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Algebra

Algebra

Forming and solving equations from words

Scenario Based Questions Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Words into algebra
  4. When the equation is a quadratic
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.5 and 2.6: forming equations from words
  • 2.5Turn a word problem into an equation: ages, angles, costs, consecutive numbers, perimeters.
  • 2.5Form quadratic equations from areas and from speed / time, including 'show that'.
  • 2.5Solve, reject impossible answers, and answer the question that was asked.

Before you start: Linear equations and solving quadratics.

Formula bank

TimeLearn it\(\text{time}=\dfrac{\text{distance}}{\text{speed}}\)Use it when a journey with an unknown speed.
Consecutive numbersLearn it\(n,\ n+1,\ n+2\); odd or even: \(n,\ n+2,\ n+4\)Use it when 'consecutive'.
RectangleLearn itarea \(=lw\), perimeter \(=2(l+w)\)Use it when a shape with sides in terms of x.

Words into algebra

WordsAlgebra
5 more than n\(n+5\)
twice n, minus 3\(2n-3\)
4 years younger than someone aged 3x\(3x-4\)
n shared equally among 4\(\dfrac n4\)
the product of x and x + 3\(x(x+3)\)
time for 30 km at x km/h\(\dfrac{30}x\)

General methodHow to form and solve an equation from words

  1. Write 'Let x = …' (choose the smallest or unknown thing).
  2. Write every other quantity in terms of x.
  3. Find the fact that gives '=': a total, a sum of angles, an area, a total time.
  4. Solve. Reject answers that are impossible (negative lengths, speeds, ages).
  5. Answer the actual question, with units.

Example Ali is x years old. His sister is 3 times as old, and his brother is 4 years younger than the sister. Their ages total 52. How old is Ali?

Write each age in x and add them to 52.

  1. \(x+3x+(3x-4)=52\)
  2. \(7x-4=52\)
  3. \(x=8\)

Answer Ali is 8 (sister 24, brother 20)

Try it yourself

1
Three consecutive numbers add up to 84. What is the smallest?
Show answer

Answer 27

2
A triangle has angles x°, (2x + 10)° and (3x − 40)°. Find x.
Show answer

Answer 35

3
A rectangle is (2x + 3) cm by x cm with perimeter 48 cm. Find x.
Show answer

Answer 7

When the equation is a quadratic

Areas (one length × another) and journeys (distance ÷ an unknown speed) usually give quadratics. In a 'show that', build the equation from the situation and show every step down to the given form.

Example A rectangle has width x cm and length (x + 4) cm. Its area is 96 cm². Show that x² + 4x − 96 = 0, and find the perimeter.

Area = width × length; then solve and reject the negative.

  1. \(x(x+4)=96\Rightarrow x^{2}+4x-96=0\)
  2. \((x+12)(x-8)=0\Rightarrow x=8\) (reject −12)
  3. sides 8 and 12: perimeter 2(8 + 12)

Answer 40 cm

Try it yourself

1
A rectangle (x + 2) cm by (x − 1) cm has area 40 cm². Find x.
Hint
\(x^{2}+x-42=0\); reject the negative.
Show answer

Answer 6

Ways they test it

Type 1 Form and solve (linear)

"A carton costs p cents and a sandwich 120 cents. 4 cartons and 5 sandwiches cost $8.40. Form an equation and find p." (3 marks)

Spot it 'form an equation'.

  1. Same units (840 cents).
  2. Write the equation.
  3. Solve.

For example \(4p+600=840\)

    Answer p = 60

    Type 2 Show that (area)

    "Show that x² + 3x − 54 = 0" for a rectangle with area 54. (2 marks)

    Spot it 'show that' and a given equation.

    1. Area = length × width.
    2. Expand and move everything to one side - every step written.

    Type 3 Show that (speed/time)

    "Sana drives 60 km at x km/h; Bilal drives at (x + 10) km/h and takes half an hour less. Show that x² + 10x − 1200 = 0." (3 marks)

    Spot it two journeys, one time compared with the other.

    1. Each time = distance ÷ speed.
    2. Write the time difference (or total) as an equation.
    3. Clear the fractions.

    For example \(\dfrac{60}x-\dfrac{60}{x+10}=\dfrac12\)

    1. \(120(x+10)-120x=x(x+10)\)
    2. \(x^{2}+10x-1200=0\)

    Answer shown

    Type 4 Solve and use

    "Hence find the perimeter / the time / the two numbers."

    Spot it the last part after solving.

    1. Reject the impossible root.
    2. Use x to answer what is asked.

    Mistakes to avoid

    • ✗ '4 years younger than his sister' = 4 − 3x✓ 3x − 4Younger: subtract 4 from her age.
    • ✗ Mixing $8.40 with cents✓ 840 centsUse one unit.
    • ✗ Keeping x = −9 for a width✓ reject itLengths are positive.
    • ✗ Stopping at x when the perimeter is asked✓ finish the questionRead the last line.

    Summary

    • Let x = …; write everything in x; find the fact that makes '='.
    • Areas and journey times often give quadratics.
    • 'Show that': build it from the situation, every step shown.
    • Reject impossible values; answer in context with units.

    If you see …

    • "form an equation"write it with the letter, then solve it.
    • speed and time with xtime = distance ÷ speed for each journey.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Three consecutive odd numbers add up to 147. What is the largest?
    Hint
    \(n+(n+2)+(n+4)=147\).
    Show answer

    Answer 51

    2
    A book costs $3.50 more than a pen. 4 pens and 3 books cost $28. Find the cost of a pen in dollars.
    Hint
    \(4p+3(p+3.5)=28\).
    Show answer

    Answer 2.50

    3
    A carton costs p cents and a sandwich 120 cents. 4 cartons and 5 sandwiches cost $8.40. Find p.
    Show answer

    Answer 60

    Medium

    1
    A rectangle's length is 3 cm more than its width. Its area is 54 cm². Find its width.
    Guide me

    What is it asking? the width

    What do we know? width x, length x + 3

    Plan \(x(x+3)=54\)

    Show working
    1. \(x^{2}+3x-54=0\)
    2. \((x+9)(x-6)=0\)
    3. x = 6 (reject −9)

    Answer 6 cm

    2
    A rectangle has width x cm, length (x + 4) cm and area 96 cm². Find its perimeter.
    Guide me

    What is it asking? the perimeter

    What do we know? \(x^{2}+4x-96=0\)

    Plan x = 8; sides 8 and 12

    Show working
    1. \((x+12)(x-8)=0\)
    2. 2(8 + 12) = 40

    Answer 40 cm

    3
    A rectangle is (2x + 3) cm by x cm with perimeter 48 cm. Find its area.
    Guide me

    What is it asking? the area

    What do we know? \(2(2x+3)+2x=48\)

    Plan x = 7, sides 17 and 7

    Show working
    1. \(6x+6=48\Rightarrow x=7\)
    2. 17 × 7 = 119

    Answer 119 cm²

    Hard

    1
    The sum of the squares of two consecutive positive integers is 113. Find the smaller integer.
    Guide me

    What is it asking? the smaller integer

    What do we know? \(x^{2}+(x+1)^{2}=113\)

    Plan \(x^{2}+x-56=0\)

    Show working
    1. \(2x^{2}+2x-112=0\)
    2. \((x+8)(x-7)=0\)
    3. x = 7

    Answer 7

    PartAnswer / workingMarksGuidance
    \(x^{2}+(x+1)^{2}=113\)M1
    \(x^{2}+x-56=0\)A1
    \((x+8)(x-7)=0\)M1
    7A1
    Total: 4 marks
    2
    Ahmed cycles 30 km at x km/h and back at (x − 5) km/h. The whole trip takes 5 hours. Find x.
    Guide me

    What is it asking? x

    What do we know? \(\tfrac{30}x+\tfrac{30}{x-5}=5\)

    Plan clear fractions; reject the root that makes a speed negative

    Show working
    1. \(30(x-5)+30x=5x(x-5)\)
    2. \(x^{2}-17x+30=0\)
    3. \((x-15)(x-2)=0\)
    4. x = 15 (x = 2 gives −3 km/h)

    Answer 15

    PartAnswer / workingMarksGuidance
    \(\tfrac{30}x+\tfrac{30}{x-5}=5\)M1
    \(x^{2}-17x+30=0\)A1
    x = 15 onlyA1
    Total: 3 marks
    3
    Sana drives 60 km at x km/h. Bilal drives 60 km at (x + 10) km/h and takes half an hour less. How long does Sana take, in hours?
    Guide me

    What is it asking? Sana's time

    What do we know? \(\tfrac{60}x-\tfrac{60}{x+10}=\tfrac12\)

    Plan \(x^{2}+10x-1200=0\)

    Show working
    1. \(120(x+10)-120x=x(x+10)\)
    2. \((x+40)(x-30)=0\Rightarrow x=30\)
    3. 60 ÷ 30 = 2 hours

    Answer 2

    PartAnswer / workingMarksGuidance
    \(\tfrac{60}x-\tfrac{60}{x+10}=\tfrac12\)M1
    \(x^{2}+10x-1200=0\)A1
    x = 30M1
    2 hoursA1
    Total: 4 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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