Secondary 4 G2 Mathematics Tutor Bukit Timah | Changing the Subject of Formulae. Did you know? The two formulas v = d/t and d = vt can describe the same relationship. One is useful when calculating speed; the other is useful when the question asks for distance. Rearranging is not inventing new mathematics. It is telling the same story with a different unknown.
For Secondary 4 G2 Mathematics students around Bukit Timah, Sixth Avenue, Beauty World and King Albert Park, making a variable the subject of a formula is a skill that connects algebra with real situations: distance, area, volume, money and changing rates. The challenge is to preserve equality through each step rather than recite “move it and change sign” without knowing which operation is being undone.
The 2027 Singapore-Cambridge SEC G2 Mathematics K210 syllabus explicitly includes changing the subject of a formula and finding unknown values from formulae under N5: Algebraic expressions and formulae. The 2026 graduating cohort should instead follow its applicable N(A)-Level Mathematics Syllabus A 4045 documentation. Families can compare SEAB’s 2026 listing with the 2027 G2 listing to ensure preparation matches the actual examination year.
Quick answer: what does ‘make x the subject’ mean?
It means write an equivalent formula with x alone on one side, so that x can be calculated once the other quantities are known. For example, if P = 2(l + w), making w the subject produces w = P/2 − l. Substitute any sensible rectangle dimensions to check that the new formula reproduces the original width.
- Circle the requested variable. Different questions may ask for l, w, h, r, time or rate.
- Group the operations acting on it. Notice brackets, coefficients, fractions, squares and added terms.
- Undo operations in a valid order. Apply the same algebraic operation to both sides while respecting non-zero denominators.
- Make the requested variable appear by itself. Do not stop at an equation where it remains multiplied, divided or squared unnecessarily.
- Check by substitution. Test numerical values that satisfy the original relation and verify sensible units.
A helpful mental image is a pair of balanced scales. Adding, subtracting, multiplying or dividing both sides by the same appropriate quantity retains balance. An unsupported “swap sides” shortcut can obscure a missing sign, denominator or bracket.
Match each algebraic obstacle to its inverse
| The target variable is… | Typical operation to undo | Example |
|---|---|---|
| In a sum | Subtract the other term | y = x + 7 becomes x = y − 7. |
| Multiplied by a constant | Divide by the non-zero constant | p = 5x becomes x = p/5. |
| Inside a bracket | Isolate the whole bracket, then remove it | A = 3(x + 2) becomes x = A/3 − 2. |
| In a denominator | Multiply and then isolate, with restrictions | v = d/t becomes t = d/v when v ≠ 0. |
| Squared | Take an appropriate square root | A = r² gives r = √A for a nonnegative length r. |
| On both sides | Collect terms containing it, then factor if needed | y = ax + bx gives x = y/(a + b), when a + b ≠ 0. |
Sixteen original worked formula-rearrangement examples
1. Distance from speed
Given v = d/t, make d the subject. Multiply by t: vt = d. d = vt for t ≠ 0. Units check: (km/h) × h = km.
2. Time from speed
Given v = d/t, make t the subject. Multiply by t and divide by v: t = d/v for v ≠ 0. In a real journey with non-zero distance, the speed used must also be nonzero.
3. Width from rectangle area
A = lw. Divide by the non-zero length l: w = A/l. Area units divided by length units yield a length.
4. Width from perimeter
P = 2(l + w). Divide by 2: P/2 = l + w. Subtract l: w = P/2 − l. Do not subtract 2l before dividing and lose the bracket.
5. x from a straight-line formula
y = mx + c. Subtract c: y − c = mx. Divide by m: x = (y − c)/m, provided m ≠ 0.
6. A temperature conversion
C = 5(F − 32)/9. Multiply by 9/5: 9C/5 = F − 32. Add 32: F = 9C/5 + 32. Check C = 0 gives F = 32.
7. Percentage-rate formula
For illustrative simple interest I = Prt/100, make r the subject: 100I = Prt, so r = 100I/(Pt), provided P and t are nonzero. Rate r is stated as a percentage here.
8. Radius from area of a circle
A = πr². Divide by π: r² = A/π. A radius is nonnegative, so r = √(A/π). The negative square root is not a physically valid radius.
9. Height from cylinder volume
V = πr²h. Divide by πr² to obtain h = V/(πr²), assuming r is nonzero. Check that the result has units of length.
10. One term from a weighted expression
P = 2a + 3b. Subtract 3b, then divide by 2: a = (P − 3b)/2. The full numerator must stay grouped.
11. Rearrange an average
M = (a + b)/2. Multiply both sides by 2: 2M = a + b; therefore b = 2M − a. Check a = 6, b = 10 gives M = 8.
12. Height from triangle area
A = (1/2)bh. Multiply by 2 and divide by b: h = 2A/b, for b ≠ 0. A length arises because area is divided by base length.
13. Unknown parallel side in a trapezium
A = (1/2)(a + b)h. Multiply by 2/h: 2A/h = a + b. Thus b = 2A/h − a, if h ≠ 0.
14. An unknown inside a numerator
Q = (x + y)/z. Multiply by z: Qz = x + y, so y = Qz − x, provided z ≠ 0.
15. Radius from the volume of a cone
V = (1/3)πr²h. Then r² = 3V/(πh), giving r = √[3V/(πh)] for a nonnegative radius and positive height. A cube root is not required: r is squared.
16. A formula with a target in a denominator
R = a/(b + c). Multiply by b + c and isolate c: R(b + c) = a, so c = a/R − b when R ≠ 0; the original requires b + c ≠ 0. Check before dividing.
17. Collect a repeated target variable
y = ax + bx. Factor the shared x: y = x(a + b), so x = y/(a + b) when a + b ≠ 0. Dividing by only a would ignore the bx term.
18. Reverse a linear-rate change
Suppose v = u + at. Make u the subject: u = v − at. Alternatively, make a the subject: a = (v − u)/t for t ≠ 0. One relationship can answer different unknowns.
What formula-rearrangement errors look like
| First wrong move | Why it fails | What to ask instead |
|---|---|---|
| P = 2(l + w) becomes w = P − 2l | The factor 2 was not removed from the whole bracket | What happens if the entire equation is divided by 2 first? |
| y = mx + c becomes x = y/m − c | The c was subtracted after the division | Is (y − c) grouped as the numerator? |
| A = πr² becomes r = A/(2π) | Squaring confused with multiplication by 2 | What operation reverses r²? |
| v = d/t becomes t = dv | The reciprocal relationship is reversed | What unit should t have when d is divided by v? |
| y = ax + bx becomes x = y/a − b | Did not first collect the terms in x | Can x be factored from both terms? |
| Uses a divided formula with zero denominator | A division-by-zero restriction was ignored | When would this coefficient be zero? |
One worked real-life story: estimate a room’s width
A rectangular study room has a perimeter of 30 m and a length of 9 m. Rearranging P = 2(l + w) gives w = P/2 − l. Substitute: w = 30/2 − 9 = 6 m. Check the original formula: 2(9 + 6) = 30 m.
Notice the two skills here. First comes rearranging the equation with symbols; second comes substituting numbers. Separating them is useful because the same formula can later handle a different perimeter or length without rebuilding the reasoning.
Changed questions for independent practice
- Make h the subject of V = lwh.
- Make x the subject of y = 5x − 4.
- Make b the subject of A = (a + b)h/2.
- Make r the subject of C = 2πr.
- Make t the subject of v = u + at.
Answers: (1) h = V/(lw) for lw ≠ 0; (2) x = (y + 4)/5; (3) b = 2A/h − a for h ≠ 0; (4) r = C/(2π); (5) t = (v − u)/a for a ≠ 0. Check each by substituting a set of sensible numerical values in both the original and rearranged formulas.
An illustrative small-group Maths tutorial near Sixth Avenue
| Minutes | Teaching decision | What shows progress |
|---|---|---|
| 0–10 | Recall inverse operations using simple equations | Student justifies every balancing action. |
| 10–25 | Short diagnostic of formulas with brackets, powers and quotients | Tutor identifies whether the problem is reading, order or algebra. |
| 25–50 | Tutor contrasts tempting wrong rearrangements with valid ones | Student corrects each unsupported transformation aloud. |
| 50–75 | Students individually rearrange unseen real-world formulas | Each produces a checked symbolic formula without a hint. |
| 75–90 | Exit question and delayed retrieval plan | Student can rework a changed formula several days later. |
The published eduKateSG Mathematics small-groups reference describes premium three-pupil, 1.5-hour weekly lessons near Sixth Avenue MRT with close feedback. The example above applies that learning format to formula rearrangement. It is not an announcement of a live Secondary 4 G2 class, tutor, fee or available seat.
A sensible one-week routine
- Day 1: practise ten simple one-step rearrangements and verify with numbers.
- Day 2: change the subject in five formulas involving brackets.
- Day 3: focus on formulas where the target is in a numerator or denominator.
- Day 4: practise two square-root rearrangements and discuss positive physical lengths.
- Day 5: mix money, distance and area formulae without giving topic hints.
- Day 6: attempt an unseen timed set and record the first wrong line.
- Day 7: retest the earlier two least secure methods after a delay.
Frequently asked questions
What does it mean to change the subject of a formula?
It means rewriting the equation so the requested letter appears alone on one side, with the other variables on the opposite side. Algebraic equivalence is preserved as long as the steps obey the applicable restrictions.
Do I have to substitute numbers immediately?
No. When the question asks only to make a letter the subject, the answer can remain symbolic. Substitution is useful as a check or a separate requested calculation.
Why are formulas with fractions harder?
There may be a target in a denominator, nested brackets or more than one term. Mark the structure and isolate groups in stages; do not try to remove every fraction by mental shortcuts.
Does this content match G2 Mathematics for 2027?
Yes. The official 2027 K210 N5 content includes changing the subject of formulae. A 2026 graduating candidate should study the 4045 documentation instead. Always use the exact entered examination syllabus.
Other Secondary 4 G2 learning guides
- Algebraic fractions and common denominators
- Equations, inequalities and simultaneous solutions
- Rates, speed and unit conversions
- 2026 N(A) and 2027 SEC revision roadmap
Official references: 2026 N(A)-Level syllabus directory · 2027 SEC G2 syllabus directory · K210 G2 Mathematics syllabus. For details of the original teaching model visit the unchanged Mathematics tuition page or eduKate Singapore. Worked examples are original educational problems, not national examination items.