GCSE Maths: Mixed numbers and improper fractions (Foundation)

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 72 means 7 pieces, each one 12 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   7 pieces of size 12
2. Work out how many pieces make one whole. The bottom number tells us this: 2 pieces make 1 whole. So we want to know how many groups of 2 we can make from 7 pieces.
   2 pieces = 1 whole
3. Divide 7 by 2 to find the whole number. Count how many whole groups of 2 fit into 7. 2 × 3 = 6, but 2 × 4 = 8 is too big.
   7 ÷ 2 = 3 remainder 1
4. Find the pieces left over. After 3 whole ones we have used 6 pieces. The rest are the remainder.
   7 − 6 = 1
5. Write the mixed number. The whole number is 3. The 1 leftover piece of size 12 make the fraction 12. The bottom number stays 2 because the pieces are the same size.
   72 = 312

Check. Convert back: 3 × 2 + 1 = 6 + 1 = 7. As decimals, 7 ÷ 2 = 3.5 and 312 = 3.5. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 114 means 11 pieces, each one 14 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   11 pieces of size 14
2. Work out how many pieces make one whole. The bottom number tells us this: 4 pieces make 1 whole. So we want to know how many groups of 4 we can make from 11 pieces.
   4 pieces = 1 whole
3. Divide 11 by 4 to find the whole number. Count how many whole groups of 4 fit into 11. 4 × 2 = 8, but 4 × 3 = 12 is too big.
   11 ÷ 4 = 2 remainder 3
4. Find the pieces left over. After 2 whole ones we have used 8 pieces. The rest are the remainder.
   11 − 8 = 3
5. Write the mixed number. The whole number is 2. The 3 leftover pieces of size 14 make the fraction 34. The bottom number stays 4 because the pieces are the same size.
   114 = 234

Check. Convert back: 2 × 4 + 3 = 8 + 3 = 11. As decimals, 11 ÷ 4 = 2.75 and 234 = 2.75. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 133 means 13 pieces, each one 13 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   13 pieces of size 13
2. Work out how many pieces make one whole. The bottom number tells us this: 3 pieces make 1 whole. So we want to know how many groups of 3 we can make from 13 pieces.
   3 pieces = 1 whole
3. Divide 13 by 3 to find the whole number. Count how many whole groups of 3 fit into 13. 3 × 4 = 12, but 3 × 5 = 15 is too big.
   13 ÷ 3 = 4 remainder 1
4. Find the pieces left over. After 4 whole ones we have used 12 pieces. The rest are the remainder.
   13 − 12 = 1
5. Write the mixed number. The whole number is 4. The 1 leftover piece of size 13 make the fraction 13. The bottom number stays 3 because the pieces are the same size.
   133 = 413

Check. Convert back: 4 × 3 + 1 = 12 + 1 = 13. As decimals, 13 ÷ 3 = 4.3333 and 413 = 4.3333. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 195 means 19 pieces, each one 15 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   19 pieces of size 15
2. Work out how many pieces make one whole. The bottom number tells us this: 5 pieces make 1 whole. So we want to know how many groups of 5 we can make from 19 pieces.
   5 pieces = 1 whole
3. Divide 19 by 5 to find the whole number. Count how many whole groups of 5 fit into 19. 5 × 3 = 15, but 5 × 4 = 20 is too big.
   19 ÷ 5 = 3 remainder 4
4. Find the pieces left over. After 3 whole ones we have used 15 pieces. The rest are the remainder.
   19 − 15 = 4
5. Write the mixed number. The whole number is 3. The 4 leftover pieces of size 15 make the fraction 45. The bottom number stays 5 because the pieces are the same size.
   195 = 345

Check. Convert back: 3 × 5 + 4 = 15 + 4 = 19. As decimals, 19 ÷ 5 = 3.8 and 345 = 3.8. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 236 means 23 pieces, each one 16 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   23 pieces of size 16
2. Work out how many pieces make one whole. The bottom number tells us this: 6 pieces make 1 whole. So we want to know how many groups of 6 we can make from 23 pieces.
   6 pieces = 1 whole
3. Divide 23 by 6 to find the whole number. Count how many whole groups of 6 fit into 23. 6 × 3 = 18, but 6 × 4 = 24 is too big.
   23 ÷ 6 = 3 remainder 5
4. Find the pieces left over. After 3 whole ones we have used 18 pieces. The rest are the remainder.
   23 − 18 = 5
5. Write the mixed number. The whole number is 3. The 5 leftover pieces of size 16 make the fraction 56. The bottom number stays 6 because the pieces are the same size.
   236 = 356

Check. Convert back: 3 × 6 + 5 = 18 + 5 = 23. As decimals, 23 ÷ 6 = 3.8333 and 356 = 3.8333. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 298 means 29 pieces, each one 18 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   29 pieces of size 18
2. Work out how many pieces make one whole. The bottom number tells us this: 8 pieces make 1 whole. So we want to know how many groups of 8 we can make from 29 pieces.
   8 pieces = 1 whole
3. Divide 29 by 8 to find the whole number. Count how many whole groups of 8 fit into 29. 8 × 3 = 24, but 8 × 4 = 32 is too big.
   29 ÷ 8 = 3 remainder 5
4. Find the pieces left over. After 3 whole ones we have used 24 pieces. The rest are the remainder.
   29 − 24 = 5
5. Write the mixed number. The whole number is 3. The 5 leftover pieces of size 18 make the fraction 58. The bottom number stays 8 because the pieces are the same size.
   298 = 358

Check. Convert back: 3 × 8 + 5 = 24 + 5 = 29. As decimals, 29 ÷ 8 = 3.625 and 358 = 3.625. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 419 means 41 pieces, each one 19 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   41 pieces of size 19
2. Work out how many pieces make one whole. The bottom number tells us this: 9 pieces make 1 whole. So we want to know how many groups of 9 we can make from 41 pieces.
   9 pieces = 1 whole
3. Divide 41 by 9 to find the whole number. Count how many whole groups of 9 fit into 41. 9 × 4 = 36, but 9 × 5 = 45 is too big.
   41 ÷ 9 = 4 remainder 5
4. Find the pieces left over. After 4 whole ones we have used 36 pieces. The rest are the remainder.
   41 − 36 = 5
5. Write the mixed number. The whole number is 4. The 5 leftover pieces of size 19 make the fraction 59. The bottom number stays 9 because the pieces are the same size.
   419 = 459

Check. Convert back: 4 × 9 + 5 = 36 + 5 = 41. As decimals, 41 ÷ 9 = 4.5556 and 459 = 4.5556. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.

1. Say what the fraction means. 507 means 50 pieces, each one 17 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.
   50 pieces of size 17
2. Work out how many pieces make one whole. The bottom number tells us this: 7 pieces make 1 whole. So we want to know how many groups of 7 we can make from 50 pieces.
   7 pieces = 1 whole
3. Divide 50 by 7 to find the whole number. Count how many whole groups of 7 fit into 50. 7 × 7 = 49, but 7 × 8 = 56 is too big.
   50 ÷ 7 = 7 remainder 1
4. Find the pieces left over. After 7 whole ones we have used 49 pieces. The rest are the remainder.
   50 − 49 = 1
5. Write the mixed number. The whole number is 7. The 1 leftover piece of size 17 make the fraction 17. The bottom number stays 7 because the pieces are the same size.
   507 = 717

Check. Convert back: 7 × 7 + 1 = 49 + 1 = 50. As decimals, 50 ÷ 7 = 7.1429 and 717 = 7.1429. ✓

Watch out. Keep the original denominator. Do not use the remainder as the denominator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 112 means 1 whole ones and 1 extra piece of size 12.
   1 wholes + 12
2. Change the whole ones into pieces. Each whole one is 2 pieces of size 12, so 1 wholes is 1 groups of 2 pieces.
   1 × 2 = 2 pieces
3. Add the extra pieces. Now add the 1 pieces from the fraction part.
   2 + 1 = 3 pieces
4. Write the total over the same denominator. Every piece is still of size 12, so the denominator does not change.
   112 = 32

Check. Divide back: 3 ÷ 2 = 1 remainder 1, which is 112. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 234 means 2 whole ones and 3 extra pieces of size 14.
   2 wholes + 34
2. Change the whole ones into pieces. Each whole one is 4 pieces of size 14, so 2 wholes is 2 groups of 4 pieces.
   2 × 4 = 8 pieces
3. Add the extra pieces. Now add the 3 pieces from the fraction part.
   8 + 3 = 11 pieces
4. Write the total over the same denominator. Every piece is still of size 14, so the denominator does not change.
   234 = 114

Check. Divide back: 11 ÷ 4 = 2 remainder 3, which is 234. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 325 means 3 whole ones and 2 extra pieces of size 15.
   3 wholes + 25
2. Change the whole ones into pieces. Each whole one is 5 pieces of size 15, so 3 wholes is 3 groups of 5 pieces.
   3 × 5 = 15 pieces
3. Add the extra pieces. Now add the 2 pieces from the fraction part.
   15 + 2 = 17 pieces
4. Write the total over the same denominator. Every piece is still of size 15, so the denominator does not change.
   325 = 175

Check. Divide back: 17 ÷ 5 = 3 remainder 2, which is 325. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 456 means 4 whole ones and 5 extra pieces of size 16.
   4 wholes + 56
2. Change the whole ones into pieces. Each whole one is 6 pieces of size 16, so 4 wholes is 4 groups of 6 pieces.
   4 × 6 = 24 pieces
3. Add the extra pieces. Now add the 5 pieces from the fraction part.
   24 + 5 = 29 pieces
4. Write the total over the same denominator. Every piece is still of size 16, so the denominator does not change.
   456 = 296

Check. Divide back: 29 ÷ 6 = 4 remainder 5, which is 456. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 578 means 5 whole ones and 7 extra pieces of size 18.
   5 wholes + 78
2. Change the whole ones into pieces. Each whole one is 8 pieces of size 18, so 5 wholes is 5 groups of 8 pieces.
   5 × 8 = 40 pieces
3. Add the extra pieces. Now add the 7 pieces from the fraction part.
   40 + 7 = 47 pieces
4. Write the total over the same denominator. Every piece is still of size 18, so the denominator does not change.
   578 = 478

Check. Divide back: 47 ÷ 8 = 5 remainder 7, which is 578. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 619 means 6 whole ones and 1 extra piece of size 19.
   6 wholes + 19
2. Change the whole ones into pieces. Each whole one is 9 pieces of size 19, so 6 wholes is 6 groups of 9 pieces.
   6 × 9 = 54 pieces
3. Add the extra pieces. Now add the 1 pieces from the fraction part.
   54 + 1 = 55 pieces
4. Write the total over the same denominator. Every piece is still of size 19, so the denominator does not change.
   619 = 559

Check. Divide back: 55 ÷ 9 = 6 remainder 1, which is 619. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 10310 means 10 whole ones and 3 extra pieces of size 110.
   10 wholes + 310
2. Change the whole ones into pieces. Each whole one is 10 pieces of size 110, so 10 wholes is 10 groups of 10 pieces.
   10 × 10 = 100 pieces
3. Add the extra pieces. Now add the 3 pieces from the fraction part.
   100 + 3 = 103 pieces
4. Write the total over the same denominator. Every piece is still of size 110, so the denominator does not change.
   10310 = 10310

Check. Divide back: 103 ÷ 10 = 10 remainder 3, which is 10310. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.

1. Say what the mixed number means. 847 means 8 whole ones and 4 extra pieces of size 17.
   8 wholes + 47
2. Change the whole ones into pieces. Each whole one is 7 pieces of size 17, so 8 wholes is 8 groups of 7 pieces.
   8 × 7 = 56 pieces
3. Add the extra pieces. Now add the 4 pieces from the fraction part.
   56 + 4 = 60 pieces
4. Write the total over the same denominator. Every piece is still of size 17, so the denominator does not change.
   847 = 607

Check. Divide back: 60 ÷ 7 = 8 remainder 4, which is 847. ✓

Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.

Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.

1. Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.
   4 parts, so each piece is 14
2. Count all the shaded parts across all the pies. This is the numerator of the improper fraction.
   5 shaded parts
3. Write the improper fraction. Shaded parts over parts in one pie.
   54
4. Write it as a mixed number. 5 ÷ 4 = 1 remainder 1, so there is 1 complete pie and 1 extra part.
   54 = 114

Check. Convert back: 1 × 4 + 1 = 5. ✓

Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.

Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.

1. Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.
   3 parts, so each piece is 13
2. Count all the shaded parts across all the pies. This is the numerator of the improper fraction.
   7 shaded parts
3. Write the improper fraction. Shaded parts over parts in one pie.
   73
4. Write it as a mixed number. 7 ÷ 3 = 2 remainder 1, so there are 2 complete pies and 1 extra part.
   73 = 213

Check. Convert back: 2 × 3 + 1 = 7. ✓

Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.

Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.

1. Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.
   6 parts, so each piece is 16
2. Count all the shaded parts across all the pies. This is the numerator of the improper fraction.
   11 shaded parts
3. Write the improper fraction. Shaded parts over parts in one pie.
   116
4. Write it as a mixed number. 11 ÷ 6 = 1 remainder 5, so there is 1 complete pie and 5 extra parts.
   116 = 156

Check. Convert back: 1 × 6 + 5 = 11. ✓

Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.

Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.

1. Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.
   5 parts, so each piece is 15
2. Count all the shaded parts across all the pies. This is the numerator of the improper fraction.
   8 shaded parts
3. Write the improper fraction. Shaded parts over parts in one pie.
   85
4. Write it as a mixed number. 8 ÷ 5 = 1 remainder 3, so there is 1 complete pie and 3 extra parts.
   85 = 135

Check. Convert back: 1 × 5 + 3 = 8. ✓

Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.

Method. A mixed number has a whole-number part and a fraction part. Check what the statement says is true.

1. Work out the value of 7/7. Any fraction with the top equal to the bottom is exactly 1 whole.
   77 = 7 ÷ 7 = 1
2. Look at what a mixed number needs. A mixed number needs a whole number and a proper fraction. 1 has no fraction part.
   1 is a whole number only
3. Decide. The statement says 7/7 is a mixed number.
   False. It is the whole number 1.

Check. If you divide 7 by 7 there is no remainder, so there is no fraction part. ✓

Watch out. Only a whole number and a proper fraction together make a mixed number.

Method. Change the mixed number to an improper fraction and compare.

1. Multiply the whole number by the denominator. Each whole one is 3 thirds.
   5 × 3 = 15
2. Add the numerator. Add the extra 2 thirds.
   15 + 2 = 17
3. Compare. The result has denominator 3, the same as the original.
   523 = 173. True.

Check. Convert back: 17 ÷ 3 = 5 remainder 2, which is 523. ✓

Watch out. Whole number × denominator, then add the numerator.

Method. Change the improper fraction to a mixed number and compare.

1. Divide the top by the bottom. This finds the whole number part and the remainder.
   11 ÷ 4 = 2 remainder 3
2. Write the mixed number. The remainder 3 goes over the denominator 4.
   114 = 234
3. Compare with the statement. The statement says 214, but the remainder is 3, not 1.
   234 is not 214. False.

Check. Convert the statement back: 214 = 94, which is not 114. ✓

Watch out. The remainder is the new numerator. Check it carefully.