Printable · GCSE Foundation · ages 14-16
Substitution into formulae worksheet — GCSE Foundation
Fifteen questions on "substitution into formulae" — DfE statement A2. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Substitution into formulae worksheet — GCSE Foundation
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- 1.A rectangle is x cm wide and twice as long as it is wide, so its area A cm² is given by A = 2x². Work out A when x = 3.
- 2.A formula is F = 4s. A pupil says that when s = 5 the value of F is 9, because they added 4 and 5 instead of multiplying. Work out the correct value of F when s = 5.
- 3.The formula y = x² + 1 gives y in terms of x. Work out the value of y when x = −2.y = x² + 1
- 4.Work out the value of 3b − 4 when b = 6
- 5.The formula for the energy stored in a stretched spring is E = (1/2)kx², where E is in joules, k is the spring constant in N/m and x is the extension in m. A spring with a spring constant of 40 N/m is stretched so that its extension is 0.3 m. Work out the energy stored in the spring.
- 6.The volume of a cuboid is given by the formula V = lwh. Work out V, in cm³, when l = 2, w = 3 and h = 6.
- 7.Work out the value of 5y + 9 when y = 0
- 8.Oliver has m stickers and gives 5 of them away. The number he has left is L = m − 5. Work out L when m = 8.
- 9.On Monday, a runner covers 15 km in 2.5 hours. On Tuesday, she covers 12 km in 1.5 hours. Using the formula speed = distance ÷ time, work out on which day she ran faster, and by how much.
- 10.A person's body mass index is given by the formula B = m ÷ h², where m is the mass in kg and h is the height in metres. Work out B, to 1 decimal place, when m = 68 and h = 1.6.
- 11.Work out the value of 3x² − 4x when x = −2.
- 12.The braking distance of a car, in metres, is estimated using the formula d = 0.4v² + 6, where v is the speed in mph. A car travelling at 35 mph brakes. Work out the estimated braking distance.
- 13.A recipe gives the cooking time, in minutes, for a chicken as T = 40m + 20, where m is the mass in kg. A chicken has a mass of 1.8 kg. Work out the cooking time in hours and minutes.
- 14.Work out the value of 4p² − 3 when p = 2.5
- 15.The formula connecting speed, distance and time is v = s ÷ t. Work out v, in km/h, when s = 180 and t = 4.
Answer key
- (a) 18 cm² — Method: substitute the width into the formula, applying the index to the letter before multiplying by the 2 in front of it. Working: x² = 3 × 3 = 9, and then A = 2 × 9 = 18, so the area is 18 cm². Answer: 18 cm². The distractors: 36 cm² comes from multiplying 2 by 3 first and squaring afterwards, giving (2 × 3)²; 12 cm² comes from doubling instead of squaring, so that x² is replaced by 2x and the calculation becomes 2 × 2 × 3; 6 cm² comes from working out 2 × 3 and never applying the index at all.
- (b) 20 — F = 4s, so when s = 5, F = 4 × 5 = 20. 9 comes from adding 4 and 5 instead of multiplying, the mistake described in the question. 25 comes from squaring s (5²) instead of multiplying by 4. 45 comes from writing the digits 4 and 5 next to each other instead of carrying out the multiplication.
- (c) 5 — Method: substitute the value into the formula, work out the index first and the addition afterwards. Working: x² = (−2) × (−2) = 4, so y = 4 + 1 = 5. Answer: 5. The distractors: −3 comes from squaring only the 2 and keeping the minus sign outside the index, giving −4 + 1; 1 comes from adding the 1 to x before squaring, giving (−2 + 1)²; 4 comes from squaring correctly but stopping there and never adding the 1.
- (c) 14 — Substitute b = 6 into 3b − 4: 3 × 6 − 4 = 18 − 4 = 14. 5 comes from treating 3b as 3 + b instead of 3 × b, giving 3 + 6 − 4. 6 comes from subtracting 4 from b before multiplying by 3, 3 × (6 − 4). 18 comes from working out 3 × 6 correctly but forgetting to subtract the 4.
- (a) 1.8 — E = (1/2)kx² = 0.5 × 40 × 0.3² = 0.5 × 40 × 0.09 = 1.8 joules. 3.6 comes from forgetting the (1/2) at the front, 40 × 0.09. 72 comes from squaring kx together instead of squaring only x, 0.5 × (40 × 0.3)² = 0.5 × 144 = 72. 6 comes from using x instead of x², 0.5 × 40 × 0.3.
- (d) 36 cm³ — V = lwh = 2 × 3 × 6 = 36 cm³. A candidate who adds all three numbers instead of multiplying gets 2 + 3 + 6 = 11 cm³. A candidate who multiplies only two of the three numbers, forgetting the length, gets w × h = 3 × 6 = 18 cm³. A candidate who multiplies l × w and w × h separately and adds the two products gets (2 × 3) + (3 × 6) = 6 + 18 = 24 cm³.
- (c) 9 — Method: replace the letter with the value given for it, then work out the multiplication before the addition. Working: 5y means 5 × y, and 5 × 0 = 0, so the expression becomes 0 + 9, which is 9. Answer: 9. The distractors: 14 comes from treating 5 × 0 as 5, a common slip with zero, and then adding the 9; 0 comes from assuming that an expression containing a zero must itself be zero, so the + 9 is ignored; 45 comes from multiplying the whole expression by 5 instead of multiplying only y, giving 5 × (0 + 9).
- (a) 3 — Method: substitute the given value into the formula and carry out the subtraction in the order the formula is written. Working: replacing m with 8 gives L = 8 − 5, and 8 − 5 = 3. Answer: 3. The distractors: 13 comes from adding 5 to 8 instead of subtracting it; 40 comes from reading m − 5 as a multiplication and working out 8 × 5; −3 comes from subtracting the wrong way round and working out 5 − 8.
- (a) Tuesday, by 2 km/h — Monday's speed is 15 ÷ 2.5 = 6 km/h and Tuesday's speed is 12 ÷ 1.5 = 8 km/h, so Tuesday was faster, by 8 − 6 = 2 km/h. A candidate who works out the correct speeds but mislabels which day is faster gets Monday, by 2 km/h. A candidate who divides 15 ÷ 2.5 incorrectly as 5 instead of 6 gets a difference of 8 − 5 = 3 km/h, still crediting Tuesday. A candidate who forgets to find Monday's speed and gives Tuesday's speed itself as the difference states Tuesday, by 8 km/h.
- (c) 26.6 — h² = 1.6² = 2.56, so B = 68 ÷ 2.56 = 26.5625, which rounds to 26.6 (1 d.p.). A candidate who forgets to square the height gets 68 ÷ 1.6 = 42.5. A candidate who squares the mass instead of the height gets 68² ÷ 1.6 = 2890.0. A candidate who truncates 26.5625 instead of rounding it gets 26.5.
- (b) 20 — 3x² − 4x = 3(−2)² − 4(−2) = 3(4) − (−8) = 12 + 8 = 20. A candidate who makes a sign error on −4x, treating −4 × −2 as −8 instead of +8, gets 12 − 8 = 4. A candidate who squares −2 but keeps the result negative, using 3 × (−4) = −12 for the first term, but correctly works out −4x = −4 × (−2) = 8, gets −12 + 8 = −4. A candidate who squares the whole term 3x, working out (3 × −2)² − 4 × (−2), gets (−6)² + 8 = 36 + 8 = 44.
- (c) 496 — d = 0.4v² + 6. First v² = 35 × 35 = 1225. Then 0.4 × 1225 = 490. Add 6: 490 + 6 = 496 m. 490 comes from forgetting to add the 6 at the end. 202 comes from squaring 0.4v together instead of squaring only v, (0.4 × 35)² + 6 = 14² + 6 = 202. 20 comes from using v instead of v², 0.4 × 35 + 6.
- (d) 1 hour 32 minutes — T = 40 × 1.8 + 20 = 72 + 20 = 92 minutes. Since 92 = 60 + 32, the cooking time is 1 hour 32 minutes. A candidate who rounds the mass to 2 kg before substituting gets 40 × 2 + 20 = 100 minutes = 1 hour 40 minutes. A candidate who forgets to add the 20 minutes gets 40 × 1.8 = 72 minutes = 1 hour 12 minutes. A candidate who multiplies the mass by (40 + 20) = 60 instead of substituting into the formula gets 1.8 × 60 = 108 minutes = 1 hour 48 minutes.
- (c) 22 — p² means p × p. Substitute p = 2.5: p² = 6.25, so 4p² = 4 × 6.25 = 25, and 25 − 3 = 22. 97 comes from squaring 4p together instead of just p, (4 × 2.5)² − 3 = 10² − 3 = 97. 7 comes from using p instead of p², 4 × 2.5 − 3. 1 comes from squaring the 3 instead of the p, 4 × 2.5 − 3².
- (d) 45 — v = s ÷ t = 180 ÷ 4 = 45 km/h. 720 comes from multiplying s and t instead of dividing, 180 × 4. 184 comes from adding s and t instead of dividing, 180 + 4. 176 comes from subtracting t from s instead of dividing, 180 − 4.
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