GCSE Maths Substitution and Formulae (Foundation): check-up and exam-style questions

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. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   5x + 2
2. Substitute: x = −3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   5x + 2 = 5 × (−3) + 2
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.
   5 × (−3) = −15 (a negative times a positive gives a negative)
   = −15 + 2
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −15 + 2 = −13
   = −13

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 5 × (−3) + 2 gives −13. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

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.

1. Write down the formula. This is the rule we are going to use.
   v = u + at
2. Substitute: u = 4, a = 3, t = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   v = 4 + 3 × 5
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.
   3 × 5 = 15
   v = 4 + 15
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   4 + 15 = 19
   v = 19
5. Write the answer with its units. The answer should say what was being worked out.
   v = 19 m/s

Check. Put the numbers straight into a calculator: 4 + 3 × 5 gives 19. ✓

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.

1. “£6 for each book, n books” Rate times number.
   6n
2. “plus £2 postage” A fixed charge, added on.
   + 2
3. Put the pieces together. Join the pieces in the same order as the words, using the operations the words describe.
   C = 6n + 2

Check. Test with a number. Let n = 3. In words: 3 books at £6 is £18, plus £2 = 20. The expression gives 6 × 3 + 2 = 18 + 2 = 20. They agree. ✓

Watch out. Read the words carefully. “5 less than n” is n − 5 (start from n), not 5 − n.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 4x − 1
   Reading from the subject outwards, what happens to x: it is multiplied by 4; then 1 is subtracted.
   To undo it we reverse the order: add 1, then divide by 4.
2. Add 1 to both sides. 1 is being subtracted. The inverse of subtracting is adding, so add 1 to both sides.
   y + 1 = 4x − 1 + 1
   y + 1 = 4x
3. Divide both sides by 4. x is being multiplied by 4 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 4.
   y + 14 = 4x4
   y + 14 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y + 14

Check. Pick numbers for the other letters: y = 3. The new formula gives
x = 3 + 14 = 44 = 1
Now put x = 1 back into the original formula. Left side: 3. Right side: 4 × 1 − 1 = 4 − 1 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until w is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get w on its own we must undo everything that has been done to it, in the reverse order.
   Formula: P = 2l + 2w
   Reading from the subject outwards, what happens to w: it is multiplied by 2; then 2l is added.
   To undo it we reverse the order: subtract 2l, then divide by 2.
2. Subtract 2l from both sides. 2l is being added to the part containing w. The inverse (opposite) of adding is subtracting, so subtract 2l from both sides. Doing the same to both sides keeps the formula true.
   P − 2l = 2l + 2w − 2l
   P − 2l = 2w
3. Divide both sides by 2. w is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   P − 2l2 = 2w2
   P − 2l2 = w
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   w = P − 2l2

Check. Pick numbers for the other letters: P = 10, l = 3. The new formula gives
w = 10 − 2 × 32 = 10 − 62 = 42 = 2
Now put w = 2 back into the original formula. Left side: 10. Right side: 2 × 3 + 2 × 2 = 6 + 4 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

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. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x² − 3x
2. Substitute: x = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   x² − 3x = (−2)² − 3 × (−2)
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−2)² = (−2) × (−2) = 4 (a negative times a negative is positive)
   = 4 − 3 × (−2)
4. 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 × (−2) = −6 (a negative times a positive gives a negative)
   = 4 − (−6)
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   4 − (−6) = 4 + 6 = 10 (subtracting a negative number is the same as adding)
   = 10

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−2)² − 3 × (−2) gives 10. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

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.

1. Write down the formula. This is the rule we are going to use.
   E = 12mv²
2. Substitute: m = 6, v = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   E = 12 × 6 × 4²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   4² = 4 × 4 = 16
   E = 12 × 6 × 16
4. 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.
   12 × 6 = 3
   E = 3 × 16
5. 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 × 16 = 48
   E = 48
6. Write the answer with its units. The answer should say what was being worked out.
   E = 48 J

Check. Put the numbers straight into a calculator: 1/2 × 6 × 4² gives 48. ✓

Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until t is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get t on its own we must undo everything that has been done to it, in the reverse order.
   Formula: v = u + at
   Reading from the subject outwards, what happens to t: it is multiplied by a; then u is added.
   To undo it we reverse the order: subtract u, then divide by a.
2. Subtract u from both sides. u is being added to the part containing t. The inverse (opposite) of adding is subtracting, so subtract u from both sides. Doing the same to both sides keeps the formula true.
   v − u = u + at − u
   v − u = at
3. Divide both sides by a. t is being multiplied by a (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by a.
   v − ua = ata
   v − ua = t
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   t = v − ua

Check. Pick numbers for the other letters: v = 10, u = 6, a = 4. The new formula gives
t = 10 − 64 = 44 = 1
Now put t = 1 back into the original formula. Left side: 10. Right side: 6 + 4 × 1 = 6 + 4 = 10. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Method. Perimeter = 2 lengths + 2 widths. Expand the brackets and collect like terms.

1. Write the perimeter.
   P = 2(2x − 1) + 2(x + 4)
2. Expand. Multiply both terms in each bracket by 2.
   P = 4x − 2 + 2x + 8
3. Collect like terms. 4x + 2x = 6x and −2 + 8 = 6.
   P = 6x + 6

Check. Try x = 2: length 3, width 6, perimeter 18. And 6 × 2 + 6 = 18. ✓

Watch out. Multiply every term in the bracket.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until r is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get r on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = πr²
   Reading from the subject outwards, what happens to r: it is squared; then it is multiplied by π.
   To undo it we reverse the order: divide by π, then take the square root.
2. Divide both sides by π. r is being multiplied by π (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by π.
   Aπ = πr²π
   Aπ = r²
3. Take the square root of both sides. r is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √Aπ = √r²
   √Aπ = r
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   r = √Aπ

Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): A = 3, π = 3. The new formula gives
r = √33 = √1 = 1
Now put r = 1 back into the original formula. Left side: 3. Right side: 3 × 1² = 3 × 1 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

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. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   5a − 2b
2. Substitute: a = 4, b = −3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   5a − 2b = 5 × 4 − 2 × (−3)
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.
   5 × 4 = 20
   2 × (−3) = −6 (a negative times a positive gives a negative)
   = 20 − (−6)
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   20 − (−6) = 20 + 6 = 26 (subtracting a negative number is the same as adding)
   = 26

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 5 × 4 − 2 × (−3) gives 26. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

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.

1. Write down the formula. This is the rule we are going to use.
   v = u + at
2. Substitute: u = 6, a = −2, t = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   v = 6 + (−2) × 4
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.
   −2 × 4 = −8 (a negative times a positive gives a negative)
   v = 6 + (−8)
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   6 + (−8) = 6 − 8 = −2 (adding a negative number is the same as subtracting)
   v = −2
5. Write the answer with its units. The answer should say what was being worked out.
   v = −2 m/s

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 6 + (−2) × 4 gives −2. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until x is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 5x − 7
   Reading from the subject outwards, what happens to x: it is multiplied by 5; then 7 is subtracted.
   To undo it we reverse the order: add 7, then divide by 5.
2. Add 7 to both sides. 7 is being subtracted. The inverse of subtracting is adding, so add 7 to both sides.
   y + 7 = 5x − 7 + 7
   y + 7 = 5x
3. Divide both sides by 5. x is being multiplied by 5 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 5.
   y + 75 = 5x5
   y + 75 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y + 75

Check. Pick numbers for the other letters: y = 3. The new formula gives
x = 3 + 75 = 105 = 2
Now put x = 2 back into the original formula. Left side: 3. Right side: 5 × 2 − 7 = 10 − 7 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.

Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until r is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get r on its own we must undo everything that has been done to it, in the reverse order.
   Formula: A = πr²
   Reading from the subject outwards, what happens to r: it is squared; then it is multiplied by π.
   To undo it we reverse the order: divide by π, then take the square root.
2. Divide both sides by π. r is being multiplied by π (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by π.
   Aπ = πr²π
   Aπ = r²
3. Take the square root of both sides. r is being squared. The inverse of squaring is taking the square root, so do that to both sides.
   √Aπ = √r²
   √Aπ = r
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   r = √Aπ

Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): A = 3, π = 3. The new formula gives
r = √33 = √1 = 1
Now put r = 1 back into the original formula. Left side: 3. Right side: 3 × 1² = 3 × 1 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Take the square root of the whole side, not of each term. And a square root can be positive or negative, but in these real-life formulae lengths and times are positive, so we use the positive root.

Method. Add two lengths and two widths. Expand and collect like terms.

1. Write the perimeter.
   P = 2(3x + 2) + 2(x − 1)
2. Expand. Multiply every term by 2.
   P = 6x + 4 + 2x − 2
3. Collect like terms. 6x + 2x = 8x and 4 − 2 = 2.
   P = 8x + 2

Check. Try x = 3: length 11, width 2, perimeter 26. And 8 × 3 + 2 = 26. ✓

Watch out. Marks: 1 for each bracket expanded correctly, 1 for the simplified answer.

Method. The speed formula needs the time in hours, so change 25 minutes to hours first.

1. Change the time to hours. There are 60 minutes in an hour.
   t = 25 ÷ 60 = 2560 = 512 hour
2. Substitute into s = d/t. d = 5 and t = 5/12.
   s = 5 ÷ 512
3. Divide by the fraction. Dividing by a fraction means multiplying by its reciprocal (Chapter 3).
   5 × 125 = 12 km/h

Check. Another way: 25 minutes is 5 km, so 5 minutes is 1 km, and 60 minutes is 12 km. ✓

Watch out. Marks: 1 for converting the time, 1 for the substitution, 1 for the answer with units.

Method. Expand both brackets, being careful with the minus sign, then collect like terms.

1. Expand the first bracket. 3 × x = 3x and 3 × 4 = 12.
   3x + 12
2. Expand the second bracket. The minus sign in front changes the sign of both terms: −2 × x = −2x and −2 × 1 = −2.
   −2x − 2
3. Collect like terms. 3x − 2x = x and 12 − 2 = 10.
   3x + 12 − 2x − 2 = x + 10

Check. Try x = 5: 3 × 9 − 2 × 6 = 27 − 12 = 15, and 5 + 10 = 15. ✓

Watch out. Marks: 1 for expanding both brackets correctly, 1 for collecting to reach x + 10.

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. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   2x² − x
2. Substitute: x = −4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   2x² − x = 2 × (−4)² − (−4)
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−4)² = (−4) × (−4) = 16 (a negative times a negative is positive)
   = 2 × 16 − (−4)
4. 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 × 16 = 32
   = 32 − (−4)
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   32 − (−4) = 32 + 4 = 36 (subtracting a negative number is the same as adding)
   = 36

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 2 × (−4)² − (−4) gives 36. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Method. Substitute with brackets and follow BIDMAS: the power is done before the minus sign in front.

1. Substitute. Put −2 in brackets.
   −(−2)²
2. Square first. (−2) × (−2) = 4.
   −(4)
3. Apply the minus sign. The minus sign in front makes it negative.
   −4
4. Conclude.
   Maya is not correct. She forgot that the minus sign comes after the squaring.

Check. −(−2)² = −4. ✓

Watch out. Marks: 1 for the correct working, 1 for the conclusion that the answer is −4.

Method. Write a cost for each firm for 4 hours and compare.

1. Cost for Firm A. Fixed £20 plus £5 for each of 4 hours.
   A = 20 + 5 × 4 = 20 + 20 = 40 pounds
2. Cost for Firm B. Fixed £8 plus £8 for each of 4 hours.
   B = 8 + 8 × 4 = 8 + 32 = 40 pounds
3. Conclude. Both are £40.
   The two firms charge the same amount for 4 hours.

Check. A: 20 + 4 × 5 = 40 and B: 8 + 4 × 8 = 40. ✓

Watch out. Marks: 1 for each firm’s cost, 1 for the conclusion.