2.3 Ratios, Proportions & Percentages

Ratios, Proportions & Unit Rate Methods

Core Concept: A ratio is a comparative expression matching two or more quantities of the same kind measured in identical units. A proportion is an equation stating that two ratios are equal ($a:b = c:d$).

1. Ratios and Simplification

A ratio comparing quantity $A$ to quantity $B$ is written as $A : B$, $\frac{A}{B}$, or '$A$ to $B$'. Because ratios represent relative comparisons, multiplying or dividing both terms by the same non-zero number produces an equivalent ratio.

Rules for Simplifying Ratios:

1. Ensure all terms are expressed in the same unit of measurement before simplifying.
2. Divide all terms by their Highest Common Factor (HCF).
3. If terms are fractions, multiply all terms by the Least Common Multiple (LCM) of their denominators to obtain whole number terms.

Worked Examples: Ratio Simplification

• Example 1 (Basic): Simplify $24 : 36$.
  • Find $\text{HCF}(24, 36) = 12$.
  • Divide both terms by $12$: $\frac{24}{12} : \frac{36}{12} = \mathbf{2 : 3}$.
• Example 2 (Unit Conversion): Simplify $45 \text{ minutes} : 2 \text{ hours}$.
  • Convert hours to minutes: $2 \text{ hours} = 2 \times 60 = 120 \text{ minutes}$.
  • Ratio: $45 : 120$.
  • Divide by $\text{HCF}(45, 120) = 15$: $\frac{45}{15} : \frac{120}{15} = \mathbf{3 : 8}$.
• Example 3 (Fractional Terms): Simplify $\frac{1}{2} : \frac{3}{4}$.
  • Multiply both terms by $\text{LCM}(2, 4) = 4$: $(\frac{1}{2} \times 4) : (\frac{3}{4} \times 4) = \mathbf{2 : 3}$.

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2. Dividing Quantities in a Given Ratio

To divide a total quantity $Q$ into a ratio $a : b : c$, follow this reliable 3-step algorithm:

3-Step Ratio Division Algorithm:

1. Calculate Total Ratio Parts: $\text{Total Parts} = a + b + c$.
2. Determine Value per Part: $\text{Value of 1 Part} = \frac{Q}{\text{Total Parts}}$.
3. Calculate Individual Shares: Multiply the value of 1 part by each ratio term ($a \times \text{Value}$, $b \times \text{Value}$, $c \times \text{Value}$).

Comprehensive Worked Example

Problem: Share $\text{\textdollar}450$ among Sarah, Marcus, and Devin in the ratio $2 : 3 : 4$.

Solution:

• Step 1: Calculate total ratio parts: $2 + 3 + 4 = 9 \text{ parts}$.
• Step 2: Find the value of $1$ part: $\text{Value of 1 Part} = \frac{\text{\textdollar}450}{9} = \text{\textdollar}50$.
• Step 3: Calculate individual shares:
  • Sarah's share ($2$ parts) $= 2 \times \text{\textdollar}50 = \mathbf{\text{\textdollar}100}$.
  • Marcus's share ($3$ parts) $= 3 \times \text{\textdollar}50 = \mathbf{\text{\textdollar}150}$.
  • Devin's share ($4$ parts) $= 4 \times \text{\textdollar}50 = \mathbf{\text{\textdollar}200}$.
• Verification Check: $\text{\textdollar}100 + \text{\textdollar}150 + \text{\textdollar}200 = \text{\textdollar}450$.

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3. Direct Proportion and the Unit Rate (Unitary) Method

Two variables are in direct proportion when an increase in one variable causes a proportional increase in the other, keeping their ratio constant ($\frac{y}{x} = k$).

The Unitary Method Framework:

1. Find Unit Rate: Divide the given total cost/amount by the given quantity to find the rate for $1$ single unit.
2. Scale to Target: Multiply the single unit rate by the desired target quantity.

Worked Example: Unit Rate Calculation

Problem: If $8$ exercise books cost $\text{\textdollar}28.00$, determine the cost of $14$ exercise books.

• Step 1 (Unit Rate): Cost of $1$ book $= \frac{\text{\textdollar}28.00}{8} = \text{\textdollar}3.50$.
• Step 2 (Target Cost): Cost of $14$ books $= 14 \times \text{\textdollar}3.50 = \mathbf{\text{\textdollar}49.00}$.

Percentages & Commercial Arithmetic

1. Percentage Principles & Percentage Change

A percentage is a fraction expressed with a fixed denominator of $100$ (symbolised by $\%$).

Basic Formulas:

• Percentage of a Quantity: $\text{Value} = \frac{\text{Percentage}}{100} \times \text{Total Quantity}$.
• Expressing $A$ as a Percentage of $B$: $\text{Percentage} = \frac{A}{B} \times 100\%$.

Percentage Increase and Percentage Decrease

To measure percentage change relative to an original starting amount:

$$\text{Percentage Change} = \frac{\text{Absolute Amount of Change}}{\text{Original Amount}} \times 100\%$$

Worked Examples

• Example 1 (Percentage Increase): A store increases the price of a bag from $\text{\textdollar}80$ to $\text{\textdollar}96$. Find the percentage increase.
  • Increase $= \text{\textdollar}96 - \text{\textdollar}80 = \text{\textdollar}16$.
  • Percentage Increase $= \frac{\text{\textdollar}16}{\text{\textdollar}80} \times 100\% = \frac{1}{5} \times 100\% = \mathbf{20\%}$.
• Example 2 (Percentage Decrease): School enrolment drops from $500$ students to $425$ students. Find the percentage decrease.
  • Decrease $= 500 - 425 = 75$.
  • Percentage Decrease $= \frac{75}{500} \times 100\% = \frac{15}{100} \times 100\% = \mathbf{15\%}$.

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2. Commercial Math: Profit, Loss, Discount & Simple Interest

Commercial arithmetic applies percentage rules to business transactions and monetary management.

Commercial Definitions and Formulas Table

Step-by-Step Worked Commercial Problems

Problem 1 (Profit Percentage): A merchant purchases a bicycle for $\text{\textdollar}150.00$ and sells it for $\text{\textdollar}195.00$. Calculate his percentage profit.

• Profit $= \text{SP} - \text{CP} = \text{\textdollar}195.00 - \text{\textdollar}150.00 = \text{\textdollar}45.00$.
• Profit $\% = \frac{\text{\textdollar}45.00}{\text{\textdollar}150.00} \times 100\% = 0.30 \times 100\% = \mathbf{30\%}$.

Problem 2 (Discount & Sale Price): A pair of shoes marked at $\text{\textdollar}120.00$ is offered at a $15\%$ discount during a clearance sale. Calculate the discount amount and the final sale price.

• Discount Amount $= \frac{15}{100} \times \text{\textdollar}120.00 = 0.15 \times 120 = \text{\textdollar}18.00$.
• Sale Price $= \text{MP} - \text{Discount} = \text{\textdollar}120.00 - \text{\textdollar}18.00 = \mathbf{\text{\textdollar}102.00}$.

Problem 3 (Simple Interest & Total Amount): Calculate the simple interest and total accumulated amount when $\text{\textdollar}2,400.00$ is deposited into a savings account at a rate of $5\%$ per annum for $3$ years.

• Identify variables: Principal $P = 2400$, Rate $R = 5$, Time $T = 3$.
• Simple Interest $I = \frac{P \times R \times T}{100} = \frac{2400 \times 5 \times 3}{100} = 24 \times 15 = \mathbf{\text{\textdollar}360.00}$.
• Total Accumulated Amount $A = P + I = \text{\textdollar}2,400.00 + \text{\textdollar}360.00 = \mathbf{\text{\textdollar}2,760.00}$.