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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
- Review
Chapter 3: Powers, roots and indices
- Overview
- 3.1 Powers and roots
- 3.2 Estimating powers and roots
- 3.3 Laws of 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
- 4.6 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
- 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
- 7.5 Standard form
- 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
- 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
- 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
- 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
- 11.5 Real-life and distance-time graphs
- Review
Chapter 12: Quadratic expressions, equations and graphs
- Overview
- 12.1 Expanding and factorising quadratics
- 12.3 Quadratic graphs and their features
- 12.4 Solving quadratic equations by factorising
- Review
Chapter 13: Other functions and graphs
- Overview
- 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
- Review
Chapter 15: Functions, algebraic fractions and proof
- Overview
- 15.4 Identities and equivalent expressions
- 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
- 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
- 18.4 Congruence and congruence criteria
- 18.5 Similarity and similar lengths
- 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
- Review
Chapter 21: Circle geometry and circle theorems
- Overview
- 21.1 Parts of a circle
- Review
Chapter 22: Vectors
- Overview
- 22.1 Vector notation and column vectors
- 22.2 Adding and subtracting vectors
- 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
- 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
- 25.4 Scatter graphs and correlation
- Review
About this book

5.1 Ratio notation, simplifying and sharing — 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.

Simplify $\frac{12}{18}$ .
Work out $-6+15$ .

Check the warm-up

$\frac{12}{18}=\frac{2}{3}$ .
$-6+15=9$ .

If an answer differs, revisit Section 4.1 for simplifying fractions and Section 1.3 for working with negative numbers.

Practise with help

1. Simplify the ratio $18:24$ .

Hint

Find a number that divides both parts exactly.

Next step

Both parts divide by 6.

Full working

$18:24$ divides by 6 to give $3:4$ . Check that $18\div24=3\div4$ .

2. Share £84 in the ratio $4:3$ .

Hint

Add the ratio parts to find the number of equal parts.

Next step

$4+3=7$ parts, so one part is $84\div7=12$ .

Full working

The shares are $4\times12=48$ and $3\times12=36$ . Check: $48+36=84$ .

3. A recipe mixes flour and sugar in the ratio $5:2$ . A baker uses 250 g of sugar. How much flour is needed?

Hint

The 2 parts stand for 250 g. Find the value of one part first.

Next step

One part is $250\div2=125$ g. Flour is 5 parts.

Full working

Flour is $5\times125=625$ g. Check: $625:250=5:2$ .

4. Earlier learning. In a class of 30 pupils, 12 walk and the rest take the bus. What fraction of the class take the bus?

Hint

The fraction is the bus pupils over the total. Find the bus number first.

Next step

$30-12=18$ pupils take the bus.

Full working

$\frac{18}{30}=\frac{3}{5}$ of the class take the bus.