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GCSE Maths · EdexcelChapter 8 · Substitution and formulae
GCSE Maths · Substitution and Formulae · Higher · Lesson 8.4

GCSE Maths: Substituting into formulae (Higher)

Part of the GCSE Maths substitution and formulae chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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8.4

Substituting into formulae

Edexcel 1MA1: A2, A5
Grade 2-6

A formula is a rule written with letters. Each letter stands for a quantity (a length, a time, a mass…). To use a formula we substitute the numbers we know and work out the one we do not. It is exactly what you did in 8.2 and 8.3, but now the letters mean something real, so the answer needs units.

In A = lw the letter on its own on the left, A, is called the subject of the formula. We use the formula to work out the subject. (In 8.6 we will learn to change which letter is the subject.)

Here are formulae you should know. Most appear in the GCSE exam without being given, so learn them.

QuantityFormulaWhat the letters mean (usual units)
Area of a rectangleA = lwl length, w width (cm², m²)
Perimeter of a rectangleP = 2l + 2wdistance round the outside (cm, m)
Area of a triangleA = 12bhb base, h perpendicular height
Area of a trapeziumA = 12(a + b)ha and b the parallel sides, h the distance between them
Volume of a cuboidV = lwhlength, width, height (cm³, m³)
Circumference of a circleC = πdd diameter
Area of a circleA = πr²r radius
Speeds = dtdistance d and time t (km/h, m/s)
DensityD = mVmass m and volume V (g/cm³)
PressureP = FAforce F and area A (N/m²)
ForceF = mamass m (kg), acceleration a (m/s²), force in newtons (N)
Final speedv = u + atu starting speed, a acceleration, t time
Kinetic energyE = 12mv²mass m (kg), speed v (m/s), energy in joules (J)
Electric potential differenceV = IRcurrent I (amps), resistance R (ohms), volts (V)
abhlhw
A trapezium (A = ½(a + b)h) and a cuboid (V = lwh). The height h of a trapezium is measured at right angles to the parallel sides.

Using a formula: four steps

  1. Write the formula.
  2. Substitute each known value (with its × sign, and brackets round negatives).
  3. Work out the answer in the correct order (BIDMAS).
  4. Write the answer with its units.

Units must match. If the speed formula is used with distance in km and time in hours, the answer is in km/h. If the time is given in minutes, change it to hours first (30 minutes is 0.5 hours). On a calculator, use brackets for fractions: the top and the bottom each go in their own brackets, for example (3 + 5) ÷ (2 × 4), and use the π button rather than typing 3.14. Do not round until the very end.

Key ideas

  • A formula links quantities. The letter on its own on the left is the subject.
  • Write the formula, substitute, work out with BIDMAS, then write the answer with units.
  • Keep units consistent: km with hours gives km/h, metres with seconds gives m/s.
  • On a calculator use brackets for the top and bottom of a fraction, and use the π button.
  • Round only at the end, and say how you rounded (decimal places or significant figures).

Watch out

In E = ½mv² the speed is squared, not the mass and not the whole product. With m = 4 and v = 10: square the 10 first (100), then ½ × 4 × 100 = 200. Squaring the whole product (½ × 4 × 10)² = 400 would be wrong.
Worked example 8.4.1
A coach travels 150 km in 2.5 hours. Use s = dt to work out its average speed.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

  1. Write down the formula. This is the rule we are going to use.
    s = dt
  2. Substitute: d = 150, t = 2.5. Substitute means replace each letter with its number.
    s = 1502.5
  3. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
    1502.5 means 150 ÷ 2.5 = 60
    s = 60
  4. Write the answer with its units. The answer should say what was being worked out.
    s = 60 km/h

Check. Put the numbers straight into a calculator: 150/2.5 gives 60. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Worked example 8.4.2
The area of a trapezium is A = ½(a + b)h. Work out A when a = 7 cm, b = 11 cm and h = 6 cm.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

  1. Write down the formula. This is the rule we are going to use.
    A = 12(a + b)h
  2. Substitute: a = 7, b = 11, h = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
    A = 12 × (7 + 11) × 6
  3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
    7 + 11 = 18
    A = 12 × 18 × 6
  4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    12 × 18 = 9
    A = 9 × 6
  5. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    9 × 6 = 54
    A = 54
  6. Write the answer with its units. The answer should say what was being worked out.
    A = 54 cm2

Check. Put the numbers straight into a calculator: 1/2 × (7 + 11) × 6 gives 54. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Worked example 8.4.3
The kinetic energy of a moving object is E = ½mv² joules. Work out E when the mass is m = 4 kg and the speed is v = 10 m/s.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

  1. Write down the formula. This is the rule we are going to use.
    E = 12mv2
  2. Substitute: m = 4, v = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
    E = 12 × 4 × 102
  3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
    102 = 10 × 10 = 100
    E = 12 × 4 × 100
  4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    12 × 4 = 2
    E = 2 × 100
  5. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    2 × 100 = 200
    E = 200
  6. Write the answer with its units. The answer should say what was being worked out.
    E = 200 J

Check. Put the numbers straight into a calculator: 1/2 × 4 × 102 gives 200. ✓

Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.

Worked example 8.4.4
Use A = πr² to work out the area of a circle of radius 6.3 cm. Use a calculator and give your answer to 1 decimal place.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, substitute the numbers, use a calculator, then round as asked.

  1. Write down the formula. This is the rule we are going to use.
    A = πr2
  2. Substitute: r = 6.3. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.
    A = π × 6.32
  3. Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.
    A = π × 6.32
    Calculator display: 124.68981242…
  4. Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 8, which is 5 or more, so we round up.
    124.68981242… = 124.7 (1 decimal place)
  5. Write the answer with its units. Say what the number means.
    A = 124.7 cm2

Check. Estimate with rounded numbers (r = 6, π = 3): 3 × 62 = 3 × 36 = 108. This is close to 124.7, so the answer is sensible. ✓

Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.

Worked example 8.4.5
The distance travelled by a braking car is s = ut + ½at². Work out s when u = 12, t = 5 and a = −2.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

  1. Write down the formula. This is the rule we are going to use.
    s = ut + 12at2
  2. Substitute: u = 12, t = 5, a = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
    s = 12 × 5 + 12 × (−2) × 52
  3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
    52 = 5 × 5 = 25
    s = 12 × 5 + 12 × (−2) × 25
  4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    12 × 5 = 60
    12 × (−2) = −1 (a negative times a positive gives a negative)
    s = 60 + (−1) × 25
  5. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
    −1 × 25 = −25 (a negative times a positive gives a negative)
    s = 60 + (−25)
  6. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
    60 + (−25) = 60 − 25 = 35 (adding a negative number is the same as subtracting)
    s = 35
  7. Write the answer with its units. The answer should say what was being worked out.
    s = 35 m

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 12 × 5 + 1/2 × (−2) × 52 gives 35. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Exercise 8D  Using formulae (non-calculator)

Non-calculator
1
FUse each formula to work out the answer.
(a)Area of a rectangle with length 12 cm and width 5 cm: A = lw
(b)Perimeter of the same rectangle: P = 2l + 2w
(c)Volume of a cuboid 6 cm by 4 cm by 3 cm: V = lwh
(d)Area of a triangle with base 10 cm and height 7 cm: A = ½bh
2
FWork out each answer. Remember the units.
(a)A cyclist travels 240 km in 3 hours. Speed: s = d/t
(b)A metal block has mass 540 g and volume 200 cm³. Density: D = m/V
(c)A force of 600 N acts on an area of 20 m². Pressure: P = F/A
(d)A mass of 12 kg has acceleration 4 m/s². Force: F = ma
3
FA car has starting speed u, acceleration a and time t. Its final speed is v = u + at. Work out v when
(a)u = 5, a = 3, t = 4
(b)u = 12, a = 2, t = 6
(c)u = 0, a = 10, t = 3
4
FThe electrical formula V = IR gives the potential difference V volts from the current I amps and the resistance R ohms. Work out V when
(a)I = 3 and R = 8
(b)I = 0.5 and R = 12
5
PSThe time, T minutes, needed to cook a joint of meat is T = 40m + 20, where m is the mass of the joint in kg. Work out the cooking time for a joint of mass
(a)1.5 kg
(b)2.5 kg
(c)3 kg
6
PSA formula to change degrees Celsius (C) to degrees Fahrenheit (F) is F = 1.8C + 32. Work out F when
(a)C = 20
(b)C = 100
(c)C = −40
7
FWork out the area of the trapezium using A = ½(a + b)h.
5 cm9 cm4 cm
8
FWork out the kinetic energy E = ½mv² (in joules) when
(a)m = 2 and v = 6
(b)m = 10 and v = 3
(c)m = 0.5 and v = 8
9
RZoe uses v = u + at with u = 4, a = 3 and t = 2. She writes v = 7 × 2 = 14. Explain her mistake and work out the correct value of v.
10
PSCar A travels 180 km in 2.5 hours. Car B travels 150 km in 2 hours. Use s = d/t.
(a)Work out the average speed of car A.
(b)Work out the average speed of car B.
(c)Which car has the greater average speed?
11
PSBlock X has mass 840 g and volume 300 cm³. Block Y has mass 620 g and volume 250 cm³. Use D = m/V.
(a)Work out the density of block X.
(b)Work out the density of block Y.
(c)Which block is denser?
12
PSThe final speed v m/s of an object is v = √(u² + 2as). Work out v when
(a)u = 4, a = 3 and s = 8
(b)u = 6, a = 2 and s = 7
13
FThe distance s metres travelled is s = ut + ½at². Work out s when
(a)u = 12, t = 5 and a = −2
(b)u = 0, t = 6 and a = 10
(c)u = −8, t = 4 and a = 4
14
RUse P = F/A to show that a force of 500 N on an area of 25 m² gives the same pressure as a force of 100 N on an area of 5 m².

Exercise 8E  Using formulae with a calculator

Calculator allowed
1
FUse A = πr² to work out the area of each circle. Give your answers to 1 decimal place.
(a)radius 4.5 cm
(b)radius 12 cm
(c)radius 0.8 cm
2
FUse C = πd to work out the circumference of each circle. Give your answers to 1 decimal place.
(a)diameter 7.3 cm
(b)diameter 15.6 cm
3
FThe volume of a cylinder is V = πr²h. Work out the volume of each cylinder, to 1 decimal place.
(a)radius 3.5 cm, height 10 cm
(b)radius 2.4 cm, height 7.5 cm
4
PSUsain Bolt ran 100 m in 9.58 seconds. Use s = d/t to work out his average speed in m/s, to 1 decimal place.
5
FA metal sample has mass 84.6 g and volume 12.5 cm³. Use D = m/V to work out its density. Give your answer to 3 significant figures.
6
PSA stone falls from a height of 12 m. Its speed on landing is v = √(2gh), where g = 9.8 and h is the height in metres. Work out v to 1 decimal place.
7
PSThe time for a pendulum of length l metres to swing is T = 2π√(l/g) seconds, where g = 9.8. Work out T when l = 1.5, to 2 decimal places.
8
FThe volume of a sphere is V = 43πr³. Work out the volume of a sphere of radius 2.4 cm, to 1 decimal place.
9
PSA circular ring has outer radius R = 8.5 cm and inner radius r = 6.2 cm. Its area is A = π(R² − r²). Work out A to 1 decimal place.
10
PSA cylindrical tin has radius 4.2 cm and height 11.5 cm. Work out its volume, to the nearest cm³.
11
FUse F = 1.8C + 32 to change a body temperature of 36.6°C into degrees Fahrenheit. Give your answer to 1 decimal place.
12
RSam works out the area of a circle with radius 5 cm by typing (π × 5)² on his calculator. He gets 246.7. Explain his mistake and work out the correct area, to 1 decimal place.

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