GCSE Maths: Inverse operations, cancelling and reciprocals (Higher)

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   ? + 7 = 20
2. Use the inverse operation: subtract. The unknown has 7 added to it. The inverse (opposite) of adding 7 is subtracting 7.
   20 − 7 = 13
3. Write the missing number.
   ? = 13

Check. Put the answer back into the question: 13 + 7 = 20. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   ? − 8 = 15
2. Use the inverse operation: add. The unknown has had 8 taken away. The inverse of subtracting 8 is adding 8.
   15 + 8 = 23
3. Write the missing number.
   ? = 23

Check. Put the answer back into the question: 23 − 8 = 15. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   35 − ? = 12
2. Use the inverse operation: subtract. The unknown is the amount taken away from 35 to leave 12. So it is the difference between 35 and 12.
   35 − 12 = 23
3. Write the missing number.
   ? = 23

Check. Put the answer back into the question: 35 − 23 = 12. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   6 × ? = 54
2. Use the inverse operation: divide. The unknown is multiplied by 6. The inverse of multiplying by 6 is dividing by 6.
   54 ÷ 6 = 9
3. Write the missing number.
   ? = 9

Check. Put the answer back into the question: 6 × 9 = 54. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   ? ÷ 4 = 9
2. Use the inverse operation: multiply. The unknown is divided by 4. The inverse of dividing by 4 is multiplying by 4.
   9 × 4 = 36
3. Write the missing number.
   ? = 36

Check. Put the answer back into the question: 36 ÷ 4 = 9. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Key words. Inverse operations undo each other: addition and subtraction are inverses, and so are multiplication and division.

Method. Use the inverse (opposite) operation to undo what was done to the missing number.

1. Spot the missing number and the operation done to it. The question mark stands for the number we have to find.
   72 ÷ ? = 8
2. Use the inverse operation: divide. 72 shared into groups of the unknown size gives 8 groups. So the unknown is 72 divided by 8.
   72 ÷ 8 = 9
3. Write the missing number.
   ? = 9

Check. Put the answer back into the question: 72 ÷ 9 = 8. ✓

Watch out. Be careful when the missing number is after a subtraction or division sign: 20 − ? = 12 means ? = 20 − 12, not 20 + 12.

Method. Work backwards through the steps, undoing each one with its inverse operation, last step first.

1. Write the steps as a chain, starting from the unknown number n. A chain (function machine) shows each operation in order.
   n → (× 3) → (− 4) → 23
2. Work backwards from the answer, using the inverse of each step in reverse order. To undo a series of steps, you must undo the last step first, like taking off your shoes before your socks if you put on the socks first.
   Undo "− 4": 23 + 4 = 27
   Undo "× 3": 27 ÷ 3 = 9
3. Write the starting number.
   The number is 9.

Check. Go forwards from 9: 9 × 3 = 27, 27 − 4 = 23. This gives 23, as in the question. ✓

Watch out. Undo the steps in reverse order, and use the opposite operation for each.

Method. Work backwards through the steps, undoing each one with its inverse operation, last step first.

1. Write the steps as a chain, starting from the unknown number n. A chain (function machine) shows each operation in order.
   n → (+ 8) → (× 2) → 30
2. Work backwards from the answer, using the inverse of each step in reverse order. To undo a series of steps, you must undo the last step first, like taking off your shoes before your socks if you put on the socks first.
   Undo "× 2": 30 ÷ 2 = 15
   Undo "+ 8": 15 − 8 = 7
3. Write the starting number.
   The number is 7.

Check. Go forwards from 7: 7 + 8 = 15, 15 × 2 = 30. This gives 30, as in the question. ✓

Watch out. Undo the steps in reverse order, and use the opposite operation for each.

Method. Work backwards through the steps, undoing each one with its inverse operation, last step first.

1. Write the steps as a chain, starting from the unknown number n. A chain (function machine) shows each operation in order.
   n → (÷ 5) → (+ 3) → 11
2. Work backwards from the answer, using the inverse of each step in reverse order. To undo a series of steps, you must undo the last step first, like taking off your shoes before your socks if you put on the socks first.
   Undo "+ 3": 11 − 3 = 8
   Undo "÷ 5": 8 × 5 = 40
3. Write the starting number.
   The number is 40.

Check. Go forwards from 40: 40 ÷ 5 = 8, 8 + 3 = 11. This gives 11, as in the question. ✓

Watch out. Undo the steps in reverse order, and use the opposite operation for each.

Method. To check a multiplication, divide the answer by one of the numbers.

1. Divide the answer by 15. If the multiplication is right, we get back the other number.
   5130 ÷ 15 = 342
2. Compare. The result is 342, the other number.
   342 = 342, so the answer is correct.

Check. 342 × 15 = 5130. ✓

Method. To check a division, multiply the answer by the divisor.

1. Multiply 627 by 12. If the division is right, we get back the number that was divided.
   627 × 12 = 7524
2. Compare.
   7524 = 7524, so the answer is correct.

Check. 7524 ÷ 12 = 627. ✓

Method. To check a subtraction, add the answer to the number that was taken away.

1. Add 542 and 468. If the subtraction is right, we get back 1000.
   542 + 468 = 1010
2. Compare. 1010 is not 1000, so the answer is wrong. The correct answer is 532.
   1000 − 468 = 532, so the answer is incorrect.

Check. 532 + 468 = 1000. ✓

Method. Use the given fact. Work out how each number has changed (a power of 10), then change the answer in the same way.

1. Write down the fact you are given. The question tells us this result so that we do not have to calculate from scratch.
   23 × 47 = 1081
2. Compare each number in the new calculation with the numbers in the given fact. Look for how many times 10 each number has been multiplied or divided by.
   23 → 230: 23 × 10 = 230
   47 → 47: no change
3. Change the answer in the same way. Each change to a number changes the answer in the same way. Together, the answer changes by × 10.
   1081 × 10 = 10 810

Check. Rounded check: 200 × 50 = 10 000, which is close to 10 810. ✓

Watch out. When a number is multiplied by 10 and another by 100, the answer is multiplied by 10 × 100 = 1000.

Method. Use the given fact. Work out how each number has changed (a power of 10), then change the answer in the same way.

1. Write down the fact you are given. The question tells us this result so that we do not have to calculate from scratch.
   23 × 47 = 1081
2. Compare each number in the new calculation with the numbers in the given fact. Look for how many times 10 each number has been multiplied or divided by.
   23 → 23: no change
   47 → 4.7: 47 ÷ 10 = 4.7
3. Change the answer in the same way. Each change to a number changes the answer in the same way. Together, the answer changes by ÷ 10.
   1081 ÷ 10 = 108.1

Check. Rounded check: 20 × 5 = 100, which is close to 108.1. ✓

Watch out. When a number is multiplied by 10 and another by 100, the answer is multiplied by 10 × 100 = 1000.

Method. Use the given fact. Work out how each number has changed (a power of 10), then change the answer in the same way.

1. Write down the fact you are given. The question tells us this result so that we do not have to calculate from scratch.
   23 × 47 = 1081
2. Compare each number in the new calculation with the numbers in the given fact. Look for how many times 10 each number has been multiplied or divided by.
   23 → 2.3: 23 ÷ 10 = 2.3
   47 → 470: 47 × 10 = 470
3. Change the answer in the same way. Each change to a number changes the answer in the same way. The changes cancel out, so the answer stays the same.
   1081

Check. Rounded check: 2 × 500 = 1000, which is close to 1081. ✓

Watch out. When a number is multiplied by 10 and another by 100, the answer is multiplied by 10 × 100 = 1000.

Method. Use the given fact and its inverse. Work out how each number has changed (a power of 10) and change the answer to match.

1. Write down the fact you are given. We will use it, so we do not have to work from scratch.
   23 × 47 = 1081
2. Use the inverse to get a division fact. Multiplication and division are inverses. If a × b = c then c ÷ b = a and c ÷ a = b.
   1081 ÷ 23 = 47
3. Compare the new numbers with the numbers in this fact. Look for the power of 10 that links each pair.
   1081 → 1081: no change
   23 → 23: no change
4. Change the answer. The number being divided has not changed. The number we divide by has not changed.
   47 = 47

Check. Multiply back: 47 × 23 = 1081, which is 1081. ✓

Watch out. Making the number you divide by bigger makes the answer smaller. 1081 ÷ 470 is a tenth of 1081 ÷ 47.

Method. Use the given fact and its inverse. Work out how each number has changed (a power of 10) and change the answer to match.

1. Write down the fact you are given. We will use it, so we do not have to work from scratch.
   23 × 47 = 1081
2. Use the inverse to get a division fact. Multiplication and division are inverses. If a × b = c then c ÷ b = a and c ÷ a = b.
   1081 ÷ 47 = 23
3. Compare the new numbers with the numbers in this fact. Look for the power of 10 that links each pair.
   1081 → 10 810: 1081 × 10 = 10 810
   47 → 47: no change
4. Change the answer. Changing the number being divided by × 10 changes the answer in the same way. The number we divide by has not changed.
   23 × 10 = 230

Check. Multiply back: 230 × 47 = 10 810, which is 10 810. ✓

Watch out. Making the number you divide by bigger makes the answer smaller. 1081 ÷ 470 is a tenth of 1081 ÷ 47.

Method. Use the given fact and its inverse. Work out how each number has changed (a power of 10) and change the answer to match.

1. Write down the fact you are given. We will use it, so we do not have to work from scratch.
   23 × 47 = 1081
2. Use the inverse to get a division fact. Multiplication and division are inverses. If a × b = c then c ÷ b = a and c ÷ a = b.
   1081 ÷ 23 = 47
3. Compare the new numbers with the numbers in this fact. Look for the power of 10 that links each pair.
   1081 → 108.1: 1081 ÷ 10 = 108.1
   23 → 23: no change
4. Change the answer. Changing the number being divided by ÷ 10 changes the answer in the same way. The number we divide by has not changed.
   47 ÷ 10 = 4.7

Check. Multiply back: 4.7 × 23 = 108.1, which is 108.1. ✓

Watch out. Making the number you divide by bigger makes the answer smaller. 1081 ÷ 470 is a tenth of 1081 ÷ 47.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value.
   5 = 51
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   51 → 15

Check. Multiply the number by its reciprocal: 5 × 15 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   23 → 32

Check. Multiply the number by its reciprocal: 23 × 32 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.25 = 14
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   14 → 41
4. Simplify. A fraction with 1 on the bottom is just a whole number.
   41 = 4

Check. Multiply the number by its reciprocal: 0.25 × 4 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.4 = 25
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   25 → 52
4. Write the answer as a decimal if it helps. Divide the top by the bottom.
   52 = 2.5

Check. Multiply the number by its reciprocal: 0.4 × 52 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   18 → 81
3. Simplify. A fraction with 1 on the bottom is just a whole number.
   81 = 8

Check. Multiply the number by its reciprocal: 18 × 8 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Method. Look for a multiplication and a division by the same number (or numbers that divide exactly). Cancel first, then multiply.

1. Look for a number that divides exactly into both 18 and 18. Multiplying by 18 and then dividing by 18 is the same as multiplying by 18 ÷ 18. Doing this division first keeps the numbers small.
   18 ÷ 18 = 1
2. Rewrite the calculation. 37 × 18 ÷ 18 = 37 × (18 ÷ 18) because multiplying and dividing can be done in either order.
   37 × 1
3. Multiply.
   37 × 1 = 37

Check. The long way: 37 × 18 = 666, then 666 ÷ 18 = 37. ✓

Watch out. Cancelling only works when you are multiplying and dividing. It does not work with addition: (7 + 3) ÷ 3 is not 7.

Method. Look for a multiplication and a division by the same number (or numbers that divide exactly). Cancel first, then multiply.

1. Look for a number that divides exactly into both 48 and 8. Multiplying by 48 and then dividing by 8 is the same as multiplying by 48 ÷ 8. Doing this division first keeps the numbers small.
   48 ÷ 8 = 6
2. Rewrite the calculation. 125 × 48 ÷ 8 = 125 × (48 ÷ 8) because multiplying and dividing can be done in either order.
   125 × 6
3. Multiply.
   125 × 6 = 750

Check. The long way: 125 × 48 = 6000, then 6000 ÷ 8 = 750. ✓

Watch out. Cancelling only works when you are multiplying and dividing. It does not work with addition: (7 + 3) ÷ 3 is not 7.

Method. Look for a multiplication and a division by the same number (or numbers that divide exactly). Cancel first, then multiply.

1. Look for a number that divides exactly into both 24 and 6. Multiplying by 24 and then dividing by 6 is the same as multiplying by 24 ÷ 6. Doing this division first keeps the numbers small.
   24 ÷ 6 = 4
2. Rewrite the calculation. 15 × 24 ÷ 6 = 15 × (24 ÷ 6) because multiplying and dividing can be done in either order.
   15 × 4
3. Multiply.
   15 × 4 = 60

Check. The long way: 15 × 24 = 360, then 360 ÷ 6 = 60. ✓

Watch out. Cancelling only works when you are multiplying and dividing. It does not work with addition: (7 + 3) ÷ 3 is not 7.

Method. Look for a multiplication and a division by the same number (or numbers that divide exactly). Cancel first, then multiply.

1. Look for a number that divides exactly into both 72 and 9. Dividing 72 by 9 first keeps the numbers small, because the order of multiplying and dividing does not matter.
   72 ÷ 9 = 8
2. Rewrite the calculation.
   8 × 35
3. Multiply.
   8 × 35 = 280

Check. The long way: 72 × 35 = 2520, then 2520 ÷ 9 = 280. ✓

Watch out. Cancelling only works when you are multiplying and dividing. It does not work with addition: (7 + 3) ÷ 3 is not 7.

Method. Divide the number part by the same number so that it cancels.

1. Write 6x as 6 × x. The number and the letter are multiplied together.
   6x ÷ 6 = 6 × x ÷ 6
2. Cancel the 6s. Multiplying by 6 and dividing by 6 cancel.
   x

Check. If x = 5: 30 ÷ 6 = 5. ✓

Method. Multiplying by 5 and dividing by 5 cancel.

1. Spot the matching × 5 and ÷ 5. They undo each other.
   (4a × 5) ÷ 5
2. Cancel. What is left is 4a.
   4a

Check. If a = 2: (8 × 5) ÷ 5 = 8. ✓

Method. Adding 9 and taking away 9 cancel.

1. Spot the + 9 and − 9. They undo each other.
   y + 9 − 9
2. Cancel.
   y

Check. If y = 4: 4 + 9 − 9 = 4. ✓

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Change the mixed number to an improper fraction. We need a single fraction to turn upside down. Multiply the whole number by the denominator, add the numerator, and keep the denominator.
   214 = 94
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   94 → 49

Check. Multiply the number by its reciprocal: 214 × 49 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.125 = 18
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   18 → 81
4. Simplify. A fraction with 1 on the bottom is just a whole number.
   81 = 8

Check. Multiply the number by its reciprocal: 0.125 × 8 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   1.6 = 85
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   85 → 58
4. Write the answer as a decimal if it helps. Divide the top by the bottom.
   58 = 0.625

Check. Multiply the number by its reciprocal: 1.6 × 58 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Method. Dividing by a number is the same as multiplying by its reciprocal (the number turned upside down).

1. Write the number you are dividing by as a fraction. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.25 = 14
2. Find its reciprocal. The reciprocal of a fraction is the fraction turned upside down.
   reciprocal of 14 is 41 = 4
3. Change the division into a multiplication. Dividing by a number is the same as multiplying by its reciprocal.
   12 ÷ 0.25 = 12 × 4
4. Multiply.
   12 × 4 = 48

Check. 48 × 0.25 = 12. ✓

Watch out. The reciprocal of a number less than 1 is bigger than 1, so dividing by 0.25 makes the answer 4 times bigger.

Method. Dividing by a number is the same as multiplying by its reciprocal (the number turned upside down).

1. Write the number you are dividing by as a fraction. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.75 = 34
2. Find its reciprocal. The reciprocal of a fraction is the fraction turned upside down.
   reciprocal of 34 is 43
3. Change the division into a multiplication. Dividing by a number is the same as multiplying by its reciprocal.
   9 ÷ 0.75 = 9 × 43
4. Multiply.
   9 × 43 = 12

Check. 12 × 0.75 = 9. ✓

Watch out. The reciprocal of a number less than 1 is bigger than 1, so dividing by 0.25 makes the answer 4 times bigger.

Method. Dividing by a number is the same as multiplying by its reciprocal (the number turned upside down).

1. Write the number you are dividing by as a fraction. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.4 = 25
2. Find its reciprocal. The reciprocal of a fraction is the fraction turned upside down.
   reciprocal of 25 is 52 = 2.5
3. Change the division into a multiplication. Dividing by a number is the same as multiplying by its reciprocal.
   20 ÷ 0.4 = 20 × 52
4. Multiply.
   20 × 52 = 50

Check. 50 × 0.4 = 20. ✓

Watch out. The reciprocal of a number less than 1 is bigger than 1, so dividing by 0.25 makes the answer 4 times bigger.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   (2.5)²
2. Indices (powers): work out (2.5)². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   (2.5)²
   (2.5)² = 2.5 × 2.5 = 6.25
   = 6.25

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 6.25. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   (1.2)³
2. Indices (powers): work out (1.2)³. A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   (1.2)³
   (1.2)³ = 1.2 × 1.2 × 1.2 = 1.728
   = 1.728

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 1.728. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   √20.25
2. Indices (roots): work out √20.25. The square root of a number is the number that multiplies by itself to give it. After brackets, powers and roots come next.
   √20.25
   √20.25 = 4.5 because 4.5 × 4.5 = 20.25
   = 4.5

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 4.5. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   ³√0.027
2. Indices (roots): work out ³√0.027. The cube root of a number is the number that multiplies by itself three times to give it. After brackets, powers and roots come next.
   ³√0.027
   ³√0.027 = 0.3 because 0.3 × 0.3 × 0.3 = 0.027
   = 0.3

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 0.3. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   (0.4)² + (0.3)²
2. Indices (powers): work out (0.4)². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   (0.4)² + (0.3)²
   (0.4)² = 0.4 × 0.4 = 0.16
   = 0.16 + (0.3)²
3. Indices (powers): work out (0.3)². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   0.16 + (0.3)²
   (0.3)² = 0.3 × 0.3 = 0.09
   = 0.16 + 0.09
4. Add: 0.16 + 0.09. Only addition or subtraction is left, so do it now.
   0.16 + 0.09
   0.16 + 0.09 = 0.25
   = 0.25

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 0.25. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   √((0.6)² + (0.8)²)
2. Work out what is under the root sign first (it acts like a bracket). Indices (powers): work out (0.6)². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   √((0.6)² + (0.8)²)
   (0.6)² = 0.6 × 0.6 = 0.36
   = √(0.36 + (0.8)²)
3. Work out what is under the root sign first (it acts like a bracket). Indices (powers): work out (0.8)². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   √(0.36 + (0.8)²)
   (0.8)² = 0.8 × 0.8 = 0.64
   = √(0.36 + 0.64)
4. Work out what is under the root sign first (it acts like a bracket). Add: 0.36 + 0.64. Only addition or subtraction is left, so do it now.
   √(0.36 + 0.64)
   0.36 + 0.64 = 1
   = √1
5. Indices (roots): work out √1. The square root of a number is the number that multiplies by itself to give it. After brackets, powers and roots come next.
   √1
   √1 = 1 because 1 × 1 = 1
   = 1

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 1. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.8 = 45
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   45 → 54
4. Write the answer as a decimal if it helps. Divide the top by the bottom.
   54 = 1.25

Check. Multiply the number by its reciprocal: 0.8 × 54 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.5 = 12
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   12 → 21
4. Simplify. A fraction with 1 on the bottom is just a whole number.
   21 = 2

Check. Multiply the number by its reciprocal: 0.5 × 2 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   1.25 = 54
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   54 → 45
4. Write the answer as a decimal if it helps. Divide the top by the bottom.
   45 = 0.8

Check. Multiply the number by its reciprocal: 1.25 × 45 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Write the decimal as a fraction in its simplest form. A decimal is a fraction with 10, 100, ... on the bottom. Simplify it.
   0.2 = 15
3. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   15 → 51
4. Simplify. A fraction with 1 on the bottom is just a whole number.
   51 = 5

Check. Multiply the number by its reciprocal: 0.2 × 5 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.