GCSE Maths: Using formulae: problems and reasoning (Higher)

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 h is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get 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.
2. 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
3. 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.

1. Write down the formula. This is the rule we are going to use.
   h = Vπr²
2. 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
3. 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
4. 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
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
6. 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. 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.

1. 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.
2. 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
3. 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
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   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.

1. Substitute C = −40. Brackets round the negative number.
   F = 9 × (−40)5 + 32 = −3605 + 32 = −72 + 32 = −40
2. 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.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   E = 12mv²
2. Substitute: m = 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
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   20² = 20 × 20 = 400
   E = 12 × 1200 × 400
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 × 1200 = 600
   E = 600 × 400
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.
   600 × 400 = 240000
   E = 240000
6. 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.

1. Write down the formula. This is the rule we are going to use.
   E = 12mv²
2. 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
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   40² = 40 × 40 = 1600
   E = 12 × 1200 × 1600
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 × 1200 = 600
   E = 600 × 1600
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.
   600 × 1600 = 960000
   E = 960000
6. 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.

1. Write the two energies. From parts (a) and (b).
   240 000 J and 960 000 J
2. 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.

1. 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.
2. 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
3. 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.

1. Write down the formula. This is the rule we are going to use.
   m = DV
2. 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
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.
   7800 × 2.5 = 19500
   m = 19500
4. 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.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   d = 0.3v + v²50
2. 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
3. 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
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.
   0.3 × 20 = 6
   40050 means 400 ÷ 50 = 8
   d = 6 + 8
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   6 + 8 = 14
   d = 14
6. 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.

1. Write down the formula. This is the rule we are going to use.
   d = 0.3v + v²50
2. 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
3. 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
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.
   0.3 × 40 = 12
   160050 means 1600 ÷ 50 = 32
   d = 12 + 32
5. 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
6. 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.

1. Double the first distance. If Tom is right, the distance at v = 40 would be twice that at v = 20.
   2 × 14 = 28 m
2. Compare with the actual distance.
   The actual distance is 44 m, which is more than 28 m.
3. 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.

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 s is on its own.

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get s on its own we must undo everything that has been done to it, in the reverse order.
   Formula: v² = u² + 2as
   Reading from the subject outwards, what happens to s: it is multiplied by 2a; then u² is added.
   To undo it we reverse the order: subtract u², then divide by 2a.
2. Subtract u² from both sides. u² is being added to the part containing s. The inverse (opposite) of adding is subtracting, so subtract u² from both sides. Doing the same to both sides keeps the formula true.
   v² − u² = u² + 2as − u²
   v² − u² = 2as
3. Divide both sides by 2a. s is being multiplied by 2a (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2a.
   v² − u²2a = 2as2a
   v² − u²2a = s
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   s = v² − u²2a

Check. Pick numbers for the other letters: v = 4, u = 2, a = 6. The new formula gives
s = 4² − 2²2 × 6 = 16 − 412 = 1212 = 1
Now put s = 1 back into the original formula. Left side: 4² = 16. Right side: 2² + 2 × 6 × 1 = 4 + 2 × 6 × 1 = 4 + 12 × 1 = 4 + 12 = 16. 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.

1. Write down the formula. This is the rule we are going to use.
   s = v² − u²2a
2. Substitute: v = 13, u = 5, a = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   s = 13² − 5²2 × 6
3. Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.
   13² = 13 × 13 = 169
   5² = 5 × 5 = 25
   2 × 6 = 12
   s = 169 − 2512
4. Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   169 − 25 = 144
   s = 14412
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.
   14412 means 144 ÷ 12 = 12
   s = 12
6. Write the answer with its units. The answer should say what was being worked out.
   s = 12 m

Check. Put the numbers straight into a calculator: (13² − 5²)/(2 × 6) gives 12. ✓

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

Method. Use the perimeter formula P = 2(l + w) with P = 20 and make w the subject.

1. Substitute P = 20.
   20 = 2(l + w)
2. Divide both sides by 2. Undo the multiplication by 2.
   10 = l + w
3. Subtract l from both sides. Undo the + l.
   w = 10 − l

Check. Try l = 6: w = 4 and 2(6 + 4) = 20. ✓

Watch out. Show every line when the question says “show that”.

Method. Substitute w = 10 − l into A = lw, then work out A for each l.

1. Write the area in terms of l. Replace w by 10 − l.
   A = l(10 − l)
2. Work out the areas. Substitute l = 1, 2, 3, …, 9.
   l = 1: 9,   l = 2: 16,   l = 3: 21,   l = 4: 24,   l = 5: 25,   l = 6: 24,   l = 7: 21,   l = 8: 16,   l = 9: 9
3. Find the largest. The numbers rise to 25 and then fall again.
   The largest area is 25 cm², when l = 5 (a square of side 5 cm).

Check. Check l = 5: w = 5 and 5 × 5 = 25. ✓

Watch out. Among rectangles with the same perimeter, the square has the greatest area.

Method. Make r the subject of the circumference formula and substitute it into the area formula.

1. Make r the subject. Divide C = 2πr by 2π.
   r = C2π
2. Substitute. Replace r in A = πr².
   A = π(C2π)² = π × C²4π²
3. Cancel. One π cancels.
   A = C²4π

Check. With π = 3 and r = 2: C = 12, A = 12; and 144 ÷ 12 = 12. ✓

Watch out. Square the whole of 2π.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   A = C²4π
2. Substitute: C = 62.8, π = 3.14. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   A = 62.8²4 × 3.14
3. Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.
   62.8² = 62.8 × 62.8 = 3943.84
   4 × 3.14 = 12.56
   A = 3943.8412.56
4. 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.
   3943.8412.56 means 3943.84 ÷ 12.56 = 314
   A = 314
5. Write the answer with its units. The answer should say what was being worked out.
   A = 314 cm²

Check. Put the numbers straight into a calculator: 62.8²/(4 × 3.14) gives 314. ✓

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