GCSE Maths: Writing formulae (Foundation)
Part of the GCSE Maths substitution and formulae chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Writing formulae
So far the formulae were given to you. Now you will write your own. Writing a formula means describing a rule in algebra, so that you can use it for any number.
Follow these steps:
- Choose letters and say what each stands for, including units (for example C = cost in pounds, n = number of tickets).
- Split the rule into a fixed part (paid once, never changes) and a changing part (a rate multiplied by the amount).
- Write the formula with the subject on its own on the left.
- Test it with a small easy number to see if it gives the right answer.
Many real rules have the shape total = fixed charge + rate × amount. A taxi that charges £3 to start and £2 per mile has cost C = 3 + 2m. A phone contract of £8 a month plus 4p for each minute has cost C = 8 + 0.04m pounds (we must write 4p as £0.04 so the units match).
Formulae from diagrams
When a shape has sides given by expressions, label every side, then add them for the perimeter or multiply for the area. Remember that a rectangle has two lengths and two widths, so its perimeter is 2 × length + 2 × width, and any bracket you write needs the 2 to multiply both terms inside it.
Key ideas
- Choose letters and say what they stand for (with units).
- Typical shape: total = fixed part + (rate × amount).
- For shapes: label the sides, then add (perimeter) or multiply (area). Use brackets when a sum is multiplied.
- Always test your formula with a simple number.
- Make sure the units match (pounds with pounds, minutes with minutes).
Watch out
Do not mix units. If the rate is in pence per minute and the total is in pounds, change the pence to pounds first. And write the fixed charge on its own, not multiplied: £3 start plus £2 per mile is 3 + 2m, not 5m.Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£8 a month” A fixed amount, paid once. It does not depend on the minutes.8
- “4p for each minute” 4p = £0.04. With m minutes the cost is 0.04 × m.0.04m
- “plus” The fixed charge and the minute charge are added.+
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 8 + 0.04m
Check. Test with a number. Let m = 100. In words: £8 plus 100 minutes at £0.04 = £8 + £4 = 12. The expression gives 8 + 0.04 × 100 = 8 + 4 = 12. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Method. The perimeter of a rectangle is two lengths plus two widths. Write it, expand the brackets, then collect like terms.
- Write the perimeter. There are 2 lengths and 2 widths.P = 2(x + 3) + 2(2x − 1)
- Expand both brackets. Multiply every term inside each bracket by the 2.2 × x = 2x and 2 × 3 = 6
2 × 2x = 4x and 2 × (−1) = −2
P = 2x + 6 + 4x − 2 - Collect like terms. 2x + 4x = 6x and 6 − 2 = 4.P = 6x + 4 cm
Check. Try x = 5: length = 8, width = 9, perimeter = 2 × 8 + 2 × 9 = 34. And 6 × 5 + 4 = 34. ✓
Watch out. Do not forget to multiply the second term in each bracket: 2(2x − 1) = 4x − 2, not 4x − 1.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “each extra triangle adds 2 sticks” Look at the pattern: 3, 5, 7. It goes up by 2 each time, so n triangles give 2 × n = 2n sticks.2n
- “the first triangle needs 3 sticks, which is one more than 2” For n = 1, 2n = 2 but we need 3, so we add 1.+ 1
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.m = 2n + 1
Check. Test with a number. Let n = 3. In words: 3 triangles use 7 sticks = 7. The expression gives 2 × 3 + 1 = 6 + 1 = 7. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Exercise 8F Writing formulae
Non-calculatorKey words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£7 each” The price of one ticket.7
- “n tickets” Multiply the price by the number of tickets.7 × n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 7n
Check. Test with a number. Let n = 4. In words: 4 tickets at £7 = 28. The expression gives 7 × 4 = 28. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£40 call-out” A fixed charge, paid once.40
- “£25 per hour for h hours” Rate times amount, added to the fixed charge.+ 25h
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.T = 40 + 25h
Check. Test with a number. Let h = 3. In words: £40 plus 3 hours at £25 = 115. The expression gives 40 + 25 × 3 = 40 + 75 = 115. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “4 equal sides of length s” The perimeter is all four sides added: s + s + s + s.4s
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.P = 4s
Check. Test with a number. Let s = 5. In words: 4 sides of 5 cm = 20. The expression gives 4 × 5 = 20. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “side times side” The area of a square is its side multiplied by itself.s × s
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.A = s2
Check. Test with a number. Let s = 5. In words: 5 × 5 = 25. The expression gives 52 = 25. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “add the two numbers” The mean needs the total first.a + b
- “divide by 2” There are two numbers.÷ 2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.m = a + b2
Check. Test with a number. Let a = 4, b = 10. In words: (4 + 10) shared between 2 = 7. The expression gives 4 + 102 = 142 = 7. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£20” The amount we start with.20
- “n books at £4 each” The spending is taken away from the £20.− 4n
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 20 − 4n
Check. Test with a number. Let n = 3. In words: £20 minus 3 books (£12) = 8. The expression gives 20 − 4 × 3 = 20 − 12 = 8. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£40 call-out plus £25 per hour” As in the previous question.40 + 25h
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.T = 40 + 25h
Check. Test with a number. Let h = 3. In words: £40 plus 3 hours at £25 = 115. The expression gives 40 + 25 × 3 = 40 + 75 = 115. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.T = 40 + 25h
- Substitute: h = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.T = 40 + 25 × 3
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.25 × 3 = 75
T = 40 + 75 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.40 + 75 = 115
T = 115 - Write the answer. The answer should say what was being worked out.T = £115
Check. Put the numbers straight into a calculator: 40 + 25 × 3 gives 115. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Work backwards: take off the call-out charge, then see how many hours that pays for.
- Take away the call-out charge. The £140 includes the fixed £40.140 − 40 = 100
- Divide by the hourly rate. Each hour costs £25.100 ÷ 25 = 4
- Answer. 4 hours
Check. Put h = 4 in the formula: T = 40 + 25 × 4 = 40 + 100 = 140. ✓
Watch out. You cannot just divide 140 by 25. The £40 is paid once, so take it away first.
Method. The perimeter is two lengths plus two widths.
- Write it. Length is x + 5, width is 3.P = 2(x + 5) + 2 × 3
- Expand the bracket. Multiply both terms by 2.P = 2x + 10 + 6
- Collect like terms. 10 + 6 = 16.P = 2x + 16
Check. Try x = 1: sides 6 and 3, perimeter 2 × 6 + 2 × 3 = 18, and 2 × 1 + 16 = 18. ✓
Watch out. The 2 multiplies both terms in the bracket.
Method. The area is length times width.
- Write it. Area = length × width.A = 3(x + 5)
- Expand the bracket. Multiply both terms by 3.3 × x = 3x and 3 × 5 = 15
- Write the answer. A = 3x + 15
Check. Try x = 1: sides 6 and 3, area 18, and 3 × 1 + 15 = 18. ✓
Watch out. Area uses multiplication, perimeter uses addition.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “square of side s: perimeter” Four equal sides.4s
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.P = 4s
Check. Test with a number. Let s = 6. In words: 4 sides of 6 = 24. The expression gives 4 × 6 = 24. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “square of side s: area” Side times side.s2
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.A = s2
Check. Test with a number. Let s = 6. In words: 6 × 6 = 36. The expression gives 62 = 36. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.m = 2n + 1
- Substitute: n = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.m = 2 × 10 + 1
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.2 × 10 = 20
m = 20 + 1 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.20 + 1 = 21
m = 21
Check. Put the numbers straight into a calculator: 2 × 10 + 1 gives 21. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. The formula m = 2n + 1 always gives an odd number. Compare with 100.
- Think about 2n. 2n is 2 multiplied by a whole number, so it is always even.Even
- Add the 1. An even number plus 1 is odd.2n + 1 is always odd.
- Compare with 100. 100 is even.So 100 sticks can never be used up exactly.
Check. Try: n = 49 gives 99 and n = 50 gives 101. Neither is 100. ✓
Watch out. Test the formula: it only produces odd numbers.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£6 per hour for h hours” Rate times number of hours.6h
- “plus a £10 deposit” The deposit is paid once.+ 10
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 6h + 10
Check. Test with a number. Let h = 4. In words: 4 hours at £6 is £24, plus £10 = 34. The expression gives 6 × 4 + 10 = 24 + 10 = 34. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.C = 6h + 10
- Substitute: h = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 6 × 4 + 10
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.6 × 4 = 24
C = 24 + 10 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.24 + 10 = 34
C = 34 - Write the answer. The answer should say what was being worked out.C = £34
Check. Put the numbers straight into a calculator: 6 × 4 + 10 gives 34. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£15 a month” Fixed charge.15
- “20p per unit, u units” 20p = £0.20. Units multiplied by the rate.+ 0.2u
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 15 + 0.2u
Check. Test with a number. Let u = 100. In words: £15 plus 100 units at £0.20 = 35. The expression gives 15 + 0.2 × 100 = 15 + 20 = 35. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.C = 15 + 0.2u
- Substitute: u = 250. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 15 + 0.2 × 250
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.0.2 × 250 = 50
C = 15 + 50 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.15 + 50 = 65
C = 65 - Write the answer. The answer should say what was being worked out.C = £65
Check. Put the numbers straight into a calculator: 15 + 0.2 × 250 gives 65. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Work out the true cost for 2 pencils, test Matt’s formula, then correct it.
- Work out the true cost. 2 pencils at 45p is 90p = £0.90, plus the £2 ruler.£0.90 + £2 = £2.90
- Test Matt’s formula with n = 2. Substitute.C = 45 + 2 × 2 = 45 + 4 = 49 (way too big)
- Say what is wrong. He has put 45 on its own and 2 multiplied by n. The 45p belongs with the pencils and the £2 is the fixed cost. He also forgot the units: 45p is £0.45.The rate 0.45 should multiply n.
- Write the correct formula. Pencils: 0.45 × n. Ruler: add 2.C = 0.45n + 2
Check. Test n = 2: C = 0.45 × 2 + 2 = 0.9 + 2 = 2.9 = £2.90, matching the true cost. ✓
Watch out. The letter goes with the thing that changes. Fixed costs are added on their own.
Method. For this L-shape, the perimeter is the same as the perimeter of the rectangle that fits round it. Find that rectangle’s width and height.
- Find the total width. The bottom bar is 3x wide, and the whole shape is no wider.width = 3x
- Find the total height. The left bar is 5 tall and sits on top of the bottom bar, which is 4 tall.height = 4 + 5 = 9
- Add round the outside. A step shape has the same perimeter as its bounding rectangle: 2 widths and 2 heights.P = 2 × 3x + 2 × 9 = 6x + 18
Check. Add the six sides one by one: 3x + 4 + 2x + 5 + x + 9 = 6x + 18. (The step is 3x − x = 2x long.) ✓
Watch out. The six sides all count. Do not forget the short step along the top of the lower bar.
Method. Split the shape into two rectangles, find each area and add.
- Bottom rectangle. It is 3x wide and 4 tall.3x × 4 = 12x
- Top rectangle. It is x wide and 5 tall.x × 5 = 5x
- Add the areas. 12x + 5x = 17x.A = 17x cm²
Check. Try x = 2: bottom 6 by 4 = 24, top 2 by 5 = 10, total 34. And 17 × 2 = 34. ✓
Watch out. Cut the shape into rectangles you know the area of.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “£120 hire” A fixed charge.120
- “£3.50 for each of g guests” Rate times number of guests.+ 3.5g
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 120 + 3.5g
Check. Test with a number. Let g = 10. In words: £120 plus 10 guests at £3.50 = 155. The expression gives 120 + 3.5 × 10 = 120 + 35 = 155. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.C = 120 + 3.5g
- Substitute: g = 60. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 120 + 3.5 × 60
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.3.5 × 60 = 210
C = 120 + 210 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.120 + 210 = 330
C = 330 - Write the answer. The answer should say what was being worked out.C = £330
Check. Put the numbers straight into a calculator: 120 + 3.5 × 60 gives 330. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. The fence goes along three sides only: the two widths and one length (the fourth side is the wall).
- Count the sides that need fencing. Two sides of width w and one of length l.w + w + l
- Write the formula. Collect the two w terms.F = l + 2w
Check. Try l = 10 and w = 3: fence = 3 + 3 + 10 = 16, and 10 + 2 × 3 = 16. ✓
Watch out. The side along the wall is only counted once, and only for the length.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.F = l + 2w
- Substitute: l = 40, w = 15. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.F = 40 + 2 × 15
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.2 × 15 = 30
F = 40 + 30 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.40 + 30 = 70
F = 70 - Write the answer with its units. The answer should say what was being worked out.F = 70 m
Check. Put the numbers straight into a calculator: 40 + 2 × 15 gives 70. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Key words. A variable is a letter that stands for a number that can change. An expression is a group of terms made from numbers and letters, with no equals sign.
Method. Take the sentence one piece at a time, write each piece in algebra, then join the pieces.
- “n shirts at £p” The full price is the number times the price of one.np
- “20% off means pay 80%” 100% − 20% = 80% = 0.8.× 0.8
- Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.C = 0.8np
Check. Test with a number. Let n = 3, p = 10. In words: full price £30, pay 80% = 24. The expression gives 0.8 × 3 × 10 = 2.4 × 10 = 24. They agree. ✓
Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.C = 0.8np
- Substitute: n = 3, p = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 0.8 × 3 × 10
- Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.0.8 × 3 = 2.4
C = 2.4 × 10 - Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.2.4 × 10 = 24
C = 24 - Write the answer. The answer should say what was being worked out.C = £24
Check. Put the numbers straight into a calculator: 0.8 × 3 × 10 gives 24. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Substitute m = 6 into both formulae.
- Firm A. A = 2 + 1.5 × 6 = 2 + 9 = 11
- Firm B. B = 5 + 6 = 11
- Compare. The answers are equal.Both firms charge £11 for 6 miles.
Check. Firm A: £2 + 6 × £1.50 = £2 + £9 = £11. Firm B: £5 + £6 = £11. ✓
Watch out. Compare the costs for the same distance.
Method. Work out both costs for 10 miles and compare.
- Firm A. A = 2 + 1.5 × 10 = 2 + 15 = 17
- Firm B. B = 5 + 10 = 15
- Compare. 15 is less than 17.Firm B is cheaper by £2.
Check. B has the larger starting charge but the smaller rate per mile, so it wins on long journeys. ✓
Watch out. A cheaper rate per mile beats a cheaper starting charge when the journey is long enough.