Rearranging Formulae Quizzes

Introduction

Rearranging formulae is a fundamental skill in GCSE Maths. It involves manipulating a formula to make a different variable the subject. This skill is essential for algebra, physics, geometry, and real-life problem-solving. Mastery of rearranging formulae allows students to solve equations, substitute values accurately, and apply formulas flexibly.

For example, the formula for the area of a rectangle is $$A = l × w$$. If we know the area and the length, we can rearrange the formula to find the width: $$w = \frac{A}{l}$$. Understanding how to rearrange formulae is crucial for solving problems efficiently in exams and real-life contexts.

Core Concepts

Identifying the Subject

The subject of a formula is the variable you want to isolate. Rearranging involves performing inverse operations to make this variable alone on one side of the equation.

Example:

• Formula: $$A = l × w$$, make w the subject
• Divide both sides by l: $$w = \frac{A}{l}$$

Inverse Operations

Use the inverse of operations in reverse order to isolate the variable:

• Addition ↔ Subtraction
• Multiplication ↔ Division
• Raising to a power ↔ Taking roots
• Example: $$y + 3 = 10$$ → subtract 3 → $$y = 7$$

Rearranging Formulas with Multiplication and Division

Example: $$F = m × a$$

• Make m the subject: $$m = \frac{F}{a}$$
• Make a the subject: $$a = \frac{F}{m}$$

Rearranging Formulas with Addition and Subtraction

Example: $$v = u + at$$

• Make u the subject: $$u = v - at$$
• Make t the subject: $$v = u + at \Rightarrow v - u = at \Rightarrow t = \frac{v - u}{a}$$

Formulas with Brackets

When the variable is inside brackets, expand or divide as needed:

Example: $$A = 2(l + w)$$

• Make w the subject: $$A = 2(l + w)$$ → divide both sides by 2: $$\frac{A}{2} = l + w$$
• Subtract l: $$w = \frac{A}{2} - l$$

Formulas with Fractions

When a variable is in the numerator or denominator, multiply or divide accordingly:

Example: $$P = \frac{F}{A}$$, make F the subject

• Multiply both sides by A: $$F = P × A$$

Formulas with Powers

When a variable is raised to a power, use roots to isolate it:

Example: $$A = πr^2$$, make r the subject

• Divide both sides by π: $$\frac{A}{π} = r^2$$
• Take square root: $$r = \sqrt{\frac{A}{π}}$$

Formulas with Multiple Variables

Rearrange systematically, one step at a time:

Example: $$y = mx + c$$, make x the subject

• Subtract c: $$y - c = mx$$
• Divide by m: $$x = \frac{y - c}{m}$$

Checking Your Rearrangement

After rearranging, substitute numbers to verify that the formula works correctly. This ensures accuracy.

Example:

• Formula: $$F = m × a$$, rearranged m = F / a
• Values: F = 20, a = 5 → m = 20 / 5 = 4
• Check: 4 × 5 = 20 ✔

Real-Life Applications

• Physics: solving formulas for speed, distance, acceleration, force
• Geometry: rearranging area, perimeter, and volume formulas
• Finance: solving formulas for interest, profit, and cost
• Engineering: using formulas to calculate power, pressure, or density
• Everyday problems: converting units, scaling recipes, or determining quantities

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Worked Examples

Example 1 (Foundation): Simple subtraction

Formula: $$v = u + at$$, make u the subject

• Subtract at: $$u = v - at$$

Example 2 (Foundation): Simple division

Formula: $$F = m × a$$, make a the subject

• Divide both sides by m: $$a = \frac{F}{m}$$

Example 3 (Higher): Brackets

Formula: $$A = 2(l + w)$$, make w the subject

• Divide by 2: $$\frac{A}{2} = l + w$$
• Subtract l: $$w = \frac{A}{2} - l$$

Example 4 (Higher): Fractions

Formula: $$P = \frac{F}{A}$$, make F the subject

• Multiply by A: $$F = P × A$$

Example 5 (Higher): Powers

Formula: $$A = πr^2$$, make r the subject

• Divide by π: $$\frac{A}{π} = r^2$$
• Take square root: $$r = \sqrt{\frac{A}{π}}$$

Example 6 (Higher): Multiple variables

Formula: $$y = mx + c$$, make x the subject

• Subtract c: $$y - c = mx$$
• Divide by m: $$x = \frac{y - c}{m}$$

Example 7 (Higher): Negative coefficient

Formula: $$-2x + 5 = y$$, make x the subject

• Subtract 5: $$-2x = y - 5$$
• Divide by -2 (reverse sign if necessary): $$x = \frac{5 - y}{2}$$

Example 8 (Real-life application)

Formula: Distance = Speed × Time, d = s × t, make t the subject

• Divide both sides by s: $$t = \frac{d}{s}$$

Example 9 (Real-life application)

Formula: Simple interest $$I = P × r × t$$, make r the subject

• Divide both sides by P × t: $$r = \frac{I}{P × t}$$

Example 10 (Higher): Rearranging with fractions and brackets

Formula: $$V = \frac{1}{3}πr^2h$$, make h the subject

• Multiply both sides by 3: $$3V = πr^2h$$
• Divide by πr^2: $$h = \frac{3V}{πr^2}$$

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Common Mistakes

• Not performing the same operation on both sides
• Forgetting to reverse inequality when dividing/multiplying by a negative (for inequalities)
• Incorrectly handling fractions
• Expanding incorrectly when brackets are involved
• Not simplifying final formula

Tips to avoid errors:

• Identify the subject clearly before starting
• Apply inverse operations systematically
• Check signs, especially for negative numbers
• Work step by step, especially with brackets or fractions
• Verify rearranged formula by substituting known values

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Applications

• Physics: Speed, distance, acceleration, force, pressure
• Geometry: Area, volume, perimeter formulas
• Finance: Interest, profit, cost calculations
• Engineering: Power, density, and other applied formulas
• Everyday problem-solving: cooking, scaling, budgeting

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Strategies & Tips

• Identify the variable you need to make the subject
• Use inverse operations carefully
• Handle brackets and fractions systematically
• Check the rearrangement by substitution
• Practice with multiple types of formulas for confidence

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Summary / Call-to-Action

Rearranging formulae is a critical algebra skill in GCSE Maths. By mastering inverse operations, brackets, fractions, powers, and multiple variables, students can manipulate formulas effectively. Regular practice ensures confidence, accuracy, and the ability to apply formulas in exams and real-life scenarios.

Next Steps:

• Attempt rearranging formulae quizzes to reinforce learning
• Practice step-by-step rearrangement for different types of formulas
• Apply formulas in real-life problems and substitutions
• Challenge yourself with higher-level formulas involving fractions, powers, and multiple variables

Consistent practice will make rearranging formulae intuitive and error-free in all GCSE Maths problems.