Formulae and Equations

A taxi firm works out fares with the formula C = 2.5m + 3, where C is the fare in pounds and m is the number of miles. What is the fare for a 14-mile journey?

• A£38.00✓
• B£42.50
• C£35.00
• D£44.50

Answer: Substitute m = 14: C = 2.5 × 14 + 3 = 35 + 3 = £38.00. Multiplication comes before addition. Working out 2.5 × (14 + 3) gives £42.50, and leaving out the fixed £3 gives £35.00.

A driving guide estimates a car's stopping distance d, in feet, at a speed of v mph with the formula d = v²/20 + v. What stopping distance does it give at 40 mph?

• A120 feet✓
• B42 feet
• C80 feet
• D44 feet

Answer: Substitute v = 40: v² = 40 × 40 = 1,600, and 1,600 ÷ 20 = 80. Then add v: 80 + 40 = 120 feet. Leaving off the + v gives 80 feet; working out 40 × 2 instead of 40² gives 44 feet.

An electrician uses the formula C = 60 + 38h to work out a bill in pounds, where h is the number of hours worked. How much is the bill for a job lasting 1.5 hours?

• A£57
• B£98
• C£117✓
• D£147

Answer: Substitute h = 1.5: 38 × 1.5 = 57, then C = 60 + 57 = £117. Multiplying the whole of 60 + 38 by 1.5 gives £147, and £57 leaves out the £60 call-out charge.

A circular table top has a radius of 0.6 m. A metal strip will be fitted all the way round its edge. Using C = 2πr and π = 3.14, what length of strip is needed, to 2 decimal places?

• A7.54 m
• B1.13 m
• C3.77 m✓
• D1.88 m

Answer: C = 2 × 3.14 × 0.6 = 3.768 m, which is 3.77 m to 2 decimal places. 1.13 m is the area (πr²) in square metres, not a length, and 1.88 m forgets the 2 in the formula.

The formula F = 9C/5 + 32 converts a temperature in degrees Celsius (C) to degrees Fahrenheit (F). What is -10°C in degrees Fahrenheit?

• A14°F✓
• B-50°F
• C39.6°F
• D50°F

Answer: Substitute C = -10: 9 × -10 = -90, and -90 ÷ 5 = -18. Then -18 + 32 = 14°F. Dropping the minus sign gives 18 + 32 = 50°F; adding 32 before multiplying gives 39.6°F.

A van hire firm charges C = 45d + 0.2m pounds, where d is the number of days and m is the number of miles driven. What is the charge for 3 days, driving 260 miles?

• A£187✓
• B£135
• C£97
• D£655

Answer: Substitute d = 3 and m = 260: 45 × 3 = 135 and 0.2 × 260 = 52. C = 135 + 52 = £187. £135 covers only the daily charge, and reading 0.2 as 2 gives £655.

The area of a trapezium is A = 1/2(a + b)h, where a and b are the parallel sides and h is the distance between them. A flower bed has parallel sides of 3.5 m and 6.5 m that are 4 m apart. What is its area?

• A20 m²✓
• B27.75 m²
• C40 m²
• D14 m²

Answer: Add the parallel sides: 3.5 + 6.5 = 10. Half of that is 5, and 5 × 4 = 20 m². Forgetting the 1/2 gives 40 m². Working out 1/2 × 3.5 and then adding 6.5 × 4 ignores the brackets and gives 27.75 m².

A delivery driver uses the formula time = distance ÷ speed. She must drive 135 miles at an average speed of 54 mph. How long will the drive take?

• A2 hours 5 minutes
• B24 minutes
• C2 hours 50 minutes
• D2 hours 30 minutes✓

Answer: Time = 135 ÷ 54 = 2.5 hours. Half an hour is 30 minutes, so the drive takes 2 hours 30 minutes. Reading 2.5 hours as 2 hours 5 minutes is a common slip; dividing the other way round (54 ÷ 135 = 0.4 hours) gives 24 minutes.

The perimeter of a rectangle is given by P = 2(l + w). A rectangular garden has a perimeter of 34 m and a length of 11 m. What is its width?

• A12 m
• B3 m
• C23 m
• D6 m✓

Answer: 34 = 2(11 + w), so dividing both sides by 2 gives 17 = 11 + w, and w = 17 - 11 = 6 m. Check: 2 × (11 + 6) = 34. Subtracting 11 twice from 34 gives 12 m, which is the total of the two widths.

A cookbook gives the turkey roasting time as T = 20W + 90, where T is in minutes and W is the weight in kilograms. How long should a 5.5 kg turkey roast for?

• A3 hours 2 minutes
• B1 hour 50 minutes
• C3 hours 20 minutes✓
• D2 hours 0 minutes

Answer: Substitute W = 5.5: 20 × 5.5 = 110, and 110 + 90 = 200 minutes. 200 minutes = 3 × 60 + 20, which is 3 hours 20 minutes. Leaving out the 90 gives 110 minutes, or 1 hour 50 minutes.

A lab technician uses the formula N = 500 × 2^t to estimate the number of bacteria after t hours. How many bacteria does it predict after 3 hours?

• A1,500
• B3,000
• C8,000
• D4,000✓

Answer: Powers first: 2³ = 2 × 2 × 2 = 8. Then N = 500 × 8 = 4,000. Reading 2³ as 2 × 3 gives 3,000, and 8,000 comes from using 2⁴ instead of 2³.

Body Mass Index is worked out as BMI = mass in kg ÷ (height in m)². What is the BMI of an adult who is 1.6 m tall and has a mass of 64 kg?

• A20
• B40
• C25✓
• D102.4

Answer: Square the height first: 1.6 × 1.6 = 2.56. Then BMI = 64 ÷ 2.56 = 25. Dividing by 1.6 without squaring gives 40, and dividing by 1.6 × 2 = 3.2 gives 20.

A mobile plan costs £12 a month plus 8p for each MB of data over 2,000 MB. The formula is C = 12 + 0.08(d - 2,000) pounds, where d is the MB used. What is the bill for a month when 2,350 MB are used?

• A£28.00
• B£40.00✓
• C£200.00
• D£292.00

Answer: Work out the bracket first: 2,350 - 2,000 = 350 MB over the limit. 0.08 × 350 = £28. C = 12 + 28 = £40.00. Charging every MB (0.08 × 2,350) gives £200.00, and £28.00 leaves out the £12 monthly charge.