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Glossary
Chapter 1: Number skills
- Overview
- 1.1 Place value and ordering numbers
- 1.2 The four operations
- 1.3 Negative numbers
- 1.4 Priority of operations and inverse operations
- 1.5 Rounding and estimation
- Review
Chapter 2: Factors, multiples and primes
- Overview
- 2.1 Factors, multiples and prime numbers
- 2.2 Prime factorisation, HCF and LCM
- 2.3 Systematic listing strategies
- 2.4 The product rule for counting
- Review
Chapter 3: Powers, roots and indices
- Overview
- 3.1 Powers and roots
- 3.2 Estimating powers and roots
- 3.3 Laws of indices
- 3.4 Negative and fractional indices
- Review
Chapter 4: Fractions, decimals and percentages
- Overview
- 4.1 Equivalent fractions and ordering
- 4.2 Adding and subtracting fractions
- 4.3 Multiplying and dividing fractions
- 4.4 Fractions and terminating decimals
- 4.5 Recurring decimals
- 4.6 Fractions and percentages as operators
- Review
Chapter 5: Ratio and proportion
- Overview
- 5.1 Ratio notation, simplifying and sharing
- 5.2 Ratio, fractions and linear functions
- 5.3 Direct proportion
- 5.4 Inverse proportion
- 5.5 Scale factors, scale diagrams and maps
- 5.6 Comparing lengths, areas and volumes
- Review
Chapter 6: Percentages and financial mathematics
- Overview
- 6.1 Percentage of an amount
- 6.2 Percentage change
- 6.3 Reverse percentages
- 6.4 Simple and compound interest
- 6.5 Growth, decay and repeated percentage change
- Review
Chapter 7: Measures, accuracy, standard form and surds
- Overview
- 7.1 Units of measure and conversions
- 7.2 Compound measures: speed, density and pressure
- 7.3 Rounding and error intervals
- 7.4 Upper and lower bounds
- 7.5 Standard form
- 7.6 Surds and exact calculations
- Review
Chapter 8: Algebraic expressions and manipulation
- Overview
- 8.1 Algebraic notation and vocabulary
- 8.2 Simplifying and collecting like terms
- 8.3 Expanding brackets
- 8.4 Factorising with common factors
- 8.5 Substitution into expressions and formulae
- 8.6 Rearranging formulae
- 8.7 Expanding products of three binomials
- Review
Chapter 9: Equations and inequalities
- Overview
- 9.1 Solving linear equations
- 9.2 Equations with unknowns on both sides
- 9.3 Forming and solving equations
- 9.4 Inequalities on a number line and set notation
- 9.5 Solving linear inequalities
- 9.6 Quadratic inequalities
- Review
Chapter 10: Sequences
- Overview
- 10.1 Term-to-term and position-to-term rules
- 10.2 Special sequences
- 10.3 The nth term of a linear sequence
- 10.4 The nth term of a quadratic sequence
- 10.5 Geometric sequences
- Review
Chapter 11: Straight-line graphs and rates of change
- Overview
- 11.1 Coordinates and plotting graphs
- 11.2 Gradient and intercept: y = mx + c
- 11.3 Finding the equation of a line
- 11.4 Parallel and perpendicular lines
- 11.5 Real-life and distance-time graphs
- 11.6 Gradients and areas under graphs
- Review
Chapter 12: Quadratic expressions, equations and graphs
- Overview
- 12.1 Expanding and factorising quadratics
- 12.2 Factorising harder quadratics
- 12.3 Quadratic graphs and their features
- 12.4 Solving quadratic equations by factorising
- 12.5 The quadratic formula
- 12.6 Completing the square
- Review
Chapter 13: Other functions and graphs
- Overview
- 13.1 Cubic, reciprocal and exponential graphs
- 13.2 Trigonometric graphs
- 13.3 Transformations of graphs
- 13.4 The equation of a circle and tangents
- Review
Chapter 14: Simultaneous equations and iteration
- Overview
- 14.1 Simultaneous equations by elimination
- 14.2 Simultaneous equations by substitution
- 14.3 Solving simultaneous equations graphically
- 14.4 Linear and quadratic simultaneous equations
- 14.5 Iteration and numerical methods
- Review
Chapter 15: Functions, algebraic fractions and proof
- Overview
- 15.1 Functions: inverse and composite
- 15.2 Simplifying algebraic fractions
- 15.3 Operations with algebraic fractions
- 15.4 Identities and equivalent expressions
- 15.5 Algebraic proof
- Review
Chapter 16: Angles, polygons and geometrical reasoning
- Overview
- 16.1 Angle facts at a point and on a line
- 16.2 Angles in parallel lines
- 16.3 Angles in triangles and quadrilaterals
- 16.4 Angles in polygons
- 16.5 Geometrical reasoning and proof
- Review
Chapter 17: Properties, constructions, loci and bearings
- Overview
- 17.1 Properties of 2D and 3D shapes
- 17.2 Plans and elevations
- 17.3 Constructions with ruler and compasses
- 17.4 Loci
- 17.5 Bearings and scale drawings
- Review
Chapter 18: Transformations, congruence and similarity
- Overview
- 18.1 Reflection, rotation and translation
- 18.2 Enlargement
- 18.3 Enlargement with fractional and negative scale factors
- 18.4 Congruence and congruence criteria
- 18.5 Similarity and similar lengths
- 18.6 Similarity: areas and volumes
- Review
Chapter 19: Perimeter, area, surface area and volume
- Overview
- 19.1 Perimeter and area of 2D shapes
- 19.2 Circumference and area of circles
- 19.3 Arc length and sector area
- 19.4 Surface area and volume of prisms and cylinders
- 19.5 Surface area and volume of spheres, cones and pyramids
- 19.6 Composite shapes and solids
- Review
Chapter 20: Pythagoras' theorem and trigonometry
- Overview
- 20.1 Pythagoras' theorem
- 20.2 Trigonometric ratios (SOHCAHTOA)
- 20.3 Angles of elevation and depression
- 20.4 Exact trigonometric values
- 20.5 3D Pythagoras and trigonometry
- 20.6 The sine rule
- 20.7 The cosine rule
- 20.8 The area of any triangle
- Review
Chapter 21: Circle geometry and circle theorems
- Overview
- 21.1 Parts of a circle
- 21.2 Circle theorems
- 21.3 Tangents and chords
- 21.4 Proof of circle theorems
- Review
Chapter 22: Vectors
- Overview
- 22.1 Vector notation and column vectors
- 22.2 Adding and subtracting vectors
- 22.3 Vectors in geometric problems and proofs
- Review
Chapter 23: Probability
- Overview
- 23.1 The probability scale and equally likely outcomes
- 23.2 Relative frequency and expected outcomes
- 23.3 Frequency trees, tables and possibility spaces
- 23.4 Combined events and tree diagrams
- 23.5 Venn diagrams and set notation
- 23.6 Conditional probability
- Review
Chapter 24: Statistics: representing and analysing data
- Overview
- 24.1 Collecting data, sampling and questionnaires
- 24.2 Tables, bar charts, pie charts and pictograms
- 24.3 Averages and range
- 24.4 Averages from frequency tables
- 24.5 Time series and line graphs
- Review
Chapter 25: Statistics: spread, grouped data and correlation
- Overview
- 25.1 Quartiles, interquartile range and box plots
- 25.2 Cumulative frequency
- 25.3 Histograms
- 25.4 Scatter graphs and correlation
- 25.5 Comparing distributions and outliers
- Review
About this book

5.3 Direct proportion — Practise with help

Try each question first. Open a hint if you are stuck, the next step for one concrete move, and the full working only to check.

Warm-up: remember

Answer these from memory. Do not look back at the section.

Work out $5^2$ .
Find the value of $k$ when $12=4k$ .

Check the warm-up

$5^2=25$ .
Since $12\div4=3$ , $k=3$ .

If an answer differs, revisit Section 3.1 for squares and Section 1.4 for inverse operations.

Practise with help

1. $y$ is proportional to $x^2$ . When $x=3$ , $y=45$ . Find $y$ when $x=5$ .

Hint

Write the model $y=kx^2$ and substitute the known pair.

Next step

$45=9k$ , so $k=5$ .

Full working

Substitute $x=5$ : $y=5\times25=125$ . Check with the scale factor: $45\times(25/9)=125$ .

2. $y$ is proportional to $x$ . When $x=7$ , $y=35$ . Find $y$ when $x=12$ .

Hint

Write $y=kx$ and substitute the known pair.

Next step

$k=35\div7=5$ .

Full working

$y=5\times12=60$ . Check: $35\div7=60\div12=5$ .

3. The number of pens is directly proportional to the number of packs. 3 packs hold 36 pens. How many pens do 8 packs hold?

Hint

Direct proportion means a constant number of pens per pack.

Next step

One pack holds $36\div3=12$ pens.

Full working

$12\times8=96$ pens. Check: $96\div8=12$ .

4. Earlier learning. Work out $6^2$ and $(-2)^3$ .

Hint

A square multiplies a number by itself. A cube multiplies it three times.

Next step

$6^2=6\times6$ , and $(-2)^3=(-2)\times(-2)\times(-2)$ .

Full working

$6^2=36$ . Also, $(-2)^3=-8$ , because three negative factors give a negative answer.