GCSE Maths: Changing the subject: one step and two steps (Foundation)

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x + 5
   Reading from the subject outwards, what happens to x: 5 is added.
   To undo it we reverse the order: subtract 5.
2. Subtract 5 from both sides. 5 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 5 from both sides. Doing the same to both sides keeps the formula true.
   y − 5 = x + 5 − 5
   y − 5 = x
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.
   x = y − 5

Check. Pick numbers for the other letters: y = 9. The new formula gives
x = 9 − 5 = 4
Now put x = 4 back into the original formula. Left side: 9. Right side: 4 + 5 = 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. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x − 3
   Reading from the subject outwards, what happens to x: 3 is subtracted.
   To undo it we reverse the order: add 3.
2. Add 3 to both sides. 3 is being subtracted. The inverse of subtracting is adding, so add 3 to both sides.
   y + 3 = x − 3 + 3
   y + 3 = x
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.
   x = y + 3

Check. Pick numbers for the other letters: y = 8. The new formula gives
x = 8 + 3 = 11
Now put x = 11 back into the original formula. Left side: 8. Right side: 11 − 3 = 8. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 4x
   Reading from the subject outwards, what happens to x: it is multiplied by 4.
   To undo it we reverse the order: divide by 4.
2. Divide both sides by 4. x is being multiplied by 4 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 4.
   y4 = 4x4
   y4 = x
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.
   x = y4

Check. Pick numbers for the other letters: y = 8. The new formula gives
x = 84 = 2
Now put x = 2 back into the original formula. Left side: 8. Right side: 4 × 2 = 8. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x3
   Reading from the subject outwards, what happens to x: it is divided by 3.
   To undo it we reverse the order: multiply by 3.
2. Multiply both sides by 3. The part containing x is being divided by 3. The inverse of dividing is multiplying, so multiply both sides by 3.
   y × 3 = x3 × 3
   3y = x
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.
   x = 3y

Check. Pick numbers for the other letters: y = 2. The new formula gives
x = 3 × 2 = 6
Now put x = 6 back into the original formula. Left side: 2. Right side: 63 = 2. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x + a
   Reading from the subject outwards, what happens to x: a is added.
   To undo it we reverse the order: subtract a.
2. Subtract a from both sides. a is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract a from both sides. Doing the same to both sides keeps the formula true.
   y − a = x + a − a
   y − a = x
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.
   x = y − a

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

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = kx
   Reading from the subject outwards, what happens to x: it is multiplied by k.
   To undo it we reverse the order: divide by k.
2. Divide both sides by k. x is being multiplied by k (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by k.
   yk = kxk
   yk = x
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.
   x = yk

Check. Pick numbers for the other letters: y = 10, k = 2. The new formula gives
x = 102 = 5
Now put x = 5 back into the original formula. Left side: 10. Right side: 2 × 5 = 10. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2x + 3
   Reading from the subject outwards, what happens to x: it is multiplied by 2; then 3 is added.
   To undo it we reverse the order: subtract 3, then divide by 2.
2. Subtract 3 from both sides. 3 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 3 from both sides. Doing the same to both sides keeps the formula true.
   y − 3 = 2x + 3 − 3
   y − 3 = 2x
3. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   y − 32 = 2x2
   y − 32 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y − 32

Check. Pick numbers for the other letters: y = 9. The new formula gives
x = 9 − 32 = 62 = 3
Now put x = 3 back into the original formula. Left side: 9. Right side: 2 × 3 + 3 = 6 + 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. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 5x − 4
   Reading from the subject outwards, what happens to x: it is multiplied by 5; then 4 is subtracted.
   To undo it we reverse the order: add 4, then divide by 5.
2. Add 4 to both sides. 4 is being subtracted. The inverse of subtracting is adding, so add 4 to both sides.
   y + 4 = 5x − 4 + 4
   y + 4 = 5x
3. Divide both sides by 5. x is being multiplied by 5 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 5.
   y + 45 = 5x5
   y + 45 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y + 45

Check. Pick numbers for the other letters: y = 6. The new formula gives
x = 6 + 45 = 105 = 2
Now put x = 2 back into the original formula. Left side: 6. Right side: 5 × 2 − 4 = 10 − 4 = 6. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = x2 + 1
   Reading from the subject outwards, what happens to x: it is divided by 2; then 1 is added.
   To undo it we reverse the order: subtract 1, then multiply by 2.
2. Subtract 1 from both sides. 1 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 1 from both sides. Doing the same to both sides keeps the formula true.
   y − 1 = x2 + 1 − 1
   y − 1 = x2
3. Multiply both sides by 2. The part containing x is being divided by 2. The inverse of dividing is multiplying, so multiply both sides by 2.
   (y − 1) × 2 = x2 × 2
   2(y − 1) = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = 2(y − 1)

Check. Pick numbers for the other letters: y = 6. The new formula gives
x = 2 × (6 − 1) = 2 × 5 = 10
Now put x = 10 back into the original formula. Left side: 6. Right side: 102 + 1 = 5 + 1 = 6. Both sides are equal, so the rearranging is correct. ✓

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

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

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

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

Check. Pick numbers for the other letters: y = 4. The new formula gives
x = 4 × (4 + 2) = 4 × 6 = 24
Now put x = 24 back into the original formula. Left side: 4. Right side: 244 − 2 = 6 − 2 = 4. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 7 + 3x
   Reading from the subject outwards, what happens to x: it is multiplied by 3; then 7 is added.
   To undo it we reverse the order: subtract 7, then divide by 3.
2. Subtract 7 from both sides. 7 is being added to the part containing x. The inverse (opposite) of adding is subtracting, so subtract 7 from both sides. Doing the same to both sides keeps the formula true.
   y − 7 = 7 + 3x − 7
   y − 7 = 3x
3. Divide both sides by 3. x is being multiplied by 3 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 3.
   y − 73 = 3x3
   y − 73 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y − 73

Check. Pick numbers for the other letters: y = 10. The new formula gives
x = 10 − 73 = 33 = 1
Now put x = 1 back into the original formula. Left side: 10. Right side: 7 + 3 × 1 = 7 + 3 = 10. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get x on its own we must undo everything that has been done to it, in the reverse order.
   Formula: y = 2x − 9
   Reading from the subject outwards, what happens to x: it is multiplied by 2; then 9 is subtracted.
   To undo it we reverse the order: add 9, then divide by 2.
2. Add 9 to both sides. 9 is being subtracted. The inverse of subtracting is adding, so add 9 to both sides.
   y + 9 = 2x − 9 + 9
   y + 9 = 2x
3. Divide both sides by 2. x is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   y + 92 = 2x2
   y + 92 = x
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   x = y + 92

Check. Pick numbers for the other letters: y = 9. The new formula gives
x = 9 + 92 = 182 = 9
Now put x = 9 back into the original formula. Left side: 9. Right side: 2 × 9 − 9 = 18 − 9 = 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. 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 a 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 a on its own we must undo everything that has been done to it, in the reverse order.
   Formula: F = ma
   Reading from the subject outwards, what happens to a: it is multiplied by m.
   To undo it we reverse the order: divide by m.
2. Divide both sides by m. a is being multiplied by m (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by m.
   Fm = mam
   Fm = a
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.
   a = Fm

Check. Pick numbers for the other letters: F = 9, m = 3. The new formula gives
a = 93 = 3
Now put a = 3 back into the original formula. Left side: 9. Right side: 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. 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: F = ma
   Reading from the subject outwards, what happens to m: it is multiplied by a.
   To undo it we reverse the order: divide by a.
2. Divide both sides by a. m is being multiplied by a (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by a.
   Fa = maa
   Fa = 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 = Fa

Check. Pick numbers for the other letters: F = 10, a = 2. The new formula gives
m = 102 = 5
Now put m = 5 back into the original formula. Left side: 10. Right side: 5 × 2 = 10. Both sides are equal, so the rearranging is correct. ✓

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

Key words. 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 I 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 I on its own we must undo everything that has been done to it, in the reverse order.
   Formula: V = IR
   Reading from the subject outwards, what happens to I: it is multiplied by R.
   To undo it we reverse the order: divide by R.
2. Divide both sides by R. I is being multiplied by R (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by R.
   VR = IRR
   VR = I
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.
   I = VR

Check. Pick numbers for the other letters: V = 6, R = 3. The new formula gives
I = 63 = 2
Now put I = 2 back into the original formula. Left side: 6. Right side: 2 × 3 = 6. Both sides are equal, so the rearranging is correct. ✓

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

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get R on its own we must undo everything that has been done to it, in the reverse order.
   Formula: V = IR
   Reading from the subject outwards, what happens to R: it is multiplied by I.
   To undo it we reverse the order: divide by I.
2. Divide both sides by I. R is being multiplied by I (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by I.
   VI = IRI
   VI = R
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.
   R = VI

Check. Pick numbers for the other letters: V = 4, I = 2. The new formula gives
R = 42 = 2
Now put R = 2 back into the original formula. Left side: 4. Right side: 2 × 2 = 4. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until d 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 d on its own we must undo everything that has been done to it, in the reverse order.
   Formula: s = dt
   Reading from the subject outwards, what happens to d: it is divided by t.
   To undo it we reverse the order: multiply by t.
2. Multiply both sides by t. The part containing d is being divided by t. The inverse of dividing is multiplying, so multiply both sides by t.
   s × t = dt × t
   ts = d
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.
   d = ts

Check. Pick numbers for the other letters: s = 9, t = 5. The new formula gives
d = 5 × 9 = 45
Now put d = 45 back into the original formula. Left side: 9. Right side: 455 = 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. The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject means to rearrange the formula so that a different letter is on its own. An inverse operation undoes another: adding and subtracting are inverses, multiplying and dividing are inverses, squaring and square rooting are inverses.

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get t on its own we must undo everything that has been done to it, in the reverse order.
   Formula: s = dt
   Reading from the subject outwards, what happens to t: it is the divisor in d ÷ t.
   To undo it we reverse the order: multiply by t, then divide by s.
2. Multiply both sides by t. t is on the bottom of a fraction. To bring it up, multiply both sides by t. (Anything divided by t, then multiplied by t, is back where it started.)
   s × t = dt × t
   st = d
3. Divide both sides by s. t is being multiplied by s (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by s.
   ds = sts
   ds = t
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   t = ds

Check. Pick numbers for the other letters: s = 4, d = 8. The new formula gives
t = 84 = 2
Now put t = 2 back into the original formula. Left side: 4. Right side: 82 = 4. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until u 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 u on its own we must undo everything that has been done to it, in the reverse order.
   Formula: v = u + at
   Reading from the subject outwards, what happens to u: at is added.
   To undo it we reverse the order: subtract at.
2. Subtract at from both sides. at is being added to the part containing u. The inverse (opposite) of adding is subtracting, so subtract at from both sides. Doing the same to both sides keeps the formula true.
   v − at = u + at − at
   v − at = u
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.
   u = v − at

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

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

Key words. 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 a 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 a on its own we must undo everything that has been done to it, in the reverse order.
   Formula: v = u + at
   Reading from the subject outwards, what happens to a: it is multiplied by t; then u is added.
   To undo it we reverse the order: subtract u, then divide by t.
2. Subtract u from both sides. u is being added to the part containing a. The inverse (opposite) of adding is subtracting, so subtract u from both sides. Doing the same to both sides keeps the formula true.
   v − u = u + at − u
   v − u = at
3. Divide both sides by t. a is being multiplied by t (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by t.
   v − ut = att
   v − ut = a
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.
   a = v − ut

Check. Pick numbers for the other letters: v = 8, u = 2, t = 6. The new formula gives
a = 8 − 26 = 66 = 1
Now put a = 1 back into the original formula. Left side: 8. Right side: 2 + 1 × 6 = 2 + 6 = 8. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until d 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 d on its own we must undo everything that has been done to it, in the reverse order.
   Formula: C = πd
   Reading from the subject outwards, what happens to d: it is multiplied by π.
   To undo it we reverse the order: divide by π.
2. Divide both sides by π. d is being multiplied by π (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by π.
   Cπ = πdπ
   Cπ = d
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.
   d = Cπ

Check. Pick numbers for the other letters (for this check only, let π = 3 to keep the numbers easy): C = 6, π = 3. The new formula gives
d = 63 = 2
Now put d = 2 back into the original formula. Left side: 6. Right side: 3 × 2 = 6. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until 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: A = 12bh
   Reading from the subject outwards, what happens to h: it is multiplied by b; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by b.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   A × 2 = 12bh × 2
   2A = bh
3. Divide both sides by b. h is being multiplied by b (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by b.
   2Ab = bhb
   2Ab = h
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.
   h = 2Ab

Check. Pick numbers for the other letters: A = 10, b = 5. The new formula gives
h = 2 × 105 = 205 = 4
Now put h = 4 back into the original formula. Left side: 10. Right side: 12 × 5 × 4 = 2.5 × 4 = 10. Both sides are equal, so the rearranging is correct. ✓

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

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

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

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

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

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

Key words. 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: P = 4s
   Reading from the subject outwards, what happens to s: it is multiplied by 4.
   To undo it we reverse the order: divide by 4.
2. Divide both sides by 4. s is being multiplied by 4 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 4.
   P4 = 4s4
   P4 = s
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.
   s = P4

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

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

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

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

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

Check. Pick numbers for the other letters: y = 9, m = 2, c = 3. The new formula gives
x = 9 − 32 = 62 = 3
Now put x = 3 back into the original formula. Left side: 9. Right side: 2 × 3 + 3 = 6 + 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. 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: y = mx + c
   Reading from the subject outwards, what happens to m: it is multiplied by x; then c is added.
   To undo it we reverse the order: subtract c, then divide by x.
2. Subtract c from both sides. c is being added to the part containing m. The inverse (opposite) of adding is subtracting, so subtract c from both sides. Doing the same to both sides keeps the formula true.
   y − c = mx + c − c
   y − c = mx
3. Divide both sides by x. m is being multiplied by x (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by x.
   y − cx = mxx
   y − cx = m
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.
   m = y − cx

Check. Pick numbers for the other letters: y = 6, x = 4, c = 2. The new formula gives
m = 6 − 24 = 44 = 1
Now put m = 1 back into the original formula. Left side: 6. Right side: 1 × 4 + 2 = 4 + 2 = 6. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until 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: y = mx + c
   Reading from the subject outwards, what happens to c: mx is added.
   To undo it we reverse the order: subtract mx.
2. Subtract mx from both sides. mx is being added to the part containing c. The inverse (opposite) of adding is subtracting, so subtract mx from both sides. Doing the same to both sides keeps the formula true.
   y − mx = mx + c − mx
   y − mx = c
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.
   c = y − mx

Check. Pick numbers for the other letters: y = 9, m = 4, x = 2. The new formula gives
c = 9 − 4 × 2 = 9 − 8 = 1
Now put c = 1 back into the original formula. Left side: 9. Right side: 4 × 2 + 1 = 8 + 1 = 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. 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 d 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 d on its own we must undo everything that has been done to it, in the reverse order.
   Formula: C = 35d + 20
   Reading from the subject outwards, what happens to d: it is multiplied by 35; then 20 is added.
   To undo it we reverse the order: subtract 20, then divide by 35.
2. Subtract 20 from both sides. 20 is being added to the part containing d. The inverse (opposite) of adding is subtracting, so subtract 20 from both sides. Doing the same to both sides keeps the formula true.
   C − 20 = 35d + 20 − 20
   C − 20 = 35d
3. Divide both sides by 35. d is being multiplied by 35 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 35.
   C − 2035 = 35d35
   C − 2035 = d
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.
   d = C − 2035

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.
   d = C − 2035
2. Substitute: C = 195. Substitute means replace each letter with its number.
   d = 195 − 2035
3. 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.
   195 − 20 = 175
   d = 17535
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.
   17535 means 175 ÷ 35 = 5
   d = 5
5. Write the answer with its units. The answer should say what was being worked out.
   d = 5 days

Check. Put the numbers straight into a calculator: (195 − 20)/35 gives 5. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

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: A = 12bh
   Reading from the subject outwards, what happens to h: it is multiplied by b; then it is multiplied by 12.
   To undo it we reverse the order: multiply by 2, then divide by b.
2. Multiply both sides by 2. 12 means "divide by 2". The inverse of dividing is multiplying, so multiply both sides by 2.
   A × 2 = 12bh × 2
   2A = bh
3. Divide both sides by b. h is being multiplied by b (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by b.
   2Ab = bhb
   2Ab = h
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.
   h = 2Ab

Check. Pick numbers for the other letters: A = 10, b = 5. The new formula gives
h = 2 × 105 = 205 = 4
Now put h = 4 back into the original formula. Left side: 10. Right side: 12 × 5 × 4 = 2.5 × 4 = 10. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. 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 = 2Ab
2. Substitute: A = 30, b = 12. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   h = 2 × 3012
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   2 × 30 = 60
   h = 6012
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.
   6012 means 60 ÷ 12 = 5
   h = 5
5. Write the answer with its units. The answer should say what was being worked out.
   h = 5 cm

Check. Put the numbers straight into a calculator: (2 × 30)/12 gives 5. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. 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 a 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 a on its own we must undo everything that has been done to it, in the reverse order.
   Formula: F = ma
   Reading from the subject outwards, what happens to a: it is multiplied by m.
   To undo it we reverse the order: divide by m.
2. Divide both sides by m. a is being multiplied by m (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by m.
   Fm = mam
   Fm = a
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.
   a = Fm

Check. Pick numbers for the other letters: F = 9, m = 3. The new formula gives
a = 93 = 3
Now put a = 3 back into the original formula. Left side: 9. Right side: 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.
   a = Fm
2. Substitute: F = 3000, m = 1200. Substitute means replace each letter with its number.
   a = 30001200
3. 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.
   30001200 means 3000 ÷ 1200 = 2.5
   a = 2.5
4. Write the answer with its units. The answer should say what was being worked out.
   a = 2.5 m/s²

Check. Put the numbers straight into a calculator: 3000/1200 gives 2.5. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

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

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

1. Find the letter we want, and see what is done to it. Treat the formula like a function machine. To get t on its own we must undo everything that has been done to it, in the reverse order.
   Formula: s = dt
   Reading from the subject outwards, what happens to t: it is the divisor in d ÷ t.
   To undo it we reverse the order: multiply by t, then divide by s.
2. Multiply both sides by t. t is on the bottom of a fraction. To bring it up, multiply both sides by t. (Anything divided by t, then multiplied by t, is back where it started.)
   s × t = dt × t
   st = d
3. Divide both sides by s. t is being multiplied by s (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by s.
   ds = sts
   ds = t
4. Write it with the subject on the left. The subject is the letter on its own. By convention we write it on the left of the equals sign. Swapping the two sides does not change the meaning.
   t = ds

Check. Pick numbers for the other letters: s = 4, d = 8. The new formula gives
t = 84 = 2
Now put t = 2 back into the original formula. Left side: 4. Right side: 82 = 4. 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.
   t = ds
2. Substitute: d = 360, s = 90. Substitute means replace each letter with its number.
   t = 36090
3. 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.
   36090 means 360 ÷ 90 = 4
   t = 4
4. Write the answer with its units. The answer should say what was being worked out.
   t = 4 hours

Check. Put the numbers straight into a calculator: 360/90 gives 4. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

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 y 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 y on its own we must undo everything that has been done to it, in the reverse order.
   Formula: x = 2y + 1
   Reading from the subject outwards, what happens to y: it is multiplied by 2; then 1 is added.
   To undo it we reverse the order: subtract 1, then divide by 2.
2. Subtract 1 from both sides. 1 is being added to the part containing y. The inverse (opposite) of adding is subtracting, so subtract 1 from both sides. Doing the same to both sides keeps the formula true.
   x − 1 = 2y + 1 − 1
   x − 1 = 2y
3. Divide both sides by 2. y is being multiplied by 2 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 2.
   x − 12 = 2y2
   x − 12 = y
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.
   y = x − 12

Check. Pick numbers for the other letters: x = 3. The new formula gives
y = 3 − 12 = 22 = 1
Now put y = 1 back into the original formula. Left side: 3. Right side: 2 × 1 + 1 = 2 + 1 = 3. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until y 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 y on its own we must undo everything that has been done to it, in the reverse order.
   Formula: x = y5 − 3
   Reading from the subject outwards, what happens to y: it is divided by 5; then 3 is subtracted.
   To undo it we reverse the order: add 3, then multiply by 5.
2. Add 3 to both sides. 3 is being subtracted. The inverse of subtracting is adding, so add 3 to both sides.
   x + 3 = y5 − 3 + 3
   x + 3 = y5
3. Multiply both sides by 5. The part containing y is being divided by 5. The inverse of dividing is multiplying, so multiply both sides by 5.
   (x + 3) × 5 = y5 × 5
   5(x + 3) = y
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.
   y = 5(x + 3)

Check. Pick numbers for the other letters: x = 5. The new formula gives
y = 5 × (5 + 3) = 5 × 8 = 40
Now put y = 40 back into the original formula. Left side: 5. Right side: 405 − 3 = 8 − 3 = 5. Both sides are equal, so the rearranging is correct. ✓

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

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

Method. Change the subject by undoing the operations one at a time, in reverse order, doing the same to both sides each time, until y 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 y on its own we must undo everything that has been done to it, in the reverse order.
   Formula: a = 3y − b
   Reading from the subject outwards, what happens to y: it is multiplied by 3; then b is subtracted.
   To undo it we reverse the order: add b, then divide by 3.
2. Add b to both sides. b is being subtracted. The inverse of subtracting is adding, so add b to both sides.
   a + b = 3y − b + b
   a + b = 3y
3. Divide both sides by 3. y is being multiplied by 3 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 3.
   a + b3 = 3y3
   a + b3 = y
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.
   y = a + b3

Check. Pick numbers for the other letters: a = 10, b = 2. The new formula gives
y = 10 + 23 = 123 = 4
Now put y = 4 back into the original formula. Left side: 10. Right side: 3 × 4 − 2 = 12 − 2 = 10. Both sides are equal, so the rearranging is correct. ✓

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

Key words. 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 y 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 y on its own we must undo everything that has been done to it, in the reverse order.
   Formula: p = q + 4y
   Reading from the subject outwards, what happens to y: it is multiplied by 4; then q is added.
   To undo it we reverse the order: subtract q, then divide by 4.
2. Subtract q from both sides. q is being added to the part containing y. The inverse (opposite) of adding is subtracting, so subtract q from both sides. Doing the same to both sides keeps the formula true.
   p − q = q + 4y − q
   p − q = 4y
3. Divide both sides by 4. y is being multiplied by 4 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 4.
   p − q4 = 4y4
   p − q4 = y
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.
   y = p − q4

Check. Pick numbers for the other letters: p = 9, q = 5. The new formula gives
y = 9 − 54 = 44 = 1
Now put y = 1 back into the original formula. Left side: 9. Right side: 5 + 4 × 1 = 5 + 4 = 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. 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: T = 40m + 20
   Reading from the subject outwards, what happens to m: it is multiplied by 40; then 20 is added.
   To undo it we reverse the order: subtract 20, then divide by 40.
2. Subtract 20 from both sides. 20 is being added to the part containing m. The inverse (opposite) of adding is subtracting, so subtract 20 from both sides. Doing the same to both sides keeps the formula true.
   T − 20 = 40m + 20 − 20
   T − 20 = 40m
3. Divide both sides by 40. m is being multiplied by 40 (letters or numbers written next to each other are multiplied). The inverse of multiplying is dividing, so divide both sides by 40.
   T − 2040 = 40m40
   T − 2040 = m
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.
   m = T − 2040

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 = T − 2040
2. Substitute: T = 180. Substitute means replace each letter with its number.
   m = 180 − 2040
3. 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.
   180 − 20 = 160
   m = 16040
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.
   16040 means 160 ÷ 40 = 4
   m = 4
5. Write the answer with its units. The answer should say what was being worked out.
   m = 4 kg

Check. Put the numbers straight into a calculator: (180 − 20)/40 gives 4. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.