Problem solving and GCSE exam practice – Year 11 GCSE maths revision notes to download as PDF

Worked example 1: a multi-step percentage problem

A coat costs £80. In a sale the price is reduced by 15%. In a second sale the new price is reduced by a further 10%. Find the final price.

Multiplier for 15% off is 0.85 and for 10% off is 0.9. \(80\times0.85\times0.9=61.20\). The final price is £61.20. (This is not 25% off, which would be £60.)

Worked example 2: algebra in context

A rectangle has length \(2x+3\) and width \(x-1\). The perimeter is 34. Find the area.

\(2(2x+3)+2(x-1)=34\Rightarrow6x+4=34\Rightarrow x=5\). Length \(=13\), width \(=4\), area \(=52\).

Worked example 3: a path around a field

A rectangular field is 40 m by 30 m. A path 2 m wide runs around the inside edge. Find the area of the path.

The inner rectangle is \((40-4)\) by \((30-4)\), which is \(36\) by \(26=936\). The field is \(1200\). Path \(=1200-936=264\) m\(^2\).

Worked example 4: proof

Prove that the sum of three consecutive integers is a multiple of 3.

Let the integers be \(n\), \(n+1\), \(n+2\). Sum \(=3n+3=3(n+1)\), which is 3 times an integer.

Worked example 5: Read, plan, do, check

A rectangle has perimeter 34 cm and length 5 cm more than its width. Find its area.

Read: unknowns are length and width. Plan: let the width be \(w\), so the length is \(w+5\) and \(2(2w+5)=34\). Do: \(4w+10=34\), \(4w=24\), \(w=6\) and length \(=11\). Area \(=66\) cm\(^2\). Check: perimeter \(=2\times17=34\) ✓. Writing the four stages shows the examiner your method and earns marks even if you slip.

Worked example 6: Using command words

What is the difference between "show that", "work out" and "explain" in an exam question?

"Work out" or "calculate" asks for a numerical answer with working. "Show that" gives you the answer and you must prove it by writing every step. "Explain" or "give a reason" needs a sentence, not only a number. "Hence" means use the previous result. Underline the command word before you start; many marks are lost by answering a different question from the one asked.

Worked example 7: A multi-step percentage problem

A shop sells a laptop for £540, which includes 20% VAT. Find the price before VAT and the amount of VAT.

The price including VAT is 120% of the original, so the multiplier is 1.2. Price before VAT \(=540\div1.2=450\). VAT \(=540-450=90\). A common error is to take 20% off £540, which gives £432. Always divide by the multiplier in a reverse problem.

Worked example 8: Working with ratio and algebra together

The ages of Ali and Ben are in the ratio \(3:5\). In 6 years the ratio will be \(2:3\). Find their ages now.

Let the ages now be \(3k\) and \(5k\). In 6 years: \(\dfrac{3k+6}{5k+6}=\dfrac23\), so \(3(3k+6)=2(5k+6)\), \(9k+18=10k+12\), so \(k=6\). Ali is 18 and Ben is 30. Check: in 6 years they are 24 and 36, and \(24:36=2:3\) ✓.

Worked example 9: Proof by algebra

Prove that the difference between the squares of two consecutive numbers is always odd.

Let the numbers be \(n\) and \(n+1\). \((n+1)^2-n^2=n^2+2n+1-n^2=2n+1\). Since \(2n\) is even, \(2n+1\) is odd. The proof works for every whole number \(n\), which is what makes it a proof rather than an example.

Worked example 10: Choosing between methods

A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side and the smaller angle.

Two sides are known and there is a right angle, so use Pythagoras: \(\sqrt{13^2-5^2}=12\) cm. For the smaller angle, opposite the 5 cm side, \(\sin x=\dfrac5{13}\), so \(x=22.6^\circ\). When you have a right angle and two sides, think first Pythagoras for a missing length and trigonometry for an angle.

Worked example 11: Estimating to check an answer

Estimate \(\dfrac{4.9\times31}{0.52}\) and explain why it is useful.

Round each number to one significant figure: \(\dfrac{5\times30}{0.5}=300\). A calculator gives about 292. The estimate shows that the answer is sensible. If you had pressed a wrong key and got 29.2 or 2920, the estimate would warn you.