Printable · GCSE Foundation · ages 14-16
Rearranging formulae worksheet — GCSE Foundation
Fifteen questions on "rearranging formulae" — DfE statement A5. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Rearranging formulae worksheet — GCSE Foundation
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- 1.The formula for the circumference of a circle is C = 2πr, where r is the radius. Make r the subject of the formula.
- 2.Make b the subject of the formula a = 5b.
- 3.Make x the subject of the formula y = 2x − 9.y = 2x − 9
- 4.The formula for the volume of a cuboid is V = lwh, where l is the length, w is the width and h is the height. Make h the subject of the formula.
- 5.Make x the subject of the formula y = 4(x − 6).
- 6.Make x the subject of the formula y = 3(x + 2).
- 7.The formula for the perimeter of a square is P = 4s, where s is the side length. Make s the subject of the formula.
- 8.Make c the subject of the formula k = 6c.
- 9.A rectangular sheep pen has perimeter P, length l and width w, connected by the formula P = 2l + 2w. A farmer has 60 m of fencing for the perimeter and sets the width at 8 m. Work out the length of the pen.
- 10.The area of a triangle is given by the formula A = bh/2, where b is the base and h is the height. Make h the subject of the formula.
- 11.Make x the subject of the formula y = 5x + 3.y = 5x + 3
- 12.Make m the subject of the formula w = m − 8.
- 13.Make x the subject of the formula y = x/2 − 5.y = x
- 14.The perimeter of a rectangle is given by the formula P = 2(l + w), where l is the length and w is the width. A rectangular garden has a perimeter of 46 m and a length of 14 m. Rearrange the formula to make w the subject, then work out the width of the garden.
- 15.Make a the subject of the formula b = 20 − a.
Answer key
- (a) r = C / (2π) — Method: undo the multiplication by 2π by dividing both sides by 2π. Working: C = 2πr, so dividing both sides by 2π gives r = C / (2π). The value r = C / π comes from dividing by π only and forgetting the factor of 2 in 2π. The value r = 2πC comes from multiplying by 2π instead of dividing. The value r = C − 2π comes from subtracting 2π instead of dividing by it.
- (b) b = a/5 — Writing 5b means 5 × b, so b has been multiplied by 5. The inverse of multiplying by 5 is dividing by 5, and dividing both sides by 5 gives a/5 = b, which is written b = a/5. Multiplying both sides by 5 instead gives 5a, which applies the operation a second time; treating the 5 as though it were added gives a − 5; writing 5/a turns the fraction upside down.
- (a) x = (y + 9) / 2 — Method: undo the subtraction of 9 first, then undo the multiplication by 2. Working: y = 2x − 9, so adding 9 to both sides gives y + 9 = 2x, then dividing both sides by 2 gives x = (y + 9) / 2. The value x = (y − 9) / 2 comes from a sign error, keeping the −9 instead of moving it to +9. The value x = 2(y + 9) comes from multiplying by 2 instead of dividing. The value x = y / 2 + 9 comes from dividing by 2 before adding 9, the wrong order of operations, instead of dividing the whole bracket.
- (c) h = V / (lw) — Method: undo the multiplication by lw by dividing both sides by lw. Working: V = lwh, so dividing both sides by lw gives h = V / (lw). The value h = V − lw comes from subtracting lw instead of dividing by it. The value h = Vlw comes from multiplying by lw instead of dividing. The value h = lw / V comes from inverting the fraction, dividing lw by V instead of V by lw.
- (c) x = (y + 24)/4 — Method: expand the bracket first, then undo the operations done to x in reverse order. Working: y = 4(x − 6) = 4x − 24, so y + 24 = 4x, so x = (y + 24)/4. Answer: x = (y + 24)/4. x = (y + 6)/4 comes from expanding the bracket incorrectly, treating 4(x − 6) as 4x − 6 instead of 4x − 24. x = 4y + 96 comes from multiplying by 4 instead of dividing by 4 to undo the multiplication, giving 4(y + 24) = 4y + 96. x = y/4 − 6 comes from dividing by 4 first without expanding the bracket, then subtracting 6 as if the bracket's operation still applied afterwards.
- (b) x = y/3 − 2 — The bracket containing x has been multiplied by 3, so divide both sides by 3 first, which gives y/3 = x + 2. Subtracting 2 from both sides then leaves y/3 − 2 = x, so x = y/3 − 2. Taking the 2 away before dividing gives (y − 2)/3, which divides the 2 by 3 as well, although the 2 was never divided in the original formula. Adding 2 rather than subtracting it gives y/3 + 2, and multiplying by 3 instead of dividing gives 3y − 2.
- (c) s = P / 4 — Method: undo the multiplication by 4 by dividing both sides by 4. Working: P = 4s, so dividing both sides by 4 gives s = P / 4. The value s = 4P comes from multiplying by 4 instead of dividing — using the wrong inverse operation. The value s = P + 4 comes from adding 4 instead of dividing. The value s = P − 4 comes from subtracting 4 instead of dividing.
- (b) c = k/6 — k = 6c means c has been multiplied by 6, so to make c the subject, divide both sides by 6: c = k/6. Writing c = 6k multiplies by 6 again instead of undoing the multiplication. Writing c = k − 6 mistakes multiplying for adding, and subtracts 6 rather than dividing. Writing c = 6 − k reverses the order as well as the operation. The correct rearrangement is c = k/6.
- (c) 22 m — Rearranging P = 2l + 2w for l gives l = (P − 2w)/2. Substituting P = 60 and w = 8: l = (60 − 16)/2 = 44/2 = 22 m. Answering 30 m comes from ignoring the width altogether and working out l = P/2, which only holds when w = 0. Answering 38 m comes from adding 2w instead of subtracting it: (60 + 16)/2 = 38. Answering 11 m comes from dividing by 4 instead of 2, as if the formula were P = 4l + 4w. The pen's length is 22 m.
- (b) h = 2A/b — The height has been multiplied by the base and the result then divided by 2, so undo the division first. Multiplying both sides by 2 gives 2A = bh. Undoing the multiplication by the base comes next: dividing both sides by b gives 2A/b = h, so h = 2A/b. Dividing by 2 instead of multiplying gives A/(2b), a quarter of the correct height; multiplying by the base instead of dividing gives 2Ab; writing b/(2A) turns the final fraction upside down.
- (b) x = (y − 3)/5 — To make x the subject of y = 5x + 3, first subtract 3 from both sides to get y − 3 = 5x, then divide both sides by 5: x = (y − 3)/5. Writing x = (y + 3)/5 keeps the division correct but does not change the sign of the 3 when moving it across. Writing x = y/5 − 3 divides only the y term by 5 and leaves the 3 as a separate subtraction, instead of subtracting first and dividing the whole expression. Writing x = 5(y − 3) applies the correct order of subtracting 3 first, but then multiplies by 5 instead of dividing — the inverse of 5x is division, not multiplication. The correct rearrangement is x = (y − 3)/5.
- (b) m = w + 8 — To make m the subject of w = m − 8, add 8 to both sides so the subtraction is undone: m = w + 8. Writing m = w − 8 forgets to change the operation at all — it just relabels the equation without moving the 8 across. Writing m = 8 − w swaps the order and puts the wrong sign on w, as if the equation had been w = 8 − m instead. Writing m = w/8 mistakes subtraction for a scaling relationship and divides by 8 instead of adding it. The correct rearrangement is m = w + 8.
- (c) x = 2y + 10 — Method: undo the operations done to x in reverse order — add 5, then multiply by 2. Working: y = x/2 − 5, so y + 5 = x/2, so x = 2(y + 5) = 2y + 10. Answer: x = 2y + 10. x = 2y + 5 comes from multiplying only the x/2 term by 2 and forgetting to multiply the 5 as well. x = 2y − 10 comes from a sign error, subtracting 5 instead of adding it before multiplying by 2. x = (y + 5)/2 comes from dividing by 2 instead of multiplying, the wrong operation to undo a division.
- (a) 9 m — Undo the multiplication by the bracket first: dividing both sides by 2 gives P/2 = l + w. Subtracting the length from both sides gives w = P/2 − l. Substituting the measurements, 46 ÷ 2 = 23, and 23 − 14 = 9, so the width is 9 m. Taking the length off before halving gives (46 − 14) ÷ 2 = 16, which halves the length as well; expanding to P = 2l + 2w and then forgetting to divide by 2 gives 46 − 28 = 18; subtracting the length in the wrong direction gives 23 + 14 = 37.
- (a) a = 20 − b — Method: get a on its own by adding a to both sides, then subtracting b from both sides. Working: b = 20 − a, so b + a = 20, so a = 20 − b. Answer: a = 20 − b. a = b − 20 comes from treating the formula as if it read b = 20 + a and then subtracting 20 from both sides. a = 20 + b comes from moving b across from b + a = 20 without changing its sign. a = −20 − b comes from a double sign error, changing the sign of the 20 as well as of b when rearranging.
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