GCSE Maths: Substituting into expressions (Higher)
Part of the GCSE Maths substitution and formulae chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Substituting into expressions
To substitute means to replace each letter with the number it stands for. Take the expression 3x + 2 when x = 5. Wherever we see x we write 5 instead, giving 3 × 5 + 2, and then we work it out: 15 + 2 = 17.
Notice that we put the × sign back in. In algebra 3x means 3 × x. If you just replaced x by 5 you would write 35, which is thirty-five and is wrong. The number 3 is next to the letter, not next to the number 5.
The order of working: BIDMAS
After substituting you have an ordinary calculation, so use the usual order of operations. Many people remember it as BIDMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. Do the top row first.
Powers
The small number in x2 is a power (an index). It tells us how many times to multiply: x2 means x × x, not 2 × x. When x = 4, x2 is 4 × 4 = 16 (a square of side 4 has 16 little squares), but 2x is just 2 × 4 = 8. In an expression like 2x2, only the x is squared. Square first, then multiply by 2.
Brackets, fractions and roots
Work out the inside of a bracket first. The line of a fraction acts like brackets too: work out the top and the bottom separately, then divide the top by the bottom. A square root sign works the same way: work out what is under the sign first, then take the root.
Key ideas
- Substitute: replace every letter with its number. Put the × sign back where a number touches a letter.
- Put brackets round negative numbers when you substitute (see 8.3).
- Then follow BIDMAS: brackets, powers, then multiply/divide, then add/subtract (left to right).
- x2 means x × x. In 5x2 only the x is squared.
- For a fraction, work out the top and the bottom first, then divide.
Watch out
3x with x = 4 is 3 × 4 = 12, not 34. And 2x2 with x = 3 is 2 × 9 = 18, not (2 × 3)2 = 36. Square first, then multiply.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.
- Write down the expression. Start by copying the expression exactly.3x + 2
- Substitute: x = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3x + 2 = 3 × 5 + 2
- 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
= 15 + 2 - 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 = 17
= 17
Check. Put the numbers straight into a calculator: 3 × 5 + 2 gives 17. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.2a2 − b
- Substitute: a = 3, b = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.2a2 − b = 2 × 32 − 4
- 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.32 = 3 × 3 = 9
= 2 × 9 − 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 × 9 = 18
= 18 − 4 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.18 − 4 = 14
= 14
Check. Put the numbers straight into a calculator: 2 × 32 − 4 gives 14. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.2a + bc
- Substitute: a = 4, b = 7, c = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.2a + bc = 2 × 4 + 75
- Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.2 × 4 = 8
= 8 + 75 - Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.8 + 7 = 15
= 155 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.155 means 15 ÷ 5 = 3
= 3
Check. Put the numbers straight into a calculator: (2 × 4 + 7)/5 gives 3. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.√x2 + y2
- Substitute: x = 6, y = 8. Substitute means replace each letter with its number.√x2 + y2 = √62 + 82
- Under the root sign first: work out the powers (indices). Inside the root sign we still use BIDMAS, so powers come first.62 = 6 × 6 = 36
82 = 8 × 8 = 64
= √36 + 64 - Under the root sign first: do the additions and subtractions. Inside the root sign we still use BIDMAS. Only adding and subtracting are left, so work from left to right.36 + 64 = 100
= √100 - Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.√100 = 10 because 10 × 10 = 100
= 10
Check. Put the numbers straight into a calculator: √62 + 82 gives 10. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.3x2 − y2(x − y)
- Substitute: x = 4, y = 2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3x2 − y2(x − y) = 3 × 42 − 22 × (4 − 2)
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.4 − 2 = 2
= 3 × 42 − 22 × 2 - Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.42 = 4 × 4 = 16
2 × 2 = 4
= 3 × 16 − 24 - Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.3 × 16 = 48
= 48 − 24 - Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.48 − 2 = 46
= 464 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.464 means 46 ÷ 4 = 11.5
= 11.5
Check. Put the numbers straight into a calculator: (3 × 42 − 2)/(2 × (4 − 2)) gives 11.5. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Exercise 8B Substituting into expressions
Non-calculatorKey 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.
- Write down the expression. Start by copying the expression exactly.3x
- 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.3x = 3 × 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 × 4 = 12
= 12
Check. Put the numbers straight into a calculator: 3 × 4 gives 12. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.x + 7
- Substitute: x = 4. Substitute means replace each letter with its number.x + 7 = 4 + 7
- Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.4 + 7 = 11
= 11
Check. Put the numbers straight into a calculator: 4 + 7 gives 11. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.10 − x
- Substitute: x = 4. Substitute means replace each letter with its number.10 − x = 10 − 4
- Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.10 − 4 = 6
= 6
Check. Put the numbers straight into a calculator: 10 − 4 gives 6. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.x2
- Substitute: x = 4. Substitute means replace each letter with its number.x2 = 42
- Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.42 means 4 ÷ 2 = 2
= 2
Check. Put the numbers straight into a calculator: 4/2 gives 2. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.5x − 3
- 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.5x − 3 = 5 × 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.5 × 4 = 20
= 20 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.20 − 3 = 17
= 17
Check. Put the numbers straight into a calculator: 5 × 4 − 3 gives 17. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.2x + 9
- 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.2x + 9 = 2 × 4 + 9
- 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
= 8 + 9 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.8 + 9 = 17
= 17
Check. Put the numbers straight into a calculator: 2 × 4 + 9 gives 17. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.a + b
- Substitute: a = 3, b = 5. Substitute means replace each letter with its number.a + b = 3 + 5
- Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.3 + 5 = 8
= 8
Check. Put the numbers straight into a calculator: 3 + 5 gives 8. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.ab
- Substitute: a = 3, b = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.ab = 3 × 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 × 5 = 15
= 15
Check. Put the numbers straight into a calculator: 3 × 5 gives 15. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.2a + b
- Substitute: a = 3, b = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.2a + b = 2 × 3 + 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.2 × 3 = 6
= 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.6 + 5 = 11
= 11
Check. Put the numbers straight into a calculator: 2 × 3 + 5 gives 11. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.3b − a
- Substitute: b = 5, a = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3b − a = 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
= 15 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.15 − 3 = 12
= 12
Check. Put the numbers straight into a calculator: 3 × 5 − 3 gives 12. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.ab + 4
- Substitute: a = 3, b = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.ab + 4 = 3 × 5 + 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 × 5 = 15
= 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.15 + 4 = 19
= 19
Check. Put the numbers straight into a calculator: 3 × 5 + 4 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. 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.
- Write down the expression. Start by copying the expression exactly.4a − 2b + 1
- Substitute: a = 3, b = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.4a − 2b + 1 = 4 × 3 − 2 × 5 + 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.4 × 3 = 12
2 × 5 = 10
= 12 − 10 + 1 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.12 − 10 = 2
= 2 + 1 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.2 + 1 = 3
= 3
Check. Put the numbers straight into a calculator: 4 × 3 − 2 × 5 + 1 gives 3. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.qp
- Substitute: q = 10, p = 2. Substitute means replace each letter with its number.qp = 102
- Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.102 means 10 ÷ 2 = 5
= 5
Check. Put the numbers straight into a calculator: 10/2 gives 5. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.pq
- Substitute: p = 2, q = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.pq = 2 × 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 × 10 = 20
= 20
Check. Put the numbers straight into a calculator: 2 × 10 gives 20. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.q − p
- Substitute: q = 10, p = 2. Substitute means replace each letter with its number.q − p = 10 − 2
- Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.10 − 2 = 8
= 8
Check. Put the numbers straight into a calculator: 10 − 2 gives 8. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.3p + q
- Substitute: p = 2, q = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3p + q = 3 × 2 + 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.3 × 2 = 6
= 6 + 10 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.6 + 10 = 16
= 16
Check. Put the numbers straight into a calculator: 3 × 2 + 10 gives 16. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.2q − 5p
- Substitute: q = 10, p = 2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.2q − 5p = 2 × 10 − 5 × 2
- 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
5 × 2 = 10
= 20 − 10 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.20 − 10 = 10
= 10
Check. Put the numbers straight into a calculator: 2 × 10 − 5 × 2 gives 10. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.x2
- Substitute: x = 3. Substitute means replace each letter with its number.x2 = 32
- 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.32 = 3 × 3 = 9
= 9
Check. Put the numbers straight into a calculator: 32 gives 9. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.2x2
- 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.2x2 = 2 × 32
- 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.32 = 3 × 3 = 9
= 2 × 9 - 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 × 9 = 18
= 18
Check. Put the numbers straight into a calculator: 2 × 32 gives 18. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.x2 + 1
- Substitute: x = 3. Substitute means replace each letter with its number.x2 + 1 = 32 + 1
- 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.32 = 3 × 3 = 9
= 9 + 1 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.9 + 1 = 10
= 10
Check. Put the numbers straight into a calculator: 32 + 1 gives 10. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.x3
- Substitute: x = 3. Substitute means replace each letter with its number.x3 = 33
- 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.33 = 3 × 3 × 3 = 9 × 3 = 27
= 27
Check. Put the numbers straight into a calculator: 33 gives 27. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.5x2 − x
- 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.5x2 − x = 5 × 32 − 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.32 = 3 × 3 = 9
= 5 × 9 − 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 × 9 = 45
= 45 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.45 − 3 = 42
= 42
Check. Put the numbers straight into a calculator: 5 × 32 − 3 gives 42. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.10 − x2
- Substitute: x = 3. Substitute means replace each letter with its number.10 − x2 = 10 − 32
- 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.32 = 3 × 3 = 9
= 10 − 9 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.10 − 9 = 1
= 1
Check. Put the numbers straight into a calculator: 10 − 32 gives 1. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.2(n + 3)
- Substitute: n = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.2(n + 3) = 2 × (5 + 3)
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.5 + 3 = 8
= 2 × 8 - 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 × 8 = 16
= 16
Check. Put the numbers straight into a calculator: 2 × (5 + 3) gives 16. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.3(n − 1)
- Substitute: n = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3(n − 1) = 3 × (5 − 1)
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.5 − 1 = 4
= 3 × 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 × 4 = 12
= 12
Check. Put the numbers straight into a calculator: 3 × (5 − 1) gives 12. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4(2n − 3)
- Substitute: n = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.4(2n − 3) = 4 × (2 × 5 − 3)
- Brackets first: do the multiplications and divisions. Inside the brackets we still use BIDMAS, so multiplying and dividing come before adding and subtracting.2 × 5 = 10
= 4 × (10 − 3) - Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.10 − 3 = 7
= 4 × 7 - 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.4 × 7 = 28
= 28
Check. Put the numbers straight into a calculator: 4 × (2 × 5 − 3) gives 28. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.(n − 2)2
- Substitute: n = 5. Substitute means replace each letter with its number.(n − 2)2 = (5 − 2)2
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.5 − 2 = 3
= 32 - 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.32 = 3 × 3 = 9
= 9
Check. Put the numbers straight into a calculator: (5 − 2)2 gives 9. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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.
- Write down the expression. Start by copying the expression exactly.(n + 2)(n − 1)
- Substitute: n = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.(n + 2)(n − 1) = (5 + 2) × (5 − 1)
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.5 + 2 = 7
5 − 1 = 4
= 7 × 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.7 × 4 = 28
= 28
Check. Put the numbers straight into a calculator: (5 + 2) × (5 − 1) gives 28. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.x + y3
- Substitute: x = 8, y = 4. Substitute means replace each letter with its number.x + y3 = 8 + 43
- Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.8 + 4 = 12
= 123 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.123 means 12 ÷ 3 = 4
= 4
Check. Put the numbers straight into a calculator: (8 + 4)/3 gives 4. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.xy
- Substitute: x = 8, y = 4. Substitute means replace each letter with its number.xy = 84
- Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.84 means 8 ÷ 4 = 2
= 2
Check. Put the numbers straight into a calculator: 8/4 gives 2. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.3xy
- Substitute: x = 8, y = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.3xy = 3 × 84
- Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.3 × 8 = 24
= 244 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.244 means 24 ÷ 4 = 6
= 6
Check. Put the numbers straight into a calculator: (3 × 8)/4 gives 6. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.x + 4y − 2
- Substitute: x = 8, y = 4. Substitute means replace each letter with its number.x + 4y − 2 = 8 + 44 − 2
- Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.8 + 4 = 12
4 − 2 = 2
= 122 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.122 means 12 ÷ 2 = 6
= 6
Check. Put the numbers straight into a calculator: (8 + 4)/(4 − 2) gives 6. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.x2y
- Substitute: x = 8, y = 4. Substitute means replace each letter with its number.x2y = 824
- Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.82 = 8 × 8 = 64
= 644 - Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.644 means 64 ÷ 4 = 16
= 16
Check. Put the numbers straight into a calculator: 82/4 gives 16. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Method. Check what 5x really means, then substitute properly.
- Say what 5x means. A number next to a letter means multiply.5x = 5 × x
- Say what Tia did. She wrote the digits 5 and 3 side by side, which makes the number fifty-three.That is the mistake.
- Substitute correctly, with the × sign. x = 3.5 × 3 = 15
Check. Another way: 5x = x + x + x + x + x = 3 + 3 + 3 + 3 + 3 = 15. ✓
Watch out. Whenever you substitute into something like 5x, write the × sign straight away.
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 = 8n + 12
- Substitute: n = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 8 × 3 + 12
- 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.8 × 3 = 24
C = 24 + 12 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.24 + 12 = 36
C = 36 - Write the answer. The answer should say what was being worked out.C = £36
Check. Put the numbers straight into a calculator: 8 × 3 + 12 gives 36. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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 = 8n + 12
- Substitute: n = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 8 × 6 + 12
- 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.8 × 6 = 48
C = 48 + 12 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.48 + 12 = 60
C = 60 - Write the answer. The answer should say what was being worked out.C = £60
Check. Put the numbers straight into a calculator: 8 × 6 + 12 gives 60. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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 = 8n + 12
- 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.C = 8 × 10 + 12
- 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.8 × 10 = 80
C = 80 + 12 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.80 + 12 = 92
C = 92 - Write the answer. The answer should say what was being worked out.C = £92
Check. Put the numbers straight into a calculator: 8 × 10 + 12 gives 92. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4x − 3
- Substitute: x = 1. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.4x − 3 = 4 × 1 − 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.4 × 1 = 4
= 4 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.4 − 3 = 1
= 1
Check. Put the numbers straight into a calculator: 4 × 1 − 3 gives 1. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4x − 3
- 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.4x − 3 = 4 × 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.4 × 2 = 8
= 8 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.8 − 3 = 5
= 5
Check. Put the numbers straight into a calculator: 4 × 2 − 3 gives 5. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4x − 3
- 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.4x − 3 = 4 × 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.4 × 3 = 12
= 12 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.12 − 3 = 9
= 9
Check. Put the numbers straight into a calculator: 4 × 3 − 3 gives 9. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4x − 3
- 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.4x − 3 = 4 × 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.4 × 4 = 16
= 16 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.16 − 3 = 13
= 13
Check. Put the numbers straight into a calculator: 4 × 4 − 3 gives 13. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.4x − 3
- Substitute: x = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.4x − 3 = 4 × 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.4 × 5 = 20
= 20 − 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.20 − 3 = 17
= 17
Check. Put the numbers straight into a calculator: 4 × 5 − 3 gives 17. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.P = 2(l + w)
- Substitute: l = 12, w = 7.5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.P = 2 × (12 + 7.5)
- Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.12 + 7.5 = 19.5
P = 2 × 19.5 - 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.2 × 19.5 = 39
P = 39 - Write the answer with its units. The answer should say what was being worked out.P = 39 cm
Check. Put the numbers straight into a calculator: 2 × (12 + 7.5) gives 39. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.A = lw
- Substitute: l = 12, w = 7.5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.A = 12 × 7.5
- 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.12 × 7.5 = 90
A = 90 - Write the answer with its units. The answer should say what was being worked out.A = 90 cm2
Check. Put the numbers straight into a calculator: 12 × 7.5 gives 90. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.√x
- Substitute: x = 49. Substitute means replace each letter with its number.√x = √49
- Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.√49 = 7 because 7 × 7 = 49
= 7
Check. Put the numbers straight into a calculator: √49 gives 7. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
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.
- Write down the expression. Start by copying the expression exactly.√2x + 1
- Substitute: x = 12. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.√2x + 1 = √2 × 12 + 1
- Under the root sign first: do the multiplications and divisions. Inside the root sign we still use BIDMAS, so multiplying and dividing come before adding and subtracting.2 × 12 = 24
= √24 + 1 - Under the root sign first: do the additions and subtractions. Inside the root sign we still use BIDMAS. Only adding and subtracting are left, so work from left to right.24 + 1 = 25
= √25 - Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.√25 = 5 because 5 × 5 = 25
= 5
Check. Put the numbers straight into a calculator: √2 × 12 + 1 gives 5. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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.
- Write down the expression. Start by copying the expression exactly.√x2 + y2
- Substitute: x = 5, y = 12. Substitute means replace each letter with its number.√x2 + y2 = √52 + 122
- Under the root sign first: work out the powers (indices). Inside the root sign we still use BIDMAS, so powers come first.52 = 5 × 5 = 25
122 = 12 × 12 = 144
= √25 + 144 - Under the root sign first: do the additions and subtractions. Inside the root sign we still use BIDMAS. Only adding and subtracting are left, so work from left to right.25 + 144 = 169
= √169 - Work out the root. A square root asks: which number multiplied by itself gives this? The root sign works like a bracket, so the number under it is worked out first.√169 = 13 because 13 × 13 = 169
= 13
Check. Put the numbers straight into a calculator: √52 + 122 gives 13. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
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 = 5h + 3
- 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 = 5 × 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.5 × 4 = 20
C = 20 + 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.20 + 3 = 23
C = 23 - Write the answer. The answer should say what was being worked out.C = £23
Check. Put the numbers straight into a calculator: 5 × 4 + 3 gives 23. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
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 = 5h + 3
- Substitute: h = 7. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.C = 5 × 7 + 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 × 7 = 35
C = 35 + 3 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.35 + 3 = 38
C = 38 - Write the answer. The answer should say what was being worked out.C = £38
Check. Put the numbers straight into a calculator: 5 × 7 + 3 gives 38. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Subtract Sam’s cost from Dev’s cost.
- Use the two costs from parts (a) and (b). Dev paid £38 and Sam paid £23.£38 − £23
- Subtract. 38 − 23 = 15.£15
Check. Dev pays for 3 extra hours at £5 each: 3 × 5 = 15. ✓
Watch out. Read what the question asks: it wants the difference, not the cost.
Method. Work out both expressions for each value of x and compare them in a table.
- Work out 2x and x2 for each value. Square first, then compare with doubling.x = 1: 2x = 2 × 1 = 2, x2 = 12 = 1
x = 2: 2x = 2 × 2 = 4, x2 = 22 = 4
x = 3: 2x = 2 × 3 = 6, x2 = 32 = 9
x = 4: 2x = 2 × 4 = 8, x2 = 42 = 16 - Find where they are equal. Look for the same number in both.Both are 4 when x = 2.
- Compare after that. For x = 3: 6 against 9. For x = 4: 8 against 16.So after x = 2, x2 is bigger: squaring grows faster than doubling.
Check. For x = 1, 2x = 2 is bigger than x2 = 1, so 2x is not always smaller. ✓
Watch out. Do not assume 2x and x2 are the same. They only agree for x = 2 (and x = 0).
Method. Try each value of n in turn and see which one gives 25.
- Try n = 4. Multiply first, then add.4 × 4 + 1 = 16 + 1 = 17 (not 25)
- Try n = 5. 4 × 5 + 1 = 20 + 1 = 21 (not 25)
- Try n = 6. 4 × 6 + 1 = 24 + 1 = 25 (this is 25)
Check. Check: 4 × 6 = 24, and 24 + 1 = 25. ✓
Watch out. This is a trial-and-improvement method. In Chapter 9 you will meet a quicker way: solving an equation.
Method. Substitute into the fraction, then compare with x + y.
- Substitute into the fraction. Work out the top and the bottom separately.72 − 327 − 3 = 49 − 94 = 404 = 10
- Work out x + y. 7 + 3 = 10
- Compare. Both are 10.The fraction gives the same value as x + y. This is because x2 − y2 = (x − y)(x + y) (difference of two squares), and (x − y) cancels.
Check. Try another pair, x = 9 and y = 5: 92 − 529 − 5 = 81 − 254 = 564 = 14 and 9 + 5 = 14. ✓
Watch out. Noticing a pattern with numbers is not a proof, but it suggests what to prove with algebra.
Method. First find the two numbers by listing pairs that multiply to 20, then substitute.
- List the pairs of whole numbers that multiply to 20. xy = 20 means x × y = 20.1 × 20, 2 × 10, 4 × 5
- Find the pair that adds to 9. 1 + 20 = 21, 2 + 10 = 12, 4 + 5 = 9.The numbers are 4 and 5.
- Substitute into x2 + y2. It does not matter which is x and which is y.42 + 52 = 16 + 25 = 41
Check. Swapping them gives the same: 52 + 42 = 25 + 16 = 41. ✓
Watch out. Check both conditions: 4 + 5 = 9 and 4 × 5 = 20.