Substitution into Simple Formulae

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What is a Formula?

A formula is a mathematical rule written using letters (called variables) to represent quantities. When you know the values of the variables, you substitute them in to calculate the result.

Example: \(P = 2l + 2w\) is the formula for the perimeter of a rectangle.
\(P\) = perimeter, \(l\) = length, \(w\) = width.

Substituting \(l = 8\) and \(w = 3\): \(P = 2(8) + 2(3) = 16 + 6 = 22\)

✓ Quick Check 1 — What is a Formula?

Question 1 of 10

Step-by-Step Substitution

1. Write the formula | 2. Replace each letter with its value | 3. Apply BODMAS to evaluate

Example 1: Using \(D = S \times T\), find \(D\) when \(S = 60\) and \(T = 3\)

Step 1

Write formula: \(D = S \times T\)

Step 2

Substitute: \(D = 60 \times 3\)

Step 3

Calculate: \(D = \mathbf{180}\) km

Example 2: Temperature conversion \(F = 1.8C + 32\). Find \(F\) when \(C = 25\)

Step 1

\(F = 1.8 \times 25 + 32\)

Step 2

Multiplication first (BODMAS): \(1.8 \times 25 = 45\)

Step 3

Addition: \(45 + 32 = \mathbf{77}\)°F

Example 3: Delivery cost formula \(C = 5 + 2n\). Find \(C\) when \(n = 12\)

Step 1

\(C = 5 + 2 \times 12\)

Step 2

Multiplication first: \(2 \times 12 = 24\)

Step 3

\(C = 5 + 24 = \mathbf{£29}\)

Example 4: Area of a triangle \(A = \frac{1}{2}bh\). Find \(A\) when \(b = 10\) and \(h = 6\)

Step 1

\(A = \frac{1}{2} \times 10 \times 6\)

Step 2

\(A = \frac{1}{2} \times 60 = \mathbf{30}\text{ cm}^2\)

⚠️ BODMAS matters! In \(F = 1.8C + 32\), multiply before adding. A common mistake is \(1.8 \times (25 + 32) = 1.8 \times 57 = 102.6\) — this is wrong. Substitute first, then follow BODMAS.

Formulae with Powers

Example 5: Area of a circle \(A = \pi r^2\). Find \(A\) when \(r = 5\) (use \(\pi \approx 3.14\))

Step 1

\(A = 3.14 \times 5^2\)

Step 2

Power first (BODMAS): \(5^2 = 25\)

Step 3

\(A = 3.14 \times 25 = \mathbf{78.5}\text{ cm}^2\)

💡 Tip: Always deal with powers before multiplication. In \(A = \pi r^2\), you square \(r\) first, then multiply by \(\pi\).

✓ Quick Check 2 — Step-by-Step Substitution

Question 1 of 10

✓ Quick Check 3 — Formulae with Powers & Mixed Practice

Question 1 of 10

Practice: Extra Questions

Substitute the values into each formula and work out the answer.

#	Question	Your answer	Result
1	P = 2l + 2w. l = 5, w = 3. Find P.
2	A = ½bh. b = 8, h = 5. Find A.
3	Pay = r × h. r = 3, h = 4. Find Pay.
4	C = 5(F − 32) ÷ 9. F = 32. Find C.
5	s = d ÷ t. d = 120, t = 3. Find s.
6	P = 2l + 2w. l = 7, w = 4. Find P.
7	Cost = 3n + 2. n = 5. Find Cost.
8	y = 3x + 2. x = 4. Find y.
9	A = bh ÷ 2. b = 10, h = 6. Find A.
10	C = πd (π ≈ 3.14). d = 8. Find C to 2 d.p.
11	F = ma. m = 4, a = 5. Find F.
12	A = ½bh. b = 6, h = 3. Find A.
13	D = S × T. S = 5, T = 4. Find D.

⚠ Things to Watch Out For

After substituting values, apply BODMAS/BIDMAS correctly — deal with indices and brackets before multiplication, division, then addition and subtraction.
Don't confuse different letters in a formula — read the formula carefully and substitute each variable with the correct given value.
Squaring a negative number gives a positive result — (−3)² = 9, not −9.
Substitution is not the same as rearranging — don't move terms around before substituting unless asked to rearrange the formula first.
Include appropriate units in your final answer — a formula for speed gives an answer in m/s or km/h, not just a bare number.

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Substitution into Simple Formulae

Learning Objectives

Real-World Applications

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Example 1: Using \(D = S \times T\), find \(D\) when \(S = 60\) and \(T = 3\)

Example 2: Temperature conversion \(F = 1.8C + 32\). Find \(F\) when \(C = 25\)

Example 3: Delivery cost formula \(C = 5 + 2n\). Find \(C\) when \(n = 12\)

Example 4: Area of a triangle \(A = \frac{1}{2}bh\). Find \(A\) when \(b = 10\) and \(h = 6\)

Example 5: Area of a circle \(A = \pi r^2\). Find \(A\) when \(r = 5\) (use \(\pi \approx 3.14\))

Practice: Extra Questions

⚠ Things to Watch Out For