Simple Interest - Key Formulas and Examples - Part Three

Question

Part 1 Part 2

In this third part of our series on simple interest, we continue to explore the remaining shortcut methods available to simplify and accelerate calculations. As you may recall, simple interest is calculated using the formula:

SI=P&times;R&times;T100, where  SI is the interest, P is the principal amount, R is the rate of interest, and T  is the time in years.

While this formula is pretty straightforward, repetitive calculations, especially with larger values or multiple terms, can become time-consuming. Shortcut methods are designed to streamline these computations, making it easier to quickly determine interest without having to perform complex mathematical steps repeatedly.

These shortcut methods often involve applying certain mathematical tricks or formulas that reduce the number of steps required for calculating interest.

For instance, if the time period or the rate of interest is constant across multiple problems, proportional methods or approximations can be used to obtain results much faster.

These techniques are particularly useful in competitive exams or in practical situations where time efficiency is crucial.

In this section, we will introduce some of these methods and provide examples to help you understand how to apply them effectively.

We would look at the following shortcuts

How to effect changes in P, R, and T on Simple Interest

Change in Rate R: 
		
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Change in Principal P
		
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Change in Rate R and Time T:

How to Borrowed Amount with Instalments

How to Determine Principal and Rate from Two Amounts

A. Effect of Changes in P, R, and T on Simple Interest

The effect of changing the parameters P (principal), R (rate), and T (time) on simple interest can be determined using the formula:

Change in Simple Interest = Fixed Paramete * Difference in Product of Variable Parameters

A1. Change in Rate R:

If the rate changes from R1 to R2 while keeping P and T fixed, the change in simple interest is:

&Delta;SI=P&times;T&times;( R 2 &minus; R 1 )/100

A2. Change in Principal P:

If the principal changes from P1 to P2 while keeping R and T fixed, the change in simple interest is:
&Delta;SI=R&times;T&times;( P 2 &minus; P 1 )/100

A3. Change in Rate R and Time T:

If both the rate and time change from R1, T1\) to R2, T2 while keeping P fixed, the change in simple interest is:
&Delta;SI= P 100 &times;( R 2 T 2 &minus; R 1 T 1 )

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Example A1:

Find Time from Rate Change

Given:
- Principal P = 600
- Change in simple interest $\Delta \text{SI} = 30$
- Change in rate R2 - R1 = 4%
- Time T = ?

Using the formula:

&Delta;SI= P&times;T&times;( R 2 &minus; R 1 ) 100

Substitute the values:

30= 600&times;T&times;4 100

Solving for $T$:

T= 30&times;100 600&times;4 =1.25 years

Thus, the time is 1.25 years.

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Example A2

Rate of Interest from a Difference in Principal

Given:
- Difference in simple interest $\Delta \text{SI} = 60$
- Difference in principal P2 - P1 = 400
- Time T = 5 years
- Rate R = ?

Using the formula:

&Delta;SI= R&times;T&times;( P 2 &minus; P 1 ) 100

Substitute the values:

60= R&times;5&times;400 100

Solving for R:

R= 60&times;100 5&times;400 =3%

Thus, the rate of interest is 3% per annum.

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Example A3

Finding the Principal from a Difference in Interest Rates and Time

Given:
- Change in simple interest &Delta;SI = 80
- R1 = 4%\, T1 = 4 years
- R2 = 5%, T2 = 3 years
- Principal P = ?

Using the formula:

&Delta;SI= P 100 &times;( R 1 T 1 &minus; R 2 T 2 )

Substitute the values:

80= P 100 &times;(4&times;4&minus;5&times;3)

Simplifying:

80= P 100 &times;(16&minus;15)= P 100

Solving for $P$:

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P = 80 * 100 = 8,000

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Thus, the principal is $8,000.

More Examples on Simple Interest

B. Borrowed Amount with Instalments

If a debt Z is paid in $n$ equal instalments, and the value of each instalment is $a$, then the borrowed amount is given by:

Z=na+ Ra 100&times;b &times; n(n&minus;1) 2

Where:
- R = rate of interest per annum
- b = number of instalments per year

Example B

Finding the Rate for Instalment Payments

Given:
- Borrowed amount Z = 2
- Instalment a = 1
- Number of instalments n = 3
- Instalments are paid monthly, so b = 12

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Using the formula:

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Z=na+ Ra 100&times;b &times; n(n&minus;1) 2

Substitute the values:

2=3&times;1+ R&times;1 100&times;12 &times; 3&times;(3&minus;1) 2

Simplifying:

2=3+ R 100&times;12 &times;3

Solving for $R$:

2&minus;3= 3R 1200
&minus;1= R 400 &emsp;&rArr;&emsp;R=400%

Thus, the rate of interest is 400% per annum.

More Examples on Simple Interest

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C. Determining Principal and Rate from Two Amounts

If a sum of money lent at simple interest amounts to A1 in T1 years and to A2in T2 years, then:

P= A 1 T 2 &minus; A 2 T 1 T 2 &minus; T 1
R= ( A 1 &minus; A 2 )&times;100 P&times;( T 2 &minus; T 1 )

Example C

Finding Principal and Rate from Two Amounts

Given:
- A1 = 5,184  in 2 years
-  A2 = 5,832  in 3 years

We want to find the principal P and the rate of interest R.

Step 1: Finding P (Principal)

Using the formula:

P= A 1 T 2 &minus; A 2 T 1 T 2 &minus; T 1

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Substitute the values:

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P= 5,184&times;3&minus;5,832&times;2 3&minus;2

Simplify:

P= 15,552&minus;11,664 1 =3,888

Thus, the principal P is $3,888.

Step 2: Finding $R$ (Rate of Interest)

Now, we calculate the rate of interest $R$ using the formula:

R= ( A 2 &minus; A 1 )&times;100 P&times;( T 2 &minus; T 1 )

Substitute the values:

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R= (5,832&minus;5,184)&times;100 3,888&times;(3&minus;2)

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Simplify:

R= 648&times;100 3,888&times;1 = 64,800 3,888 &asymp;16.67%

Thus, the rate of interest R is approximately 16.67% per annum.

Final Answer:
- The principal P is $3,888.
- The rate of interest R is approximately 16.67% per annum.

More Examples on Simple Interest

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Exercises

Find Time from Rate Change, given: Principal P = 600, Change in simple interest &Delta;SI = 30, Change in rate R2 - R1 = 4%.
	Find the Rate of Interest from a Difference in Principal, Given:  Difference in simple interest &Delta;SI = 60, - Difference in principal P2 - P1 = 400, Time T = 5 years 
	Findthe Principal from a Difference in Interest Rates and Time, Given: Change in simple interest &Delta;SI = 80, R1 = 4%\, T1 = 4 years, R2 = 5%, T2 = 3 years 
	Find the Rate for Instalment Payments. Given: Borrowed amount Z = 2, Instalment a = 1, Number of instalments n = 3, Instalments are paid monthly.
	Determine Principal and Rate from Two Amounts Given: A1 = 21,500 in 4.5 years, A2 = 82,900  in 7 years

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Part 2

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