Ratio and Proportion

Ratio is a comparison of two quantities of the same kind. It is expressed as a fraction or using a colon. For example, if we have 5 apples and 3 oranges, the ratio of apples to oranges is 5/3 or 5:3.

Proportion is a statement that two ratios are equal. If a/b = c/d, then a, b, c, and d are in proportion. This can also be written as a:b = c:d. In a proportion, the product of the means (the inner terms, b and c) is equal to the product of the extremes (the outer terms, a and d). So, ad = bc.

Types of Ratios

Continued Ratio: When three or more quantities are compared, like a:b:c.

Duplicate Ratio: The ratio of the squares of the terms, like a²:b².

Sub-duplicate Ratio: The ratio of the square roots of the terms, like √a:√b.

Triplicate Ratio: The ratio of the cubes of the terms, like a³:b³.

Sub-triplicate Ratio: The ratio of the cube roots of the terms, like ³√a:³√b.

Inverted Ratio: The ratio obtained by interchanging the antecedent (first term) and the consequent (second term), like b:a.

Properties of Ratios

• The ratio a:b is equal to ka:kb for any non-zero k.
• If a:b = c:d, then a+b/b = c+d/d (Componendo).
• If a:b = c:d, then a-b/b = c-d/d (Dividendo).
• If a:b = c:d, then a+b/a-b = c+d/c-d (Componendo and Dividendo).
• If a:b = c:d, then b:a = d:c (Invertendo).
• If a:b = c:d, then a:c = b:d (Alternendo).
• If a:b = c:d, then ad = bc (Cross-multiplication).

Applications of Ratio and Proportion

Ratio and proportion are fundamental concepts used in many real-world scenarios, including:

• Sharing quantities: Dividing money, property, or any other quantity among people in a given ratio.
• Ages: Calculating ages of people based on given ratios at different times.
• Partnership: Distributing profits or losses among partners based on their investment ratios.
• Mixtures and Alligations: Determining the proportions of different ingredients in a mixture.
• Map scales: Representing distances on a map using a ratio.

Example: Sharing Profit

A, B, and C invest ₹50,000, ₹75,000, and ₹1,00,000 respectively in a business. If the total profit is ₹2,00,000, find the share of each partner.

First, find the ratio of their investments: 50,000 : 75,000 : 1,00,000 Divide by 25,000: 2 : 3 : 4

The sum of the ratio parts is 2 + 3 + 4 = 9.

A's share = (2/9) * ₹2,00,000 = ₹44,444.44 (approx.) B's share = (3/9) * ₹2,00,000 = ₹66,666.67 (approx.) C's share = (4/9) * ₹2,00,000 = ₹88,888.89 (approx.)

Simple Interest

Simple Interest (SI) is a method of calculating the interest charged on a sum of money (principal) for a given period. The interest is calculated on the original principal amount only, and it remains constant for the entire duration of the loan or investment.

The formula for Simple Interest is: SI = (P * R * T) / 100

Where:

• P = Principal amount (the initial sum of money borrowed or invested)
• R = Rate of interest per annum (in percentage)
• T = Time period (in years)

The Amount (A) after T years is the sum of the Principal and the Simple Interest: A = P + SI A = P + (P * R * T) / 100 A = P * (1 + (R * T) / 100)

Key Terms

• Principal (P): The initial amount of money.
• Rate (R): The percentage of interest charged per year.
• Time (T): The duration for which the money is borrowed or invested, usually in years. If the time is given in months, convert it to years by dividing by 12. If in days, convert to years by dividing by 365 (or 366 for a leap year).
• Simple Interest (SI): The interest earned or paid on the principal amount.
• Amount (A): The total sum of money after adding the interest to the principal.

Example: Calculating Simple Interest

Calculate the simple interest on ₹5,000 at a rate of 8% per annum for 3 years. Also, find the total amount.

Given: P = ₹5,000 R = 8% per annum T = 3 years

SI = (P * R * T) / 100 SI = (5000 * 8 * 3) / 100 SI = (50 * 8 * 3) SI = 400 * 3 SI = ₹1,200

Amount (A) = P + SI A = ₹5,000 + ₹1,200 A = ₹6,200

Calculating Unknown Variables

You can rearrange the SI formula to find any of the variables if the others are known:

• To find Principal (P): P = (SI * 100) / (R * T)
• To find Rate (R): R = (SI * 100) / (P * T)
• To find Time (T): T = (SI * 100) / (P * R)

Example: Finding Time

In what time will ₹6,000 amount to ₹7,500 at 5% simple interest per annum?

Given: P = ₹6,000 A = ₹7,500 R = 5% per annum

First, find the Simple Interest (SI): SI = A - P SI = ₹7,500 - ₹6,000 SI = ₹1,500

Now, use the formula for T: T = (SI * 100) / (P * R) T = (1500 * 100) / (6000 * 5) T = 150000 / 30000 T = 5 years

Compound Interest

Compound Interest (CI) is the interest calculated on the initial principal and also on the accumulated interest from previous periods. In simpler terms, it's "interest on interest." This means the amount on which interest is calculated grows over time, leading to a higher final amount compared to simple interest for the same principal, rate, and time.

The formula for the Amount (A) with compound interest is: A = P * (1 + R/100)ⁿ

Where:

• A = The future value of the investment/loan, including interest.
• P = Principal amount (the initial sum of money).
• R = Annual interest rate (in percentage).
• n = Number of years the money is invested or borrowed for.

The Compound Interest (CI) is the difference between the Amount (A) and the Principal (P): CI = A - P CI = P * (1 + R/100)ⁿ - P CI = P * [(1 + R/100)ⁿ - 1]

Compounding Frequency

The interest can be compounded more frequently than annually. Common frequencies include:

• Half-yearly (or semi-annually): Interest is calculated twice a year. The rate becomes R/2, and the number of periods becomes 2n.
• Quarterly: Interest is calculated four times a year. The rate becomes R/4, and the number of periods becomes 4n.
• Monthly: Interest is calculated twelve times a year. The rate becomes R/12, and the number of periods becomes 12n.

The general formula for compound interest with compounding frequency 'k' times per year is: A = P * (1 + (R/k)/100)^nk CI = P * [(1 + (R/k)/100)^nk - 1]

Example: Calculating Compound Interest Annually

Calculate the compound interest on ₹10,000 at a rate of 5% per annum for 2 years.

Given: P = ₹10,000 R = 5% per annum n = 2 years

A = P * (1 + R/100)ⁿ A = 10000 * (1 + 5/100)² A = 10000 * (1 + 0.05)² A = 10000 * (1.05)² A = 10000 * 1.1025 A = ₹11,025

CI = A - P CI = ₹11,025 - ₹10,000 CI = ₹1,025

Example: Calculating Compound Interest Half-Yearly

Calculate the compound interest on ₹8,000 at a rate of 10% per annum, compounded half-yearly for 1 year.

Given: P = ₹8,000 R = 10% per annum n = 1 year Compounded half-yearly, so k = 2.

Rate per period = R/k = 10%/2 = 5% Number of periods = n*k = 1 * 2 = 2

A = P * (1 + (R/k)/100)^nk A = 8000 * (1 + 5/100)² A = 8000 * (1 + 0.05)² A = 8000 * (1.05)² A = 8000 * 1.1025 A = ₹8,820

CI = A - P CI = ₹8,820 - ₹8,000 CI = ₹820

Difference between CI and SI for 2 years

For 2 years, the difference between CI and SI can be calculated using a shortcut: Difference = P * (R/100)²

For 3 years: Difference = P * (R/100)² * (3 + R/100)

Example: Difference between CI and SI

Find the difference between the compound interest and simple interest on ₹10,000 for 2 years at 8% per annum.

P = ₹10,000 R = 8% n = 2 years

Using the shortcut: Difference = P * (R/100)² Difference = 10000 * (8/100)² Difference = 10000 * (0.08)² Difference = 10000 * 0.0064 Difference = ₹64

Area and Volume Calculations

This section covers the calculation of areas and volumes of various basic geometric shapes. Understanding these concepts is crucial for solving problems related to measurements, construction, and design.

Area Calculations

Area is the measure of the space occupied by a two-dimensional shape.

1. Rectangle

A rectangle is a quadrilateral with four right angles.

• Let length = l and breadth = b
• Area = l * b
• Perimeter = 2 * (l + b)

2. Square

A square is a rectangle with all sides equal.

• Let side = s
• Area = s²
• Perimeter = 4 * s

Also, Area = (diagonal)² / 2

3. Triangle

A triangle is a polygon with three sides and three angles.

• General Triangle: Area = (1/2) * base * height
• Equilateral Triangle: Side = a. Area = (√3 / 4) * a²
• Isosceles Triangle: Base = b, Equal sides = a. Height = √(a² - (b/2)²). Area = (1/2) * b * √(a² - (b/2)²)
• Heron's Formula (for any triangle with sides a, b, c):
• Semi-perimeter (s) = (a + b + c) / 2
• Area = √[s(s-a)(s-b)(s-c)]

4. Parallelogram

A quadrilateral with opposite sides parallel.

• Let base = b and height = h
• Area = b * h
• Perimeter = 2 * (sum of adjacent sides)

5. Rhombus

A parallelogram with all sides equal.

• Let diagonals be d1 and d2
• Area = (1/2) * d1 * d2
• Side (a) = √[(d1/2)² + (d2/2)²]
• Perimeter = 4 * a

6. Trapezium (or Trapezoid)

A quadrilateral with one pair of parallel sides.

• Let parallel sides be a and b, and height = h
• Area = (1/2) * (a + b) * h

7. Circle

A set of points equidistant from a central point.

• Let radius = r
• Area = π * r²
• Circumference = 2 * π * r
• Diameter (d) = 2 * r
• Circumference = π * d

(Note: π (pi) is approximately 22/7 or 3.14159)

Example: Area of a Circular Field

Find the area of a circular field with a radius of 7 meters.

Given: r = 7 meters π = 22/7

Area = π * r² Area = (22/7) * (7)² Area = (22/7) * 49 Area = 22 * 7 Area = 154 square meters

Volume Calculations

Volume is the measure of the three-dimensional space occupied by a solid object.

1. Cuboid

A rectangular box.

• Let length = l, breadth = b, height = h
• Volume = l * b * h
• Surface Area = 2 * (lb + bh + hl)

2. Cube

A cuboid with all sides equal.

• Let side = a
• Volume = a³
• Surface Area = 6 * a²

3. Cylinder

A solid with two parallel circular bases connected by a curved surface.

• Let radius = r, height = h
• Volume = π * r² * h
• Curved Surface Area (CSA) = 2 * π * r * h
• Total Surface Area (TSA) = 2 * π * r * h + 2 * π * r² = 2 * π * r * (h + r)

4. Cone

A solid with a circular base and a vertex.

• Let radius of base = r, height = h, slant height = l
• Relationship: l² = r² + h²
• Volume = (1/3) * π * r² * h
• Curved Surface Area (CSA) = π * r * l
• Total Surface Area (TSA) = π * r * l + π * r² = π * r * (l + r)

5. Sphere

A perfectly round geometrical object in three-dimensional space.

• Let radius = r
• Volume = (4/3) * π * r³
• Surface Area = 4 * π * r²

6. Hemisphere

Half of a sphere.

• Let radius = r
• Volume = (2/3) * π * r³
• Curved Surface Area (CSA) = 2 * π * r²
• Total Surface Area (TSA) = 2 * π * r² (curved) + π * r² (base) = 3 * π * r²

Example: Volume of a Cylindrical Tank

Find the volume of a cylindrical tank with a radius of 3 meters and a height of 7 meters.

Given: r = 3 meters h = 7 meters π = 22/7

Volume = π * r² * h Volume = (22/7) * (3)² * 7 Volume = (22/7) * 9 * 7 Volume = 22 * 9 Volume = 198 cubic meters