Year 5 Equivalent Fractions and Mixed Numbers: check-up and SATs-style questions

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

1. Look at the bottom numbers. What do we multiply 4 by to get 20? This tells us how the fraction has been changed. We must do the same to the other number.
   4 × 5 = 20, so the multiplier is 5.
2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be multiplied by 5.
   3 × 5 = 15
3. Write the equivalent fractions. Now both numbers are filled in.
   34 = 1520

Check. 34 and 1520 both simplify to 34. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

1. List the factors of 18 and of 30. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
   Factors of 18: 1, 2, 3, 6, 9, 18
   Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
   Common factors: 1, 2, 3, 6
2. Pick the biggest common factor: 6. Divide the top and the bottom by 6. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
   1830 = 18 ÷ 630 ÷ 6 = 35
3. Check it is in its simplest form. If 3 and 5 still shared a factor, we could go on.
   The only common factor of 3 and 5 is 1, so 35 is the simplest form.

Check. Multiply back: 3 × 6 = 18 and 5 × 6 = 30. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Make the denominators the same. Then the fraction with the bigger numerator is the bigger fraction.

1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 10, so 10 will do.
   10 ÷ 5 = 2, so 5 goes into 10
2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time, so the value does not change.
   710
   35 = 3 × 25 × 2 = 610
3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
   7 > 6
4. Write the answer. Use the original fractions in your answer.
   710 > 35

Check. Look at the changed fractions again: 710 and 610 have the same denominator, so the bigger numerator is the bigger fraction. ✓

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Key words. To order fractions is to put them in a line from the smallest to the largest (ascending) or from the largest to the smallest (descending). Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change every fraction to the same denominator. Then order the numerators.

1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 12, so 12 will do.
   12 ÷ 3 = 4, so 3 goes into 12
   12 ÷ 6 = 2, so 6 goes into 12
2. Change every fraction to the common denominator. Multiply the top and the bottom by the same number each time.
   23 = 2 × 43 × 4 = 812
   712 (already in twelfths)
   56 = 5 × 26 × 2 = 1012
3. Put the top numbers in order. With the same bottom number, the bigger the top number, the bigger the fraction. We want smallest to largest.
   7, 8, 10 (out of 12)
4. Write the original fractions in that order. The answer uses the fractions from the question, not the changed ones.
   712,  23,  56

Check. Read the numerators again: 7, 8, 10 go up each time. ✓

Watch out. Write down the original fractions in your final answer.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 23 by 6. The fraction line means divide. The answer tells us how many whole ones we can make.
   23 ÷ 6 = 3 remainder 5
2. The whole number is 3. 3 lots of 6 sixths make 3 whole ones: 3 × 6 = 18.
   23 − 18 = 5 sixths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 6.
   3 and 56 = 356

Check. Change back: 3 × 6 + 5 = 23 = 23. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

1. Change the 4 wholes into eighths. One whole is 88, so 4 wholes are 4 lots of 8.
   4 × 8 = 32, so 4 = 328
2. Add the 3 extra eighths. The fraction part is already in the same size parts.
   32 + 3 = 35
3. Write the improper fraction. The denominator stays the same.
   438 = 358

Check. Change back: 35 ÷ 8 = 4 remainder 3, so 358 = 438. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 10, so 10 will do.
   10 ÷ 5 = 2, so 5 goes into 10
2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
   45 = 4 × 25 × 2 = 810
3. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
   710 + 810 = 7 + 810 = 1510
4. The answer is 1 or more, so change it to a mixed number or a whole number. 1510 means 15 ÷ 10.
   1510 = 1510
5. Simplify the fraction part. 5 and 10 can both be divided by 5.
   510 = 5 ÷ 510 ÷ 5 = 12
   So the answer is 112.
6. Write the answer.
   710 + 45 = 112

Check. Take away: 112 − 45 = 710. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

1. Check the denominators. To subtract fractions, the parts must be the same size.
   Both denominators are 4, so we can subtract straight away.
2. The first fraction part (1) is smaller than the second (3), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 4 quarters, so the numbers become fractions over 4.
   314 = 3 × 4 + 14 = 134
   134 = 1 × 4 + 34 = 74
3. Subtract the numerators. Now both numbers are fractions with the same denominator.
   134 − 74 = 13 − 74 = 64
4. The answer is 1 or more, so change it to a mixed number or a whole number. 64 means 6 ÷ 4.
   64 = 124
5. Simplify the fraction part. 2 and 4 can both be divided by 2.
   24 = 2 ÷ 24 ÷ 2 = 12
   So the answer is 112.
6. Write the answer.
   314 − 134 = 112

Check. Add back: 112 + 134 = 314. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

1. Think of 5 × 23 as 5 lots of 23. Multiplying by a whole number is repeated addition.
   23 + 23 + 23 + 23 + 23
2. Multiply the numerator by 5. Keep the denominator. We are counting thirds. 5 lots of 2 thirds is 10 thirds.
   5 × 23 = 5 × 23 = 103
3. The answer is 1 or more, so change it to a mixed number or a whole number. 103 means 10 ÷ 3.
   103 = 313
4. Write the answer.
   5 × 23 = 313

Check. Divide back: 313 ÷ 5 = 23. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. To find a fraction of an amount, divide by the denominator to find one part. Then multiply by the numerator.

1. Divide the amount by the denominator. The denominator 3 tells us to cut the amount into 3 equal parts. One part is 13 of the amount.
   90 minutes ÷ 3 = 30 minutes
2. Multiply by the numerator. The numerator 2 tells us how many of those parts we want.
   2 × 30 minutes = 60 minutes
3. Write the answer.
   23 of 90 minutes = 60 minutes

Check. Other way: 2 × 90 minutes = 180 minutes, then ÷ 3 = 60 minutes. ✓

Watch out. Divide by the bottom number first. Then multiply by the top number.

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Check each option. An equivalent fraction simplifies to the same fraction as 35.

1. Write 35 in its simplest form. It is already as small as it can be.
   35
2. Simplify each option. Equivalent fractions have the same simplest form.
   A: 610 = 35 → equal
   B: 58 (already simplest) → not equal
   C: 310 (already simplest) → not equal
   D: 65 (already simplest) → not equal
3. Pick the equal one. Only one option is equivalent.
   A (610)

Check. 35 = 610. ✓

Watch out. Check the top and the bottom change by the same number.

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

1. Look at the bottom numbers. What do we multiply 6 by to get 18? This tells us how the fraction has been changed. We must do the same to the other number.
   6 × 3 = 18, so the multiplier is 3.
2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be multiplied by 3.
   5 × 3 = 15
3. Write the equivalent fractions. Now both numbers are filled in.
   56 = 1518

Check. 56 and 1518 both simplify to 56. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

1. Divide 17 by 5. The fraction line means divide. The answer tells us how many whole ones we can make.
   17 ÷ 5 = 3 remainder 2
2. The whole number is 3. 3 lots of 5 fifths make 3 whole ones: 3 × 5 = 15.
   17 − 15 = 2 fifths are left over.
3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 5.
   3 and 25 = 325

Check. Change back: 3 × 5 + 2 = 17 = 17. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Key words. To order fractions is to put them in a line from the smallest to the largest (ascending) or from the largest to the smallest (descending). Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change every fraction to the same denominator. Then order the numerators.

1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 16, so 16 will do.
   16 ÷ 2 = 8, so 2 goes into 16
   16 ÷ 4 = 4, so 4 goes into 16
   16 ÷ 8 = 2, so 8 goes into 16
2. Change every fraction to the common denominator. Multiply the top and the bottom by the same number each time.
   58 = 5 × 28 × 2 = 1016
   34 = 3 × 44 × 4 = 1216
   12 = 1 × 82 × 8 = 816
   916 (already in sixteenths)
3. Put the top numbers in order. With the same bottom number, the bigger the top number, the bigger the fraction. We want smallest to largest.
   8, 9, 10, 12 (out of 16)
4. Write the original fractions in that order. The answer uses the fractions from the question, not the changed ones.
   12,  916,  58,  34

Check. Read the numerators again: 8, 9, 10, 12 go up each time. ✓

Watch out. Write down the original fractions in your final answer.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Add the whole numbers, then add the numerators. Change the answer to a mixed number and simplify if you can.

1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 6, so 6 will do.
   6 ÷ 3 = 2, so 3 goes into 6
2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
   113 → 13 = 1 × 23 × 2 = 26
3. Add the whole numbers. Add the wholes first, then deal with the fractions.
   3 + 1 = 4
4. Add the fractions. Add the numerators. The denominator stays the same because the parts are the same size.
   56 + 26 = 5 + 26 = 76
5. The fraction part is 1 or more, so change it to a whole number and a fraction. 76 means 7 ÷ 6.
   76 = 116
   4 + 116 = 516
6. Write the answer.
   356 + 113 = 516

Check. Take away: 516 − 113 = 356. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

1. Check the denominators. To subtract fractions, the parts must be the same size.
   Both denominators are 6, so we can subtract straight away.
2. The first fraction part (1) is smaller than the second (5), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 6 sixths, so the numbers become fractions over 6.
   416 = 4 × 6 + 16 = 256
   256 = 2 × 6 + 56 = 176
3. Subtract the numerators. Now both numbers are fractions with the same denominator.
   256 − 176 = 25 − 176 = 86
4. The answer is 1 or more, so change it to a mixed number or a whole number. 86 means 8 ÷ 6.
   86 = 126
5. Simplify the fraction part. 2 and 6 can both be divided by 2.
   26 = 2 ÷ 26 ÷ 2 = 13
   So the answer is 113.
6. Write the answer.
   416 − 256 = 113

Check. Add back: 113 + 256 = 416. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. To compare fractions that have different denominators, change them to the same denominator. Then compare the numerators.

1. Find a common denominator. We need a number that all the denominators go into. List the multiples of each one until you see a number in every list.
   Multiples of 5: 5, 10, 15, 20, 25, 30, …, 55, 60
   Multiples of 12: 12, 24, 36, 48, 60
   The first number in every list is 60.
2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time.
   712 = 7 × 512 × 5 = 3560
   35 = 3 × 125 × 12 = 3660
3. Compare the top numbers. With the same denominator, the bigger numerator is the bigger fraction.
   35 < 36, so 3560 < 3660
4. Write a sentence to prove it. In a “prove it” question the last line must say what you have shown.
   No. 712 = 3560 and 35 = 3660, and 35 is less than 36. So 712 is less than 35.

Check. The bars: the 35 bar is longer than the 712 bar. ✓

Watch out. Do not guess from the numbers. Show the common denominator and the two new fractions.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

1. Think of 4 × 38 as 4 lots of 38. Multiplying by a whole number is repeated addition.
   38 + 38 + 38 + 38
2. Multiply the numerator by 4. Keep the denominator. We are counting eighths. 4 lots of 3 eighths is 12 eighths.
   4 × 38 = 4 × 38 = 128
3. The answer is 1 or more, so change it to a mixed number or a whole number. 128 means 12 ÷ 8.
   128 = 148
4. Simplify the fraction part. 4 and 8 can both be divided by 4.
   48 = 4 ÷ 48 ÷ 4 = 12
   So the answer is 112.
5. Write the answer.
   4 × 38 = 112

Check. Divide back: 112 ÷ 4 = 38. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. Find how many stickers each child gets. Add them and take the total away from 72.

1. Find the stickers Leo gets. 14 of 72.
   72 ÷ 4 = 18
2. Find the stickers Mei gets. 13 of 72.
   72 ÷ 3 = 24
3. Add the stickers given away. Both amounts leave the pile.
   18 + 24 = 42
4. Take them from 72. The rest are left.
   72 − 42 = 30

Check. 14 + 13 = 712, so 512 is left, and 512 of 72 = 30. ✓

Watch out. Marks: 1 for each fraction of 72, 1 for the answer.

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. Find the fraction she has read, then the fraction left. The pages left tell us the value of that fraction.

1. Add the fractions she has read. Change to eighths first.
   14 + 38 = 28 + 38 = 58
2. Find the fraction left. Take it away from 1 whole book.
   1 − 58 = 88 − 58 = 38
3. Find one eighth of the book. 38 of the book is 21 pages, so divide by 3.
   21 ÷ 3 = 7 pages
4. Find the whole book. There are 8 eighths.
   8 × 7 = 56 pages

Check. 58 of 56 = 35 and 35 + 21 = 56. ✓

Watch out. Marks: 1 for the fraction left, 1 for one eighth, 1 for the answer.