Ratios, Rates, Algebra, and Word Problems

• These notes cover ratios, rates, algebra, and word problems.

Ratios

• To simplify, divide each term by their highest common factor (HCF).
• Example: 6:9 simplifies to 2:3 by dividing both terms by 3.

Rates & Proportions

• Hourly Rate = Total money ÷ Hours worked
• Example: If someone earns $180 for 4 hours, their hourly rate is $45.
• To find earnings for a different number of hours, multiply the hourly rate by the new number of hours.
• Example: Earning for 6 hours can be calculated by $45 x 6 = $270

Equivalent Forms of a Fraction

• Fractions can be expressed as decimals or percentages.
• Example: 1/2 = 0.5 = 50%

Expanding Brackets

• Distribute multiplication over addition.
• Example: 2(x + 3) = 2x + 6

Speed Formula

• Speed = Distance ÷ Time
• Convert minutes to hours when needed for calculations.
• Example: 45 km in 90 minutes is 45 km in 1.5 hours, so Speed = 45 ÷ 1.5 = 30 km/h

Algebra Basics: Key Terms

• Pronumeral: A letter representing an unknown number (e.g., x, y).
• Substituting: Replacing the pronumeral with a known value.
• Factorised Form: Writing an expression as a product of factors.
• Expanded Form: Removing brackets.

Algebraic Expressions

• Triple the sum of m and n is expressed as 3(m + n).
• Double y, then subtract 5 is expressed as 2y - 5.
• 12 less than x is expressed as x - 12.

Evaluating Expressions

• 2a + b², given a=3 and b=4 is 2(3) + 4² = 6 + 16 = 22

Like Terms

• Terms can be combined if they have the same variables and exponents.
• Example: 3x + 5x = 8x, but 3x + 4y cannot be combined.

Simplifying Expressions

• 5x - 4 stays as is.
• 7a - 4b stays as is.

Factorising

• Take out the common factor.
• Example: 2(2x + 3)

Expanding and Simplifying

• Multiply out the brackets.
• Example: 3(3y - 1) = 9y - 3

Application: Feeding Formula

• Formula: F = 3c + 2d + 5b
• Substituting values: If c=6, d=3, b=10, then F = 3(6) + 2(3) + 5(10) = 18 + 6 + 50 = 74

Application: Theme Park Cost

• Expression: 50n + 10, where n is the number of people
• If 4 friends share the cost the Total cost = 50(4) + 10 = 200 + 10 = 210
• Each friend pays: 210 ÷ 4 = $52.50

Application: Gym Membership

• Expression: 200 + 25x, where x is the number of months
• Gym Membership: If x=6 months: 200 + 25(6) = 200 + 150 = $350

Study Plan

• Read notes and highlight formulas (10 min).
• Solve one example from each section (10 min).
• Write down key formulas on a flashcard (15 min):
  • Speed = Distance ÷ Time
  • Expand: a(b + c) = ab + ac
  • Factorise: Take out common factors
• Practice 5 multiple-choice questions & 3 algebra problems (15 min).
• Test yourself: Cover the notes & recall formulas (10 min).

Explore ratios, rates, and basic algebra concepts. Learn to simplify ratios, calculate hourly rates, and convert fractions. Understand expanding brackets and the speed formula with practical examples.