Forming Equations Quizzes
Visual overview of Forming Equations.
Introduction
Forming equations bridges real-life problems and algebraic solutions. It means translating words, situations, or relationships into algebraic expressions and equations. Once the equation is formed, we can solve it to find unknown values. Mastering this skill builds confidence with worded problems across algebra, geometry, finance, and science.
Example: “A number is increased by 7 to give 15” → \(x+7=15\) → \(x=8\).
Core Concepts
Understanding the Problem
Read carefully and identify:
- The unknown (variable).
- The operations applied ( +, −, ×, ÷ ).
- Relationships like total, difference, product, ratio, percentage.
Choosing a Variable
Pick a clear symbol (often \(x\), \(y\), or \(n\)) for the unknown.
- “A number increased by 5 is 12” → let \(x\) be the number → \(x+5=12\).
Translating Words into Algebra
- sum → \(+\), difference → \(−\), product → \(×\), quotient → \(÷\)
- is, equals → \(=\)
Example
“Twice a number decreased by 3 is 11” → let \(x\) be the number → \(2x-3=11\).
Multi-Step Situations
Example
“A number is multiplied by 3, then increased by 7 to give 19.”
- Let \(x\) be the number → \(3x+7=19\) → \(x=4\).
Fractions and Parts
Example
“A number is one–third of 15.” → \(x=\tfrac{1}{3}\times15=5\).
Ratios
Represent parts as multiples of a common unit.
Example
“Boys:Girls = \(3:2\). Total \(=25\).” Let boys \(=3x\), girls \(=2x\).
- \(3x+2x=25\Rightarrow 5x=25\Rightarrow x=5\)
- Boys \(=15\), Girls \(=10\).
Percentages
Convert % to decimals or fractions of 100.
- “20% of a number is 50” → \(0.2x=50\Rightarrow x=250\).
Checking Your Equation
Solve, then substitute back into the wording to verify it fits the story.
- “Number plus 7 equals 15” → \(x=8\). Check: \(8+7=15\) ✓
Real-Life Applications
- Finance: profit, cost, interest.
- Geometry: lengths, areas, volumes.
- Science: mixtures, scaling, unit conversion.
- Everyday: age, distance–time, rates.
Worked Examples
Example 1 (Foundation): One-step
“A number increased by 5 is 12.” → \(x+5=12\Rightarrow x=7\).
Example 2 (Foundation): Two-step
“Twice a number minus 3 is 11.” → \(2x-3=11\Rightarrow x=7\).
Example 3 (Higher): Multiple operations
“A number ×3, then +7 gives 19.” → \(3x+7=19\Rightarrow x=4\).
Example 4 (Higher): Fractions
“A number is one-fourth of 20.” → \(x=\tfrac{1}{4}\times20=5\).
Example 5 (Higher): Ratios
“Boys:Girls \(=3:2\). Total \(=25\).” → \(3x+2x=25\Rightarrow x=5\) → Boys \(=15\), Girls \(=10\).
Example 6 (Higher): Percentages
“20% of a number is 50.” → \(0.2x=50\Rightarrow x=250\).
Example 7 (Higher): Age problem
“John is 5 more than twice Mary’s age. John is 29.” → \(2x+5=29\Rightarrow x=12\) (Mary).
Example 8 (Higher): Distance/Speed
“A car travels at \(x\) km/h for 2 hours and covers 150 km. Find \(x\).” → \(2x=150\Rightarrow x=75\) km/h.
Example 9 (Higher): Mixture
“A solution has \(x\) L acid. Add 3 L water to make 10 L total.” → \(x+3=10\Rightarrow x=7\) L acid.
Example 10 (Higher): Combined operations
“Three times a number, minus 4, equals 11.” → \(3x-4=11\Rightarrow x=5\).
Common Mistakes
- Misreading the problem or defining the wrong variable.
- Translating words to algebra incorrectly.
- Forgetting to perform operations on both sides when solving.
- Errors with fractions, percentages, or units.
- Not checking the solution against the original context.
Applications
- Algebraic problem-solving and modelling.
- Finance (interest, profit, budgets).
- Geometry (unknown sides/angles via formulas).
- Physics (speed, distance, time; force and work).
- Everyday contexts (ages, mixtures, rates).
Strategies & Tips
- Underline key phrases (“twice”, “difference”, “of”, “total”).
- Define variables clearly before writing equations.
- Draw quick diagrams or tables for ratios and mixtures.
- Convert % to decimals before forming the equation.
- Finish by checking with substitution and units.
Summary / Call-to-Action
Forming equations turns words into maths. By defining variables, translating phrases accurately, and handling ratios, fractions, and percentages, you can model and solve a wide range of problems.
- Practise translating varied word problems to equations.
- Use diagrams for ratio/mixture contexts.
- Always verify the solution makes sense in context.