Simple Interest

Simple Interest (SI) is a method of calculating the interest charged on a loan. It is calculated on the principal amount, or on that portion of the principal amount that remains unpaid at each additional time that payment is due. In simple interest, the interest amount is the same for every year or time period. It is a straightforward way to calculate interest, making it easy to understand and compute.

Key Concepts and Formulas

The calculation of simple interest involves four main components: Principal, Rate of Interest, Time Period, and the Simple Interest itself.

• Principal (P): This is the initial sum of money borrowed or invested.
• Rate of Interest (R): This is the percentage of the principal charged as interest per annum. It is usually expressed as a percentage.
• Time (T): This is the duration for which the money is borrowed or invested. It is usually expressed in years. If the time is given in months, it needs to be converted into years by dividing by 12.
• Simple Interest (SI): This is the amount of interest earned or paid over the time period.
• Amount (A): This is the total sum of money after the interest is added to the principal.

The fundamental formula for calculating Simple Interest is:

$$ SI = \frac{P \times R \times T}{100} $$

Where:

• P = Principal amount
• R = Rate of interest per annum (in %)
• T = Time period (in years)

The Amount (A) after T years is calculated as:

$$ A = P + SI $$

Alternatively, the Amount can be calculated directly using:

$$ A = P \left(1 + \frac{R \times T}{100}\right) $$

Calculating Different Variables

Sometimes, you might be given the SI, A, R, or T and asked to find the Principal or Rate or Time. The formulas can be rearranged accordingly:

• To find Principal (P): $$ P = \frac{SI \times 100}{R \times T} $$
• To find Rate (R): $$ R = \frac{SI \times 100}{P \times T} $$
• To find Time (T): $$ T = \frac{SI \times 100}{P \times R} $$

Examples

Example 1: Calculate the Simple Interest on ₹5,000 for 3 years at an annual interest rate of 8%.

Here, P = ₹5,000, R = 8%, T = 3 years.

$$ SI = \frac{5000 \times 8 \times 3}{100} = 50 \times 8 \times 3 = 1200 $$

So, the Simple Interest is ₹1,200. The total amount will be ₹5,000 + ₹1,200 = ₹6,200.

Example 2: In what time will ₹1,000 amount to ₹1,200 at a simple interest rate of 5% per annum?

Here, P = ₹1,000, A = ₹1,200, R = 5%.

First, calculate the SI: SI = A - P = ₹1,200 - ₹1,000 = ₹200.

Now, use the formula for T:

$$ T = \frac{SI \times 100}{P \times R} = \frac{200 \times 100}{1000 \times 5} = \frac{20000}{5000} = 4 $$

So, the time required is 4 years.

Important Considerations for Simple Interest

When the interest is compounded annually, the interest earned each year is added to the principal for the next year's calculation. However, in simple interest, the interest is always calculated on the original principal amount. This means the interest earned is constant over the period.

Time in Months: If the time is given in months, convert it to years by dividing by 12. For example, 6 months is 6/12 = 0.5 years.

Rate Changes: If the rate of interest changes over different periods, you calculate the SI for each period separately and then sum them up.

Difference between SI and CI: For the same principal, rate, and time, Compound Interest (CI) will always be greater than Simple Interest (SI) for any period greater than one year. The difference between CI and SI for 2 years is calculated on the first year's interest.

Compound Interest

Compound Interest (CI) is the interest calculated on the initial principal amount and also on the accumulated interest of previous periods. In simpler terms, it's "interest on interest." This concept is fundamental in finance and economics as it explains how investments grow over time. The power of compounding is often referred to as the "eighth wonder of the world."

Key Concepts and Formulas

Similar to simple interest, compound interest involves Principal, Rate, Time, and the Amount. The crucial difference lies in how interest is calculated and added.

• Principal (P): The initial sum of money borrowed or invested.
• Rate of Interest (R): The annual interest rate.
• Time (T): The duration for which the money is invested or borrowed, usually in years.
• Compounding Frequency: This indicates how often the interest is calculated and added to the principal. Common frequencies include annually, semi-annually, quarterly, and monthly.
• Amount (A): The total sum of money after interest is compounded.
• Compound Interest (CI): The total interest earned over the period.

Calculating Compound Interest

The formula for the amount (A) when interest is compounded annually is:

$$ A = P \left(1 + \frac{R}{100}\right)^T $$

And the Compound Interest (CI) is:

$$ CI = A - P $$ $$ CI = P \left(1 + \frac{R}{100}\right)^T - P $$ $$ CI = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right] $$

Compounding Frequency

When the interest is compounded more than once a year, the formula needs to be adjusted.

• Let 'n' be the number of times interest is compounded per year.
• The rate per compounding period becomes R/n.
• The total number of compounding periods becomes T * n.

The formula for Amount (A) with compounding frequency 'n' is:

$$ A = P \left(1 + \frac{R}{100n}\right)^{Tn} $$

And the Compound Interest (CI) is:

$$ CI = P \left[ \left(1 + \frac{R}{100n}\right)^{Tn} - 1 \right] $$

Examples

Example 1: Calculate the Compound Interest on ₹10,000 for 2 years at 10% per annum, compounded annually.

Here, P = ₹10,000, R = 10%, T = 2 years, n = 1 (annually).

$$ A = 10000 \left(1 + \frac{10}{100}\right)^2 = 10000 \left(1 + 0.1\right)^2 = 10000 (1.1)^2 = 10000 \times 1.21 = 12100 $$

$$ CI = A - P = 12100 - 10000 = 2100 $$

The Compound Interest is ₹2,100.

Example 2: Calculate the Compound Interest on ₹5,000 for 1 year at 8% per annum, compounded semi-annually.

Here, P = ₹5,000, R = 8%, T = 1 year. Since it's compounded semi-annually, n = 2.

Rate per period = R/n = 8%/2 = 4%

Number of periods = T*n = 1 * 2 = 2

$$ A = 5000 \left(1 + \frac{8}{100 \times 2}\right)^{1 \times 2} = 5000 \left(1 + \frac{8}{200}\right)^2 = 5000 \left(1 + 0.04\right)^2 $$ $$ A = 5000 (1.04)^2 = 5000 \times 1.0816 = 5408 $$

$$ CI = A - P = 5408 - 5000 = 408 $$

The Compound Interest is ₹408.

Difference between SI and CI

The difference between Compound Interest and Simple Interest is a crucial concept, especially for competitive exams.

• For 1 year: SI = CI
• For 2 years: Difference (CI - SI) = $$ P \left(\frac{R}{100}\right)^2 $$
• For 3 years: Difference (CI - SI) = $$ P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right) $$

Depreciation

Compound interest principles can also be applied to calculate depreciation, which is the decrease in value of an asset over time. The formula is similar, but the rate is subtracted instead of added, and it represents a decrease in value.

Value after depreciation = $$ P \left(1 - \frac{R}{100}\right)^T $$

Ratio and Proportion

Ratio and Proportion are fundamental mathematical concepts used to compare quantities and establish relationships between them. They are widely used in various fields, including mathematics, science, finance, and everyday life. Understanding these concepts is crucial for solving many problems in quantitative aptitude.

Ratio

A ratio is a comparison of two or more quantities of the same kind by division. It expresses how many times one quantity contains another. A ratio is usually expressed in its simplest form.

If 'a' and 'b' are two quantities, their ratio is written as 'a : b' or $$ \frac{a}{b} $$.

Terms of a Ratio: In a ratio a : b, 'a' is called the antecedent (the first term) and 'b' is called the consequent (the second term).

Types of Ratios:

• Continued Ratio: A ratio involving three or more quantities, like a : b : c.
• Duplicate Ratio: The ratio of the squares of the terms, i.e., $$ a^2 : b^2 $$.
• Sub-duplicate Ratio: The ratio of the square roots of the terms, i.e., $$ \sqrt{a} : \sqrt{b} $$.
• Triplicate Ratio: The ratio of the cubes of the terms, i.e., $$ a^3 : b^3 $$.
• Sub-triplicate Ratio: The ratio of the cube roots of the terms, i.e., $$ \sqrt[3]{a} : \sqrt[3]{b} $$.
• Ratio of Equality: When the antecedent and consequent are equal (e.g., 5 : 5, which simplifies to 1 : 1).
• Ratio of Inequality: When the antecedent and consequent are not equal.

Proportion

A proportion is an equation stating that two ratios are equal. If the ratios a : b and c : d are equal, then they are in proportion. This is written as a : b :: c : d.

This can also be written as $$ \frac{a}{b} = \frac{c}{d} $$.

Terms of a Proportion: In a proportion a : b :: c : d,

• 'a' and 'd' are called the extreme terms (or extremes).
• 'b' and 'c' are called the mean terms (or means).

The fundamental property of proportion is that the product of the means is equal to the product of the extremes:

$$ a \times d = b \times c $$

Types of Proportion

• Direct Proportion: Two quantities are said to be in direct proportion if they increase or decrease together in the same ratio. If x is directly proportional to y, then $$ x \propto y $$ or $$ x = ky $$, where k is a constant. This means $$ \frac{x_1}{y_1} = \frac{x_2}{y_2} $$.
• Inverse Proportion: Two quantities are said to be in inverse proportion if, when one quantity increases, the other quantity decreases in the same ratio, and vice versa. If x is inversely proportional to y, then $$ x \propto \frac{1}{y} $$ or $$ xy = k $$, where k is a constant. This means $$ x_1 y_1 = x_2 y_2 $$.

Continued Proportion

Three quantities a, b, and c are said to be in continued proportion if $$ a : b = b : c $$. This can be written as $$ \frac{a}{b} = \frac{b}{c} $$, which implies $$ b^2 = ac $$. Here, 'b' is called the mean proportional between 'a' and 'c'.

Examples

Example 1: The ratio of two numbers is 3 : 5. If the sum of the numbers is 80, find the numbers.

Let the numbers be 3x and 5x.

Sum = 3x + 5x = 8x

Given sum = 80.

So, 8x = 80, which means x = 10.

The numbers are 3 * 10 = 30 and 5 * 10 = 50.

Example 2: If 12 : 18 :: x : 24, find the value of x.

Using the property of proportion (product of means = product of extremes):

$$ 18 \times x = 12 \times 24 $$ $$ 18x = 288 $$ $$ x = \frac{288}{18} $$ $$ x = 16 $$

Example 3: The cost of 5 kg of wheat is ₹40. What is the cost of 12 kg of wheat?

This is a problem of direct proportion. As the quantity of wheat increases, the cost also increases proportionally.

Let the cost of 12 kg of wheat be ₹y.

$$ \frac{\text{Cost}_1}{\text{Quantity}_1} = \frac{\text{Cost}_2}{\text{Quantity}_2} $$ $$ \frac{40}{5} = \frac{y}{12} $$ $$ 8 = \frac{y}{12} $$ $$ y = 8 \times 12 = 96 $$

So, the cost of 12 kg of wheat is ₹96.

Example 4: If 10 men can complete a piece of work in 15 days, in how many days can 12 men complete the same work?

This is a problem of inverse proportion. As the number of men increases, the number of days required to complete the work decreases.

Let the number of days required by 12 men be 'd'.

$$ \text{Men}_1 \times \text{Days}_1 = \text{Men}_2 \times \text{Days}_2 $$ $$ 10 \times 15 = 12 \times d $$ $$ 150 = 12d $$ $$ d = \frac{150}{12} = 12.5 $$

So, 12 men can complete the work in 12.5 days.

Partnership Problems

Ratio and proportion are heavily used in partnership problems. When two or more partners invest money in a business, their profits are usually divided in the ratio of their investments, assuming they invest for the same duration. If the durations are different, the profit is divided in the ratio of the product of investment and time.

Profit Ratio = $$ (\text{Investment}_1 \times \text{Time}_1) : (\text{Investment}_2 \times \text{Time}_2) : \dots $$