GCSE Maths: Substituting into formulae (Foundation)
Part of the GCSE Maths substitution and formulae chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Substituting into formulae
A formula is a rule written with letters. Each letter stands for a quantity (a length, a time, a mass…). To use a formula we substitute the numbers we know and work out the one we do not. It is exactly what you did in 8.2 and 8.3, but now the letters mean something real, so the answer needs units.
In A = lw the letter on its own on the left, A, is called the subject of the formula. We use the formula to work out the subject. (In 8.6 we will learn to change which letter is the subject.)
Here are formulae you should know. Most appear in the GCSE exam without being given, so learn them.
| Quantity | Formula | What the letters mean (usual units) |
|---|---|---|
| Area of a rectangle | A = lw | l length, w width (cm², m²) |
| Perimeter of a rectangle | P = 2l + 2w | distance round the outside (cm, m) |
| Area of a triangle | A = 12bh | b base, h perpendicular height |
| Area of a trapezium | A = 12(a + b)h | a and b the parallel sides, h the distance between them |
| Volume of a cuboid | V = lwh | length, width, height (cm³, m³) |
| Circumference of a circle | C = πd | d diameter |
| Area of a circle | A = πr² | r radius |
| Speed | s = dt | distance d and time t (km/h, m/s) |
| Density | D = mV | mass m and volume V (g/cm³) |
| Pressure | P = FA | force F and area A (N/m²) |
| Force | F = ma | mass m (kg), acceleration a (m/s²), force in newtons (N) |
| Final speed | v = u + at | u starting speed, a acceleration, t time |
| Kinetic energy | E = 12mv² | mass m (kg), speed v (m/s), energy in joules (J) |
| Electric potential difference | V = IR | current I (amps), resistance R (ohms), volts (V) |
Using a formula: four steps
- Write the formula.
- Substitute each known value (with its × sign, and brackets round negatives).
- Work out the answer in the correct order (BIDMAS).
- Write the answer with its units.
Units must match. If the speed formula is used with distance in km and time in hours, the answer is in km/h. If the time is given in minutes, change it to hours first (30 minutes is 0.5 hours). On a calculator, use brackets for fractions: the top and the bottom each go in their own brackets, for example (3 + 5) ÷ (2 × 4), and use the π button rather than typing 3.14. Do not round until the very end.
Key ideas
- A formula links quantities. The letter on its own on the left is the subject.
- Write the formula, substitute, work out with BIDMAS, then write the answer with units.
- Keep units consistent: km with hours gives km/h, metres with seconds gives m/s.
- On a calculator use brackets for the top and bottom of a fraction, and use the π button.
- Round only at the end, and say how you rounded (decimal places or significant figures).
Watch out
In E = ½mv² the speed is squared, not the mass and not the whole product. With m = 4 and v = 10: square the 10 first (100), then ½ × 4 × 100 = 200. Squaring the whole product (½ × 4 × 10)² = 400 would be wrong.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.s = dt
- Substitute: d = 150, t = 2.5. Substitute means replace each letter with its number.s = 1502.5
- 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.1502.5 means 150 ÷ 2.5 = 60
s = 60 - Write the answer with its units. The answer should say what was being worked out.s = 60 km/h
Check. Put the numbers straight into a calculator: 150/2.5 gives 60. ✓
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. 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 = 12(a + b)h
- Substitute: a = 7, b = 11, h = 6. 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 + 11) × 6
- 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.7 + 11 = 18
A = 12 × 18 × 6 - 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 × 18 = 9
A = 9 × 6 - 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.9 × 6 = 54
A = 54 - Write the answer with its units. The answer should say what was being worked out.A = 54 cm2
Check. Put the numbers straight into a calculator: 1/2 × (7 + 11) × 6 gives 54. ✓
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.E = 12mv2
- Substitute: m = 4, v = 10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 4 × 102
- 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.102 = 10 × 10 = 100
E = 12 × 4 × 100 - 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 × 4 = 2
E = 2 × 100 - 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 × 100 = 200
E = 200 - Write the answer with its units. The answer should say what was being worked out.E = 200 J
Check. Put the numbers straight into a calculator: 1/2 × 4 × 102 gives 200. ✓
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.A = πr2
- Substitute: r = 6.3. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.A = π × 6.32
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.A = π × 6.32
Calculator display: 124.68981242… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 8, which is 5 or more, so we round up.124.68981242… = 124.7 (1 decimal place)
- Write the answer with its units. Say what the number means.A = 124.7 cm2
Check. Estimate with rounded numbers (r = 6, π = 3): 3 × 62 = 3 × 36 = 108. This is close to 124.7, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
Exercise 8D Using formulae (non-calculator)
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. 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 = 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 × 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 × 5 = 60
A = 60 - Write the answer with its units. The answer should say what was being worked out.A = 60 cm2
Check. Put the numbers straight into a calculator: 12 × 5 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.P = 2l + 2w
- Substitute: l = 12, w = 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 + 2 × 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 × 12 = 24
2 × 5 = 10
P = 24 + 10 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.24 + 10 = 34
P = 34 - Write the answer with its units. The answer should say what was being worked out.P = 34 cm
Check. Put the numbers straight into a calculator: 2 × 12 + 2 × 5 gives 34. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Key words. 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.V = lwh
- Substitute: l = 6, w = 4, h = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.V = 6 × 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.6 × 4 = 24
V = 24 × 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.24 × 3 = 72
V = 72 - Write the answer with its units. The answer should say what was being worked out.V = 72 cm3
Check. Put the numbers straight into a calculator: 6 × 4 × 3 gives 72. ✓
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 = 12bh
- Substitute: b = 10, h = 7. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.A = 12 × 10 × 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.12 × 10 = 5
A = 5 × 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.5 × 7 = 35
A = 35 - Write the answer with its units. The answer should say what was being worked out.A = 35 cm2
Check. Put the numbers straight into a calculator: 1/2 × 10 × 7 gives 35. ✓
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.s = dt
- Substitute: d = 240, t = 3. Substitute means replace each letter with its number.s = 2403
- 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.2403 means 240 ÷ 3 = 80
s = 80 - Write the answer with its units. The answer should say what was being worked out.s = 80 km/h
Check. Put the numbers straight into a calculator: 240/3 gives 80. ✓
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. 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.D = mV
- Substitute: m = 540, V = 200. Substitute means replace each letter with its number.D = 540200
- 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.540200 means 540 ÷ 200 = 2.7
D = 2.7 - Write the answer with its units. The answer should say what was being worked out.D = 2.7 g/cm3
Check. Put the numbers straight into a calculator: 540/200 gives 2.7. ✓
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. 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 = FA
- Substitute: F = 600, A = 20. Substitute means replace each letter with its number.P = 60020
- 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.60020 means 600 ÷ 20 = 30
P = 30 - Write the answer with its units. The answer should say what was being worked out.P = 30 N/m2
Check. Put the numbers straight into a calculator: 600/20 gives 30. ✓
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. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.F = ma
- Substitute: m = 12, a = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.F = 12 × 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 × 4 = 48
F = 48 - Write the answer with its units. The answer should say what was being worked out.F = 48 N
Check. Put the numbers straight into a calculator: 12 × 4 gives 48. ✓
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.v = u + at
- Substitute: u = 5, a = 3, t = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.v = 5 + 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
v = 5 + 12 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.5 + 12 = 17
v = 17 - Write the answer with its units. The answer should say what was being worked out.v = 17 m/s
Check. Put the numbers straight into a calculator: 5 + 3 × 4 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.v = u + at
- Substitute: u = 12, a = 2, t = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.v = 12 + 2 × 6
- 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 × 6 = 12
v = 12 + 12 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.12 + 12 = 24
v = 24 - Write the answer with its units. The answer should say what was being worked out.v = 24 m/s
Check. Put the numbers straight into a calculator: 12 + 2 × 6 gives 24. ✓
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.v = u + at
- Substitute: u = 0, a = 10, t = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.v = 0 + 10 × 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.10 × 3 = 30
v = 0 + 30 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.0 + 30 = 30
v = 30 - Write the answer with its units. The answer should say what was being worked out.v = 30 m/s
Check. Put the numbers straight into a calculator: 0 + 10 × 3 gives 30. ✓
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.V = IR
- Substitute: I = 3, R = 8. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.V = 3 × 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.3 × 8 = 24
V = 24 - Write the answer with its units. The answer should say what was being worked out.V = 24 V
Check. Put the numbers straight into a calculator: 3 × 8 gives 24. ✓
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.V = IR
- Substitute: I = 0.5, R = 12. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.V = 0.5 × 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.0.5 × 12 = 6
V = 6 - Write the answer with its units. The answer should say what was being worked out.V = 6 V
Check. Put the numbers straight into a calculator: 0.5 × 12 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. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.T = 40m + 20
- Substitute: m = 1.5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.T = 40 × 1.5 + 20
- 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.40 × 1.5 = 60
T = 60 + 20 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.60 + 20 = 80
T = 80 - Write the answer with its units. The answer should say what was being worked out.T = 80 minutes
Check. Put the numbers straight into a calculator: 40 × 1.5 + 20 gives 80. ✓
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.T = 40m + 20
- Substitute: m = 2.5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.T = 40 × 2.5 + 20
- 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.40 × 2.5 = 100
T = 100 + 20 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.100 + 20 = 120
T = 120 - Write the answer with its units. The answer should say what was being worked out.T = 120 minutes
Check. Put the numbers straight into a calculator: 40 × 2.5 + 20 gives 120. ✓
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.T = 40m + 20
- Substitute: m = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.T = 40 × 3 + 20
- 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.40 × 3 = 120
T = 120 + 20 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.120 + 20 = 140
T = 140 - Write the answer with its units. The answer should say what was being worked out.T = 140 minutes
Check. Put the numbers straight into a calculator: 40 × 3 + 20 gives 140. ✓
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.F = 1.8C + 32
- Substitute: C = 20. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.F = 1.8 × 20 + 32
- 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.1.8 × 20 = 36
F = 36 + 32 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.36 + 32 = 68
F = 68 - Write the answer with its units. The answer should say what was being worked out.F = 68°F
Check. Put the numbers straight into a calculator: 1.8 × 20 + 32 gives 68. ✓
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.F = 1.8C + 32
- Substitute: C = 100. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.F = 1.8 × 100 + 32
- 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.1.8 × 100 = 180
F = 180 + 32 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.180 + 32 = 212
F = 212 - Write the answer with its units. The answer should say what was being worked out.F = 212°F
Check. Put the numbers straight into a calculator: 1.8 × 100 + 32 gives 212. ✓
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.F = 1.8C + 32
- Substitute: C = −40. 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.F = 1.8 × (−40) + 32
- 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.1.8 × (−40) = −72 (a negative times a positive gives a negative)
F = −72 + 32 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.−72 + 32 = −40
F = −40 - Write the answer with its units. The answer should say what was being worked out.F = −40°F
Check. Put the numbers straight into a calculator, with brackets round negative numbers: 1.8 × (−40) + 32 gives −40. ✓
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.
- Write down the formula. This is the rule we are going to use.A = 12(a + b)h
- Substitute: a = 5, b = 9, h = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.A = 12 × (5 + 9) × 4
- 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 + 9 = 14
A = 12 × 14 × 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 × 14 = 7
A = 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
A = 28 - Write the answer with its units. The answer should say what was being worked out.A = 28 cm2
Check. Put the numbers straight into a calculator: 1/2 × (5 + 9) × 4 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. 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.E = 12mv2
- Substitute: m = 2, v = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 2 × 62
- 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.62 = 6 × 6 = 36
E = 12 × 2 × 36 - 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 × 2 = 1
E = 1 × 36 - 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.1 × 36 = 36
E = 36 - Write the answer with its units. The answer should say what was being worked out.E = 36 J
Check. Put the numbers straight into a calculator: 1/2 × 2 × 62 gives 36. ✓
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.E = 12mv2
- Substitute: m = 10, v = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 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
E = 12 × 10 × 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.12 × 10 = 5
E = 5 × 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.5 × 9 = 45
E = 45 - Write the answer with its units. The answer should say what was being worked out.E = 45 J
Check. Put the numbers straight into a calculator: 1/2 × 10 × 32 gives 45. ✓
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.E = 12mv2
- Substitute: m = 0.5, v = 8. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 0.5 × 82
- 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.82 = 8 × 8 = 64
E = 12 × 0.5 × 64 - 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 × 0.5 = 0.25
E = 0.25 × 64 - Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.0.25 × 64 = 16
E = 16 - Write the answer with its units. The answer should say what was being worked out.E = 16 J
Check. Put the numbers straight into a calculator: 1/2 × 0.5 × 82 gives 16. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Method. Check the order of working. Multiplication comes before addition.
- Look at what Zoe did. She added 4 and 3 first, then multiplied by 2.(4 + 3) × 2 = 14
- Say why that is wrong. In u + at the multiplication at must be done first (BIDMAS), because at is a single term.The formula means 4 + (3 × 2).
- Do it correctly. Multiply, then add.v = 4 + 3 × 2 = 4 + 6 = 10
Check. Another way: speed starts at 4 and goes up by 3 each second for 2 seconds: 4 + 3 + 3 = 10. ✓
Watch out. A number next to a letter, like at, is a multiplication, and multiplication is done before addition.
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.s = dt
- Substitute: d = 180, t = 2.5. Substitute means replace each letter with its number.s = 1802.5
- 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.1802.5 means 180 ÷ 2.5 = 72
s = 72 - Write the answer with its units. The answer should say what was being worked out.s = 72 km/h
Check. Put the numbers straight into a calculator: 180/2.5 gives 72. ✓
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. 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.s = dt
- Substitute: d = 150, t = 2. Substitute means replace each letter with its number.s = 1502
- 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.1502 means 150 ÷ 2 = 75
s = 75 - Write the answer with its units. The answer should say what was being worked out.s = 75 km/h
Check. Put the numbers straight into a calculator: 150/2 gives 75. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
Method. Compare the two speeds.
- Write the two speeds. They are both in km/h, so they can be compared.Car A: 72 km/h Car B: 75 km/h
- Compare. 75 is bigger than 72.Car B is faster.
Check. In 10 hours car B would go 750 km but car A only 720 km. ✓
Watch out. Compare speeds, not distances. Car A travelled further but took longer.
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.D = mV
- Substitute: m = 840, V = 300. Substitute means replace each letter with its number.D = 840300
- 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.840300 means 840 ÷ 300 = 2.8
D = 2.8 - Write the answer with its units. The answer should say what was being worked out.D = 2.8 g/cm3
Check. Put the numbers straight into a calculator: 840/300 gives 2.8. ✓
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. 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.D = mV
- Substitute: m = 620, V = 250. Substitute means replace each letter with its number.D = 620250
- 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.620250 means 620 ÷ 250 = 2.48
D = 2.48 - Write the answer with its units. The answer should say what was being worked out.D = 2.48 g/cm3
Check. Put the numbers straight into a calculator: 620/250 gives 2.48. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
Method. The denser block has the greater density.
- Compare. Block X is 2.8 g/cm³ and block Y is 2.48 g/cm³.2.8 > 2.48
- Answer. Block X is denser.
Check. Block X packs 2.8 g into every cm³, more than block Y’s 2.48 g. ✓
Watch out. Density compares mass for the same amount of space, so you must divide by volume before comparing.
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.v = √u2 + 2as
- Substitute: u = 4, a = 3, s = 8. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.v = √42 + 2 × 3 × 8
- Under the root sign first: work out the powers (indices). Inside the root sign we still use BIDMAS, so powers come first.42 = 4 × 4 = 16
v = √16 + 2 × 3 × 8 - 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 × 3 = 6
v = √16 + 6 × 8 - 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.6 × 8 = 48
v = √16 + 48 - 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.16 + 48 = 64
v = √64 - 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.√64 = 8 because 8 × 8 = 64
v = 8 - Write the answer with its units. The answer should say what was being worked out.v = 8 m/s
Check. Put the numbers straight into a calculator: √42 + 2 × 3 × 8 gives 8. ✓
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.v = √u2 + 2as
- Substitute: u = 6, a = 2, s = 7. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.v = √62 + 2 × 2 × 7
- 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
v = √36 + 2 × 2 × 7 - 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 × 2 = 4
v = √36 + 4 × 7 - 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.4 × 7 = 28
v = √36 + 28 - 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 + 28 = 64
v = √64 - 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.√64 = 8 because 8 × 8 = 64
v = 8 - Write the answer with its units. The answer should say what was being worked out.v = 8 m/s
Check. Put the numbers straight into a calculator: √62 + 2 × 2 × 7 gives 8. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Method. Work out the pressure in each case and compare. A “show that” question needs both calculations and a conclusion.
- Pressure for 500 N on 25 m². Substitute F = 500 and A = 25.P = 50025 = 20 N/m²
- Pressure for 100 N on 5 m². Substitute F = 100 and A = 5.P = 1005 = 20 N/m²
- Conclude. Both are the same.Both give 20 N/m², so the pressures are equal.
Check. 500 ÷ 25 and 100 ÷ 5 are equivalent fractions: 50025 = 1005 (divide top and bottom by 5). ✓
Watch out. End a “show that” answer with a sentence that says what you have shown.
Exercise 8E Using formulae with a calculator
Calculator allowedKey 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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.A = πr2
- Substitute: r = 4.5. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.A = π × 4.52
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.A = π × 4.52
Calculator display: 63.617251235… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 1, which is 4 or less, so we round down.63.617251235… = 63.6 (1 decimal place)
- Write the answer with its units. Say what the number means.A = 63.6 cm2
Check. Estimate with rounded numbers (r = 5, π = 3): 3 × 52 = 3 × 25 = 75. This is close to 63.6, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.A = πr2
- Substitute: r = 12. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.A = π × 122
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.A = π × 122
Calculator display: 452.38934211… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 8, which is 5 or more, so we round up.452.38934211… = 452.4 (1 decimal place)
- Write the answer with its units. Say what the number means.A = 452.4 cm2
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.A = πr2
- Substitute: r = 0.8. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.A = π × 0.82
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.A = π × 0.82
Calculator display: 2.0106192982… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 1, which is 4 or less, so we round down.2.0106192982… = 2.0 (1 decimal place)
- Write the answer with its units. Say what the number means.A = 2.0 cm2
Check. Estimate with rounded numbers (r = 0.8, π = 3): 3 × 0.82 = 3 × 0.64 = 1.92. This is close to 2.0, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.C = πd
- Substitute: d = 7.3. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.C = π × 7.3
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.C = π × 7.3
Calculator display: 22.933626371… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 3, which is 4 or less, so we round down.22.933626371… = 22.9 (1 decimal place)
- Write the answer with its units. Say what the number means.C = 22.9 cm
Check. Estimate with rounded numbers (d = 7, π = 3): 3 × 7 = 21. This is close to 22.9, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.C = πd
- Substitute: d = 15.6. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.C = π × 15.6
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.C = π × 15.6
Calculator display: 49.008845396… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 0, which is 4 or less, so we round down.49.008845396… = 49.0 (1 decimal place)
- Write the answer with its units. Say what the number means.C = 49.0 cm
Check. Estimate with rounded numbers (d = 20, π = 3): 3 × 20 = 60. This is close to 49.0, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.V = πr2h
- Substitute: r = 3.5, h = 10. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.V = π × 3.52 × 10
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.V = π × 3.52 × 10
Calculator display: 384.84510006… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 4, which is 4 or less, so we round down.384.84510006… = 384.8 (1 decimal place)
- Write the answer with its units. Say what the number means.V = 384.8 cm3
Check. Estimate with rounded numbers (r = 4, h = 10, π = 3): 3 × 42 × 10 = 3 × 16 × 10 = 48 × 10 = 480. This is close to 384.8, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.V = πr2h
- Substitute: r = 2.4, h = 7.5. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.V = π × 2.42 × 7.5
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.V = π × 2.42 × 7.5
Calculator display: 135.71680263… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 1, which is 4 or less, so we round down.135.71680263… = 135.7 (1 decimal place)
- Write the answer with its units. Say what the number means.V = 135.7 cm3
Check. Estimate with rounded numbers (r = 2, h = 8, π = 3): 3 × 22 × 8 = 3 × 4 × 8 = 12 × 8 = 96. This is close to 135.7, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.s = dt
- Substitute: d = 100, t = 9.58. Replace each letter with its number.s = 1009.58
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.s = 100/9.58
Calculator display: 10.438413361… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 3, which is 4 or less, so we round down.10.438413361… = 10.4 (1 decimal place)
- Write the answer with its units. Say what the number means.s = 10.4 m/s
Check. Estimate with rounded numbers (d = 100, t = 10): 10010 = 10. This is close to 10.4, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.D = mV
- Substitute: m = 84.6, V = 12.5. Replace each letter with its number.D = 84.612.5
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.D = 84.6/12.5
Calculator display: 6.768 - Round to 3 significant figures. Look at the digit just after the last one we keep. Here it is 8, which is 5 or more, so we round up.6.768 = 6.77 (3 significant figures)
- Write the answer with its units. Say what the number means.D = 6.77 g/cm3
Check. Estimate with rounded numbers (m = 80, V = 10): 8010 = 8. This is close to 6.77, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.v = √2gh
- Substitute: g = 9.8, h = 12. Replace each letter with its number.v = √2 × 9.8 × 12
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.v = √2 × 9.8 × 12
Calculator display: 15.336231610… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 3, which is 4 or less, so we round down.15.336231610… = 15.3 (1 decimal place)
- Write the answer with its units. Say what the number means.v = 15.3 m/s
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.T = 2π√lg
- Substitute: l = 1.5, g = 9.8. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.T = 2π√1.59.8
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.T = 2π√1.5/9.8
Calculator display: 2.4581730898… - Round to 2 decimal places. Look at the digit just after the last one we keep. Here it is 8, which is 5 or more, so we round up.2.4581730898… = 2.46 (2 decimal places)
- Write the answer with its units. Say what the number means.T = 2.46 seconds
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.V = 43πr3
- Substitute: r = 2.4. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.V = 43π × 2.43
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.V = 4/3π × 2.43
Calculator display: 57.905835790… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 0, which is 4 or less, so we round down.57.905835790… = 57.9 (1 decimal place)
- Write the answer with its units. Say what the number means.V = 57.9 cm3
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.A = π(R2 − r2)
- Substitute: R = 8.5, r = 6.2. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.A = π(8.52 − 6.22)
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.A = π(8.52 − 6.22)
Calculator display: 106.21724761… - Round to 1 decimal place. Look at the digit just after the last one we keep. Here it is 1, which is 4 or less, so we round down.106.21724761… = 106.2 (1 decimal place)
- Write the answer with its units. Say what the number means.A = 106.2 cm2
Check. Estimate with rounded numbers (R = 9, r = 6, π = 3): 3 × (92 − 62) = 3 × (81 − 36) = 3 × 45 = 135. This is close to 106.2, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
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, substitute the numbers, use a calculator, then round as asked.
- Write down the formula. This is the rule we are going to use.V = πr2h
- Substitute: r = 4.2, h = 11.5. Replace each letter with its number. Keep π as it is: the calculator has a π button that is more accurate than 3.14.V = π × 4.22 × 11.5
- Work it out on the calculator. Type it in carefully. Use brackets for the top and bottom of a fraction, and use the power key for squares. The calculator follows BIDMAS for you, but only if you type it correctly.V = π × 4.22 × 11.5
Calculator display: 637.30348570… - Round to the nearest whole number. Look at the digit just after the last one we keep. Here it is 3, which is 4 or less, so we round down.637.30348570… = 637 (the nearest whole number)
- Write the answer with its units. Say what the number means.V = 637 cm3
Check. Estimate with rounded numbers (r = 4, h = 10, π = 3): 3 × 42 × 10 = 3 × 16 × 10 = 48 × 10 = 480. This is close to 637, so the answer is sensible. ✓
Watch out. Do not round until the very end. Rounding in the middle of a calculation makes the final answer less accurate.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.F = 1.8C + 32
- Substitute: C = 36.6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.F = 1.8 × 36.6 + 32
- 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.1.8 × 36.6 = 65.88
F = 65.88 + 32 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.65.88 + 32 = 97.88
F = 97.88 - Round to 1 decimal place. Look at the digit after the last digit we keep. If it is 5 or more, round up; if it is 4 or less, round down.97.88 = 97.9 to 1 d.p.
- Write the answer with its units. The answer should say what was being worked out.F = 97.9°F
Check. Put the numbers straight into a calculator: 1.8 × 36.6 + 32 gives 97.88, which is 97.9 to 1 d.p. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Compare what the formula says with what Sam typed.
- Read the formula. A = πr² means π times (r squared). Only the r is squared.π × 5² = π × 25
- Say what Sam did. The brackets in (π × 5)² square the π as well.(π × 5)² = 15.708…² = 246.7 (wrong)
- Do it correctly. Square 5 first, then multiply by π.25 × π = 78.539… = 78.5 cm² (1 d.p.)
Check. Estimate: π is about 3, and 3 × 25 = 75, close to 78.5. Sam’s 246.7 is far too big. ✓
Watch out. Only the letter directly under the power is squared. Do not add extra brackets round π and r.