Problem-solving practice 2 Revision Notes for AQA GCSE Maths

Problem-solving practice 2

This collection of algebra problems covers key topics you'll encounter in your GCSE exam. Each question type requires different problem-solving strategies and mathematical skills.

Triangle perimeter problems

When working with triangles that have algebraic expressions for their sides, you need to set up and solve equations using the given information.

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Key concept: The perimeter of any shape is the total distance around its edges, found by adding all side lengths together.

Method for solving triangle perimeter problems:

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Step-by-step approach:

Write an expression for the perimeter by adding all three sides
Set this equal to the given perimeter value
Collect like terms and simplify your equation
Solve to find the unknown value

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Worked Example: Triangle Perimeter

For a triangle with sides $(t+2)$ , $(2t-2)$ , and $(t-1)$ , where the perimeter is 13cm:

Add the expressions: $(t+2) + (2t-2) + (t-1) = 13$
Simplify by collecting like terms: $4t - 1 = 13$
Solve: $4t = 14$ , so $t = 3.5$

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Always remember to simplify your equation by collecting like terms before solving. This makes the algebra much easier to handle.

Arithmetic sequences

An arithmetic sequence is a sequence where consecutive terms have a constant difference between them.

Finding if a number belongs to a sequence:

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Method for checking sequence membership:

Work out the common difference between consecutive terms
Find the nth term formula using the pattern
Set the nth term equal to your target number
Solve the equation - if you get a whole number answer, the target is in the sequence

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Worked Example: Sequence Membership

For the sequence 9, 15, 21, 27...:

The common difference is 6 (15-9 = 6, 21-15 = 6, etc.)
The nth term formula would be $6n + 3$
To check if 65 is a term: $6n + 3 = 65$ , so $6n = 62$ , giving $n = 10.33...$
Since this isn't a whole number, 65 is not a term in this sequence

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You can use any successful strategy to answer sequence questions. You could work out consecutive terms on either side of your target number, or use the nth term formula method.

Speed calculations

Speed problems often involve converting between different units and applying the fundamental relationship: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ .

Working with speed problems:

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Speed calculation method:

Identify what units you're working with
Convert if necessary (especially minutes to hours)
Apply the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
Check your answer makes sense

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Worked Example: Speed Calculation

For a train travelling $y$ km in 3 minutes:

First convert 3 minutes to hours: $3 \div 60 = 0.05$ hours
Apply the formula: $\text{Speed} = y \div 0.05 = 20y$ km/h

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Watch out for unit conversions! Always convert minutes to hours by dividing by 60, then simplify your fraction.

Straight-line graphs and equations

Linear equations describe straight lines and can be written in the form $y = mx + c$ , where $m$ is the gradient and $c$ is the y-intercept.

Finding equations of straight lines:

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Method for finding line equations:

Use the gradient (given or calculated from two points)
Substitute a known point into $y = mx + c$
Solve to find the y-intercept ( $c$ value)
Write the complete equation

Checking if points lie on lines:

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Point verification method:

Substitute the x-coordinate into your equation
Calculate what the y-coordinate should be
Compare with the given y-coordinate

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Worked Example: Linear Equation

For a line with gradient 8 passing through (5, 10):

Use $y = 8x + c$
Substitute the point: $10 = 8(5) + c$ , so $10 = 40 + c$ , giving $c = -30$
The equation is $y = 8x - 30$
To check if (10, 50) lies on this line: $y = 8(10) - 30 = 50$ ✓

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When commenting on reliability of methods, consider that algebraic solutions are generally more reliable than drawing diagrams to scale, as diagrams can introduce measurement errors.

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Key Points to Remember:

Collect like terms before solving algebraic equations to simplify your working
Check your units carefully in speed problems - convert minutes to hours when needed
Whole number solutions in sequence problems indicate the target number is actually a term
Substitute coordinates into linear equations to verify if points lie on the line
Show all working clearly to demonstrate your problem-solving method and gain maximum marks

Problem-solving practice 2 (AQA GCSE Maths): Revision Notes