exam to have a better command on this concept.

E.g. 2:- Raja invested Rs. 15000 at the rate of 10% per annum for 1 year. If the interest is compounded half yearly, then found the amount received by Raja at the end of the year.

We have, P = Rs. 15000, R = 5% per half years, T = 2 half years

For CI,

A = P × {1 + R/100}T

    = 15000 × {1 + 5/100}2

    = 15000 × {1 + 1/20}2

    = 15000 × {21/20}2

    = 15000 × 441/400

    = 150 × 441/4

    = 37.5 × 441

    = Rs. 16,537.5 (Ans).

Let’s solve it using a b principle,

Here, also we have rate of interest 10% p.a. which is to be compounded half yearly.

So we get, R = 5%

Using a b Principle,

Net rate of interest = 10.25%

So, the amount that Raja will receive after 1 year will be 10.25% more than 15000.

Therefore, Required Amount = 15000 + 10.25% of 15000

                                              = 15000 + 10.25/100 of 15000

                                              = 15000 + 10.25× 150

                                              = 15000 + 1537.5

                                              = 16537.50 (Ans).

E.g. 3:- If the rate of interest is 8% per annum and Rs. 10,000 lent at the compound interest half yearly then calculate the equivalent simple interest for the first year?

Equivalent simple interest for the first year = Equivalent compound interest per half year for first year.

Here, R = 8%, T = 1 year, P = Rs. 10,000

RHalf Yearly = R/2 = 8/2 % = 4%

THalf Yearly = T × 2 = 1 × 2 half years = 2 half years

Applying a b principle,

Equivalent rate of Change = 4 + 4 + 4 × 4xz/100

                                     = 8 + 16/100

                                     = 8 + 0.16 %

                                     = 8.16 % (Ans).

E.g. 4: Find the compound interest for a sum of Rs. 9000 in a year if the rate of interest is 10 % compounded half yearly

Equivalent Percentage Change = 10 + 10 + (10 × 10)/100

                                                   = 21 %

Therefore,

Compound Interest = 21 % of 9000

                                = 21/100 × 9000

                                = 21 × 90

                                = Rs. 1890 (Ans).