GCSE Maths: Using formulae: problems and reasoning (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)
Using formulae: problems and reasoning
Exam questions often use a formula in a short story, and you must decide what to do. Here is a reliable plan.
There are two sound ways to find an unknown that is not the subject. Suppose A = ½(a + b)h and we know A, a and b but want h.
- Substitute first, then solve the equation you get: 54 = ½ × 18 × h = 9h, so h = 54 ÷ 9. This is quick for one calculation.
- Rearrange first, then substitute: h = 2A ÷ (a + b). This is better when you need to do many calculations with different numbers.
Other useful skills in this section:
- Checking a formula by putting in simple test values to see if it gives the right answer.
- Showing two expressions are equal by expanding or substituting, and ending with a clear conclusion.
- Combining formulae (Higher): if one formula gives r in terms of C, substitute it into another formula to get A in terms of C.
Key ideas
- Read carefully and write down what you know and what you want.
- Either substitute first and solve, or rearrange first and substitute.
- Check units (convert minutes to hours, pence to pounds) before you start.
- To check a formula, try simple numbers. To show something, give a calculation and a conclusion.
- Does the answer make sense? A speed of 600 km/h for a bike does not.
Watch out
Average speed is total distance ÷ total time. It is not the average of the two speeds. If you ride 12 km at 15 km/h and 8 km at 10 km/h, the average speed is not (15 + 10) ÷ 2 = 12.5 by luck: work out each time first.Method. Substitute the known values, simplify, then solve the simple equation for h.
- Write the formula and substitute. Put in A = 54, a = 7 and b = 11.54 = ½ × (7 + 11) × h
- Work out the bracket, then ½ of it. Brackets first. 7 + 11 = 18 and ½ of 18 is 9.54 = 9h
- Divide both sides by 9. h is multiplied by 9, so the inverse is dividing by 9.h = 54 ÷ 9 = 6 cm
Check. Put h = 6 back: A = 12 × (7 + 11) × 6 = 12 × 18 × 6 = 9 × 6 = 54 cm². ✓
Watch out. Do not forget the ½. A common slip is to forget it and get h = 3.
Method. Average speed = total distance ÷ total time. Use t = d/s to find each time first.
- Find the time for the first part. Time = distance ÷ speed.t = 1215 = 0.8 hours (= 0.8 h)
- Find the time for the second part. Same formula.t = 810 = 0.8 hours (= 0.8 h)
- Add the distances and the times. The whole journey is both parts together.distance = 12 + 8 = 20 km
time = 0.8 + 0.8 = 1.6 hours - Work out the average speed. Use s = d/t with the totals.s = 201.6 = 12.5 km/h
Check. In 1.6 hours at 12.5 km/h she would travel 1.6 × 12.5 = 20 km. ✓
Watch out. The average of the two speeds is 12.5 here only by chance; try 12 km at 12 km/h and 4 km at 4 km/h to see it fail.
Method. The area needs the radius. Use the circumference formula to find r first, then use it in the area formula.
- Make r the subject of C = 2πr. Divide both sides by 2π.r = C2π
- Substitute to find r. 2 × 3.14 = 6.28 and 31.4 ÷ 6.28 = 5.r = 31.42 × 3.14 = 31.46.28 = 5 m
- Use r in the area formula. Square first, then multiply by π.A = 3.14 × 52 = 3.14 × 25 = 78.5 m²
Check. Estimate: π ≈ 3 and 3 × 25 = 75, close to 78.5. ✓
Watch out. Find the radius first. Do not put the circumference straight into the area formula.
Exercise 8J Using formulae: problems and reasoning
Non-calculatorKey words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.
Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until h is on its own.
- Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get h on its own we must undo everything that has been done to it, in the reverse order.Formula: V = πr2h
Reading from the subject outwards, what happens to h: it is multiplied by πr2.
To undo it we reverse the order: divide by πr2. - Divide both sides by πr2. h is being multiplied by πr2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by πr2.Vπr2 = πr2hπr2
Vπr2 = h - Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.h = Vπr2
Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): V = 9, r = 1, π = 3. The new formula gives
h = 93 × 12 = 93 × 1 = 93 = 3
Now put h = 3 back into the original formula. Left side: 9. Right side: 3 × 12 × 3 = 3 × 1 × 3 = 3 × 3 = 9. Both sides are equal, so the rearranging is correct. ✓
Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. 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.h = Vπr2
- Substitute: V = 157, π = 3.14, r = 5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.h = 1573.14 × 52
- Bottom of the fraction first: work out the powers (indices). Inside the bottom of the fraction we still use BIDMAS, so powers come first.52 = 5 × 5 = 25
h = 1573.14 × 25 - Bottom of the fraction first: do the multiplications and divisions. Inside the bottom of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.3.14 × 25 = 78.5
h = 15778.5 - 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.15778.5 means 157 ÷ 78.5 = 2
h = 2 - Write the answer with its units. The answer should say what was being worked out.h = 2 cm
Check. Put the numbers straight into a calculator: 157/(3.14 × 52) gives 2. ✓
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.d = Cπ
- Substitute: C = 220, π = 227. Substitute means replace each letter with its number.d = 220227
- 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.220227 means 220 ÷ 227 = 70
d = 70 - Write the answer with its units. The answer should say what was being worked out.d = 70 cm
Check. Put the numbers straight into a calculator: 220/(227) gives 70. ✓
Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.
Method. Find the time for each part, add the times and the distances, then use average speed = total distance ÷ total time.
- Time for the first part. Time = distance ÷ speed.t = 3060 = 0.5 h (= 0.5 h)
- Time for the second part. Same formula.t = 6040 = 1.5 h (= 1.5 h)
- Totals. Add distances and times.distance = 30 + 60 = 90 km
time = 0.5 + 1.5 = 2 hours - Average speed. Total distance ÷ total time.s = 902 = 45 km/h
Check. In 2 hours at 45 km/h the distance is 2 × 45 = 90 km. ✓
Watch out. The average of 60 and 40 is 50, but the car spends more time at the slower speed, so the true average is less.
Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.
Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until C is on its own.
- Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get C on its own we must undo everything that has been done to it, in the reverse order.Formula: F = 95C + 32
Reading from the subject outwards, what happens to C: it is multiplied by 95; then 32 is added.
To undo it we reverse the order: subtract 32, then multiply by 59. - Subtract 32 from both sides. 32 is being added to the part containing C. The inverse (opposite) of adding is subtracting, so subtract 32 from both sides. Doing the same to both sides keeps the formula true.F − 32 = 95C + 32 − 32
F − 32 = 95C - Multiply both sides by 59. C is being multiplied by 95. To undo that, multiply by its reciprocal 59 (the fraction turned upside down), because 95 × 59 = 1.(F − 32) × 59 = 95C × 59
59(F − 32) = C - Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.C = 59(F − 32)
Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.
Method. Substitute C = −40 into the original formula and see if F is also −40.
- Substitute C = −40. Brackets round the negative number.F = 9 × (−40)5 + 32 = −3605 + 32 = −72 + 32 = −40
- Conclude. The answer is −40 again.−40°C and −40°F are the same temperature.
Check. Using the rearranged formula with F = −40: C = 5 × (−40 − 32)9 = 5 × (−72)9 = −3609 = −40. ✓
Watch out. This is the only temperature at which the two scales agree.
Method. Write the area for radius 2r, simplify, and compare with the original area πr².
- Replace r by 2r in the formula. The new radius is twice the old one.Anew = π(2r)²
- Square the bracket. Both the 2 and the r are squared: (2r)² = 22 × r² = 4r².Anew = π × 4r² = 4πr²
- Compare. The original area is πr².4πr² = 4 × (πr²), so the new area is 4 times the old area.
Check. Try r = 1 and π = 3: old area 3, new radius 2, new area 3 × 4 = 12 = 4 × 3. ✓
Watch out. (2r)² is 4r², not 2r².
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 = 1200, v = 20. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 1200 × 202
- 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.202 = 20 × 20 = 400
E = 12 × 1200 × 400 - 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 × 1200 = 600
E = 600 × 400 - 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.600 × 400 = 240000
E = 240000 - Write the answer with its units. The answer should say what was being worked out.E = 240000 J
Check. Put the numbers straight into a calculator: 1/2 × 1200 × 202 gives 240000. ✓
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 = 1200, v = 40. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.E = 12 × 1200 × 402
- 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.402 = 40 × 40 = 1600
E = 12 × 1200 × 1600 - 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 × 1200 = 600
E = 600 × 1600 - 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.600 × 1600 = 960000
E = 960000 - Write the answer with its units. The answer should say what was being worked out.E = 960000 J
Check. Put the numbers straight into a calculator: 1/2 × 1200 × 402 gives 960000. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Method. Divide the two energies.
- Write the two energies. From parts (a) and (b).240 000 J and 960 000 J
- Divide. 960 000 ÷ 240 000 = 4.The energy is 4 times bigger.
Check. The speed is squared, so doubling it multiplies the energy by 2² = 4. ✓
Watch out. Energy depends on the square of the speed, so doubling the speed does not just double the energy.
Key words. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.
Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until m is on its own.
- Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get m on its own we must undo everything that has been done to it, in the reverse order.Formula: D = mV
Reading from the subject outwards, what happens to m: it is divided by V.
To undo it we reverse the order: multiply by V. - Multiply both sides by V. The part containing m is being divided by V. The inverse of dividing is multiplying, so multiply both sides by V.D × V = mV × V
VD = m - Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.m = VD
Check. Pick numbers for the other letters: D = 3, V = 10. The new formula gives
m = 10 × 3 = 30
Now put m = 30 back into the original formula. Left side: 3. Right side: 3010 = 3. Both sides are equal, so the rearranging is correct. ✓
Watch out. Do the same thing to both sides, and treat every term on a side as a whole. For example, if you divide by 2, the whole side is divided by 2, not just one term.
Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.
- Write down the formula. This is the rule we are going to use.m = DV
- Substitute: D = 7800, V = 2.5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.m = 7800 × 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.7800 × 2.5 = 19500
m = 19500 - Write the answer with its units. The answer should say what was being worked out.m = 19500 kg
Check. Put the numbers straight into a calculator: 7800 × 2.5 gives 19500. ✓
Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.
Method. Expand the bracket and collect like terms. If everything cancels, the expression is always 0.
- Expand the bracket. Multiply both terms in the bracket by 3.3(x + 2) = 3x + 6
- Write the whole expression. Replace the bracket.3x + 6 − 3x − 6
- Collect like terms. 3x − 3x = 0 and 6 − 6 = 0.0
Check. Try x = 10: 3 × (10 + 2) − 3 × 10 − 6 = 3 × 12 − 3 × 10 − 6 = 36 − 30 − 6 = 6 − 6 = 0. ✓
Watch out. Try one number as a check, but the algebra is what proves it for every x.
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 = 0.3v + v250
- Substitute: v = 20. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.d = 0.3 × 20 + 20250
- Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.202 = 20 × 20 = 400
d = 0.3 × 20 + 40050 - 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.0.3 × 20 = 6
40050 means 400 ÷ 50 = 8
d = 6 + 8 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.6 + 8 = 14
d = 14 - Write the answer with its units. The answer should say what was being worked out.d = 14 m
Check. Put the numbers straight into a calculator: 0.3 × 20 + 202/50 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. 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 = 0.3v + v250
- Substitute: v = 40. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.d = 0.3 × 40 + 40250
- Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.402 = 40 × 40 = 1600
d = 0.3 × 40 + 160050 - 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.0.3 × 40 = 12
160050 means 1600 ÷ 50 = 32
d = 12 + 32 - Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.12 + 32 = 44
d = 44 - Write the answer with its units. The answer should say what was being worked out.d = 44 m
Check. Put the numbers straight into a calculator: 0.3 × 40 + 402/50 gives 44. ✓
Watch out. In 5x² only the x is squared, not the 5. Square first, then multiply by 5.
Method. Double the first distance and compare with the second.
- Double the first distance. If Tom is right, the distance at v = 40 would be twice that at v = 20.2 × 14 = 28 m
- Compare with the actual distance. The actual distance is 44 m, which is more than 28 m.
- Conclude. Tom is wrong: the stopping distance more than doubles.
Check. The v² term grows much faster than the 0.3v term. ✓
Watch out. Compare actual values; do not guess from the shape of the formula.
Method. Replace x in 3x + 2 by (y − 2)/3 and simplify.
- Substitute. Put the whole fraction in place of x.3 × y − 23 + 2
- Multiply. The 3 on the outside cancels the 3 on the bottom.(y − 2) + 2
- Simplify. −2 + 2 = 0.y
Check. Test with y = 17: x = 5 and 3 × 5 + 2 = 17. ✓
Watch out. This is how you can be sure a rearrangement is correct: it undoes the original formula.