Common factors and formula subjects

Factorisation reverses distribution

Factoring reverses distribution: 6a + 9ab = 3a(2 + 3b). Expanding again checks the factorisation. The common factor must divide every term.

Make a different quantity the subject

From v = u + at, subtract u from both sides: v-u = at. If a ≠ 0, divide by a: t = (v-u)/a. The brackets show that the whole difference is divided by a.

From P = 2l + 2w, subtract 2w, then divide by 2: l = (P-2w)/2 = P/2-w. A valid rearrangement preserves every pair of values satisfying the original, subject to the stated division conditions.

Find an unknown value from the given measurements

For a rectangle, P = 2l + 2w. Suppose the perimeter is 34 cm and the width is 6 cm. The unknown is the length, so substitution gives 34 = 2l + 12. Subtract 12 from both sides, then divide by 2: l = 11 cm.

The rearranged formula gives the same value: l = (34-2(6))/2 = 11. Check in the original: 2(11) + 2(6) = 34. Merely writing a new subject formula would not finish a question asking for a numerical length and its unit.

Check your understanding 1 Find the missing quantity, then check the formula

Use the supplied cuboid formula V = lwh. The volume is 180 cm³, the width is 5 cm and the height is 3 cm. Find the length. Show how you use all the given values and check your result in the original formula.

Your equation, unknown value and check, or work on paper

Show answer

Substitute to obtain 180 = l(5)(3) = 15l, so l = 180/15 = 12 cm. Equivalently, first rearrange to l = V/(wh), then substitute. Checking gives 12(5)(3) = 180 cm³. The divisor is the whole product of width and height; dividing by only one given dimension would not isolate the length.

A formula may have a restriction

Rearrange A = bh/2 for h. Why is b = 0 excluded from the rearranged quotient?