Comparing Quantities: Ratios and Proportions

• Comparing quantities involves methods to determine the relationships between two or more quantities.
• Ratios and percentages are fundamental tools used for comparison.

Ratios

• A ratio compares two quantities, indicating how much of one quantity there is compared to another.
• Ratios can be expressed in several ways, such as fractions, decimals, or using a colon.
• The ratio of a quantity 'a' to a quantity 'b' is written as a:b, where 'a' is the antecedent and 'b' is the consequent.
• Ratios can only be compared if the quantities being compared have the same units; convert to the same unit if necessary.
• Equivalent ratios represent the same comparison, achieved by multiplying or dividing both terms by the same non-zero number.
• To find the simplest form of a ratio, divide both terms by their highest common factor (HCF).

Proportions

• A proportion states that two ratios are equal.
• If a:b = c:d, then a, b, c, and d are said to be in proportion.
• In the proportion a:b = c:d, 'a' and 'd' are called extremes, while 'b' and 'c' are called means.
• The fundamental property of proportion: if a:b = c:d, then ad = bc (product of means equals product of extremes).
• This property is used to solve problems involving proportions, such as finding a missing term.

Percentages

• A percentage expresses a number as a fraction of 100, denoted by the symbol '%'.
• Percentages are used to compare quantities and express proportions in a standardized way.
• To convert a fraction or a decimal to a percentage, multiply it by 100.
• To convert a percentage to a fraction or decimal, divide it by 100.
• Percentages can calculate percentage increase or decrease, the relative change in quantity expressed as a percentage.

Percentage Increase

• Calculated as: ((New Value - Original Value) / Original Value) * 100.
• Indicates how much a quantity has grown relative to its initial amount.

Percentage Decrease

• Calculated as: ((Original Value - New Value) / Original Value) * 100.
• Indicates how much a quantity has reduced relative to its initial amount.

Applications of Percentages

• Percentages are widely used in real-life scenarios including calculating profit and loss, discounts, interest rates, and taxes.
• Profit is the gain made after selling a product for more than its cost price.
• Loss is the shortfall incurred when selling a product for less than its cost price.

Calculating Profit and Loss

• Profit = Selling Price (SP) - Cost Price (CP), when SP > CP.
• Loss = Cost Price (CP) - Selling Price (SP), when SP < CP.
• Profit Percentage = (Profit / CP) * 100.
• Loss Percentage = (Loss / CP) * 100.

Discounts

• A discount is a reduction in the marked price of an item.
• Discount = Marked Price (MP) - Selling Price (SP).
• Discount Percentage = (Discount / MP) * 100.

Simple Interest

• Simple interest is a method of calculating interest on a principal amount.
• Simple Interest (SI) = (P * R * T) / 100, where P is the principal amount, R is the rate of interest per annum, and T is the time in years.
• Amount (A) after T years is given by A = P + SI.

Compound Interest

• Compound interest is calculated on the principal and accumulated interest from previous periods.
• Amount (A) after n years = P(1 + R/100)^n, where P is the principal, R is the rate of interest per annum, and n is the number of years.
• Compound Interest (CI) = A - P.
• When interest is compounded half-yearly, the rate is halved, and the time is doubled.
• When interest is compounded quarterly, the rate is divided by four, and the time is multiplied by four.

Conversions

• Quantities are compared by converting them into different units for a better understanding.
• Units of length, mass, volume, and time are converted using appropriate conversion factors.
• For example, kilometers to meters, or kilograms to grams to facilitate comparison.

Using Ratios to Divide Quantities

• A ratio can divide a quantity into parts.
• If a quantity is divided in the ratio a:b, the first part is (a / (a+b)) * Total Quantity, and the second part is (b / (a+b)) * Total Quantity.
• This concept is often used in problems involving sharing or distributing items in a specific proportion.

Comparing Quantities with Different Units

• When comparing quantities with different units, convert them into a common unit.
• To compare speeds, convert both to either km/h or m/s.
• Understanding unit conversions is crucial for accurate comparisons.

Problems Involving Combined Concepts

• Many problems combine ratios, percentages, and proportions.
• These problems may require multiple steps to solve.
• Understanding the underlying concepts and applying them methodically is crucial for solving such problems.