Practice Maths

Substitution and Formulas

Key Ideas

Key Terms

Substitution
Means replacing a variable with a given number value.
order of operations
(BODMAS) after substituting.
negative numbers
Use brackets when substituting negatives.

Substituting into Formulas

Replace each variable with its given value, then evaluate using order of operations. Always include units in your final answer.

Hot Tip When substituting a negative number, always place it in brackets first. For example, if x = −3, then x² = (−3)² = 9, NOT −9.

Worked Example

Question: Find V when r = 3 using the formula V = 4⁄3πr³.

Step 1 — Substitute r = 3.
V = 4⁄3 × π × 3³

Step 2 — Evaluate the power first (BODMAS).
3³ = 27
V = 4⁄3 × π × 27

Step 3 — Simplify.
V = 4⁄3 × 27 × π = 4 × 9 × π = 36π

Step 4 — Evaluate numerically.
V = 36π ≈ 113.1 cm³

What is Substitution?

Substitution means replacing a variable (letter) with a number. For example, if a = 3 and b = −2, then the expression 2a + b becomes 2(3) + (−2) = 6 − 2 = 4. The key step is to replace every occurrence of the variable and to use brackets around negative numbers.

Substitution is used whenever you know the values of variables and want to calculate the result of an expression or formula.

Order of Operations (BODMAS)

When evaluating an expression after substituting, always follow the correct order of operations: Brackets, Orders (powers and roots), Division and Multiplication (left to right), then Addition and Subtraction (left to right).

Example: Find the value of 3x2 − 2x + 1 when x = 4.

= 3(4)2 − 2(4) + 1 = 3(16) − 8 + 1 = 48 − 8 + 1 = 41.

A common mistake is calculating 3 × 42 as (3 × 4)2 = 122 = 144. The power applies only to the 4, not to the 3×4 combination.

Substituting Negative Numbers

Always put brackets around a negative number when you substitute it. This prevents sign errors, especially with powers.

Example: Evaluate x2 − 3x when x = −5.

= (−5)2 − 3(−5) = 25 − (−15) = 25 + 15 = 40.

Without brackets, you might write −52 = −25 (incorrect!) instead of (−5)2 = 25. The brackets make all the difference.

Scientific and Geometric Formulas

Many formulas from science and geometry require substitution. Common examples:

  • Area of a circle: A = πr2. If r = 6, then A = π × 36 = 36π ≈ 113.1 cm2.
  • Kinetic energy: KE = 1/2 × m × v2. If m = 4 kg and v = 10 m/s, KE = 1/2 × 4 × 100 = 200 J.
  • Simple interest: I = PRT/100. If P = 5000, R = 4, T = 3, then I = 5000 × 4 × 3 ÷ 100 = $600.

Always state the units in your final answer.

Evaluating Multi-Variable Expressions

When a formula has several variables, substitute all values before simplifying. Work neatly, line by line.

Example: The formula for the area of a trapezium is A = 1/2 × (a + b) × h. Find A when a = 5, b = 9, h = 4.

A = 1/2 × (5 + 9) × 4 = 1/2 × 14 × 4 = 1/2 × 56 = 28 cm2.

Key tip: When substituting negative numbers, always use brackets: write (−3) not just −3. This single habit eliminates the most common errors in substitution questions. Show every substitution step in your working — if you get a wrong answer with working shown, you may still earn method marks.

Mastery Practice

  1. Substitute the given values into each formula and evaluate. Leave answers in exact form where indicated, otherwise round to 1 decimal place. Fluency

    1. A = lw, find A when l = 12 and w = 7.
    2. P = 2(l + w), find P when l = 9 and w = 5.
    3. A = ½bh, find A when b = 8 and h = 11.
    4. C = 2πr, find C when r = 6. Give an exact answer in terms of π and a decimal approximation.
    5. A = πr², find A when r = 5. Give an exact answer and a decimal approximation.
    6. V = lwh, find V when l = 4, w = 3 and h = 7.
    7. V = πr²h, find V when r = 2 and h = 9. Give your answer in terms of π and as a decimal.
    8. v = u + at, find v when u = 0, a = 3 and t = 8.
    9. I = Prt ÷ 100, find I when P = 2000, r = 5 and t = 3.
    10. s = d ÷ t, find s when d = 360 and t = 4.

  2. Evaluate each algebraic expression for the given values. Fluency

    1. 3x + 7, when x = 4
    2. 2a² − 5, when a = 3
    3. 4m − 2n, when m = −2 and n = 3
    4. p² + 2pq, when p = 5 and q = −1
    5. (x + y) ÷ (x − y), when x = 9 and y = 3
    6. 3c² − 4c + 1, when c = −2
    7. √(a² + b²), when a = 3 and b = 4
    8. 2x + 1⁄x − 1, when x = 4

  3. Work backwards from the output to find the missing input. Show all working. Understanding

    1. Using A = lw, if A = 48 and w = 6, find l.
    2. Using v = u + at, if v = 35, u = 5 and t = 6, find a.
    3. Using A = πr², if A = 78.54 cm² (use π ≈ 3.14), find r.
    4. Using I = Prt ÷ 100, if I = 300, P = 1000 and r = 5, find t.
    5. Using V = lwh, if V = 120, l = 5 and h = 4, find w.

  4. For each scenario, identify the formula being used, substitute the values, and clearly show your working. Understanding

    1. The perimeter of a rectangle is 34 cm. The length is 10 cm. What is the width? Use P = 2(l + w).
    2. A car travels at a speed of 80 km/h for 2.5 hours. How far does it travel? Use d = vt.
    3. Simple interest of $240 is earned on a $1200 investment over 2 years. What is the annual interest rate? Use I = Prt ÷ 100.
    4. The kinetic energy formula is E = ½mv². Find E (in joules) when m = 5 kg and v = 12 m/s.

  5. Multi-formula problems. Choose the correct formula and apply it to solve each problem. Show all working and include units. Problem Solving

    1. A cylindrical water tank has radius 1.5 m and height 3 m.
      1. Find the volume of the tank using V = πr²h. Give your answer in terms of π and as a decimal (round to 2 d.p.).
      2. Water is leaking at 0.5 m³ per day. How many full days before the tank is empty?
    2. A ball is thrown and its height h (metres) after t seconds is given by h = −5t² + 20t + 1.
      1. Find the height after 1 second.
      2. Find the height after 3 seconds.
      3. When t = 4, what is the height? What does this tell you about the ball?
    3. The surface area of a sphere is given by A = 4πr².
      1. Find the surface area of a sphere with radius 7 cm. Round to 1 decimal place.
      2. A second sphere has surface area 200.96 cm². Find its radius (use π ≈ 3.14).
    4. Josie invests $5000 at a simple interest rate of 4% per annum. At the same time, Max invests $4000 at 6% per annum. After 5 years, who has earned more interest? By how much? Use I = Prt ÷ 100.
  6. Physics formulas. Apply the given formulas to solve each problem. Show all substitution steps and include units.
    Problem Solving
    1. Newton’s second law: F = ma. Find the force (in newtons) when m = 12 kg and a = 3.5 m/s².
    2. Using v² = u² + 2as, find v when u = 0, a = 10 and s = 45. Give your answer as an exact surd and as a decimal.
    3. The formula for momentum is p = mv. Find the momentum when m = 0.5 kg and v = 60 m/s. Then find the new velocity if the mass doubles but the momentum stays the same.
    4. The gravitational potential energy formula is E = mgh where g = 9.8. Find E when m = 3 kg and h = 15 m.
    5. Using the formula s = ut + ½at², find the distance s when u = 5, t = 4 and a = 2.
  7. Negative and fractional values. Take care with brackets when substituting negatives or fractions.
    Problem Solving
    1. Evaluate T = −2x² + 6x + 8 for x = −1.
    2. Evaluate R = (a − b) ÷ (a + b) for a = −3 and b = −5.
    3. The formula C = 5⁄9(F − 32) converts Fahrenheit to Celsius. Find C when F = 14. What does this temperature represent in practical terms?
    4. Evaluate y = 3x² − 2x + 1 for x = −1⁄2. Show all steps.
    5. The formula for the sum of an arithmetic series is S = n⁄2(a + l). Find S when n = 12, a = 3 and l = 47.
  8. Choose and apply. For each problem, identify the correct formula from those listed, then apply it.
    Problem Solving

    Formulas:   A = πr²   •   V = 4⁄3πr³   •   v = u + at   •   s = d/t   •   I = Prt/100

    1. A ball starts from rest and accelerates at 5 m/s² for 6 seconds. Find its final speed.
    2. A train travels 480 km in 3 hours. Find its average speed in km/h.
    3. A spherical balloon has radius 4 cm. Find its volume in terms of π and as a decimal (1 d.p.).
    4. An investor puts $8000 into an account at 3.5% simple interest for 4 years. Find the total interest earned.
    5. A circular pizza has diameter 30 cm. Find its area to the nearest cm².
  9. Compare and decide. Use the same formula with different inputs to make a comparison or find which scenario gives a larger/smaller result.
    Problem Solving
    1. Two circular pools have radii 3 m and 5 m. Using A = πr², find the area of each. How many times larger is the bigger pool’s area? Round to 1 decimal place.
    2. Using E = ½mv², compare the kinetic energy of Object A (m = 4 kg, v = 6 m/s) and Object B (m = 9 kg, v = 4 m/s). Which has more kinetic energy and by how much?
    3. Using I = Prt/100, find which gives more interest: $3000 at 6% for 4 years, or $4500 at 4% for 3 years. Show all working.
    4. A rocket’s height in metres is modelled by h = 80t − 5t². Evaluate h for t = 2, t = 8, and t = 16. What do the results tell you about the rocket’s journey?
  10. Real-world formulas. These problems involve formulas used in science, finance, and engineering.
    Problem Solving
    1. Einstein’s energy formula is E = mc², where c = 3 × 10&sup8; m/s. Find the energy released (in joules, in scientific notation) when m = 2 × 10³ kg. Show how you handle the powers of 10.
    2. The body mass index (BMI) formula is BMI = m ÷ h², where m = mass in kg and h = height in metres. Find the BMI of a person with m = 72 kg and h = 1.75 m. Round to 1 decimal place.
    3. The compound interest formula is A = P(1 + r)³ (for 3 years), where r is the decimal interest rate. Find A when P = 1000 and r = 0.05. Round to 2 decimal places and find the interest earned.
    4. Ohm’s Law: V = IR. (i) Find V when I = 2.5 A and R = 40 Ω. (ii) Find I when V = 120 V and R = 30 Ω. (iii) Find R when V = 9 V and I = 0.3 A.
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