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This page sets out the method for writing, simplifying & ordering fractions in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Writing, simplifying & ordering fractions topic page.
When you see 'simplify' or 'write in its simplest form', divide top and bottom by the same number until you cannot go further.
When you see 'put these fractions in order', find a denominator every fraction can be changed to.
A fraction is part of one whole that has been split equally.
The pieces have to be the same size, not just the same number of pieces.
The denominator is how many equal parts the whole was split into.
The numerator is how many of those parts you have.
Four quarters, three thirds and two halves are all one whole.
If the top and bottom match, the fraction is 1.
Divide the top and the bottom by the same number.
6/8 becomes 3/4 because both were divided by 2.
A unit fraction has 1 on top. 1/5 of 40 is 40 ÷ 5.
Once you know one part, you can find two or three parts by multiplying.
Compare by giving the fractions the same denominator, or the same whole.
3/4 of a bar is more than 1/2 of the same bar.
Multiplying the top and the bottom by the same number cuts each part into smaller equal parts and keeps the same amount shaded. 3/4 multiplied top and bottom by 2 is 6/8. Multiplied by 3 it is 9/12. The value has not changed. You can check by dividing: 3 ÷ 4 = 0.75 and 9 ÷ 12 = 0.75.
Dividing top and bottom by the same number runs the picture backwards. 9/12 divided by 3 on the top and the bottom returns 3/4. Whatever you do to the top, do to the bottom. Dividing only the top, or only the bottom, makes a different fraction.
Use an equivalent fraction when a question asks you to fill a missing number. 3/5 = (?)/20. The bottom has been multiplied by 4, because 5 × 4 = 20, so the top must be multiplied by 4 as well: 3 × 4 = 12. The missing numerator is 12.
Method: 1. Multiply the numerator and the denominator by the same whole number. 2. The shaded amount stays the same, so the fractions are equal. 3. Divide top and bottom by the same number to run the step backwards.
A fraction is in its simplest form when the top and the bottom have no common factor except 1. Find a common factor, divide both by it, and look again. 18/24 divides by 2 to give 9/12, and 9/12 still divides by 3 to give 3/4. You can divide by 6 in one step, because 6 is the highest common factor of 18 and 24. Either route is fine if you finish.
9/12 is equivalent to 3/4, but it is not fully simplified. A question that says 'simplify fully' or 'write in its simplest form' wants the last step. Leaving a common factor of 3 on the top and the bottom drops the mark for the finished answer. Check the end: 3 and 4 share no factor except 1.
42/56 divides by 2 to give 21/28, then by 7 to give 3/4. Or divide once by 14: 42 ÷ 14 = 3 and 56 ÷ 14 = 4. If a prime still divides both the top and the bottom, you have not finished.
Method: 1. Find a common factor of the numerator and the denominator. 2. Divide both by that factor. 3. Stop when the only common factor left is 1.
A mixed number is wholes plus a proper fraction. 23/8 means 2 + 3/8. Each whole is 8/8, so two wholes are 16/8. Add the extra 3/8 to get 19/8. The short rule is: multiply the whole number by the denominator, add the numerator, and keep the denominator. 2 × 8 + 3 = 19, so the improper fraction is 19/8.
Do not glue the digits. 23/8 is not 23/8 and it is not 5/8. The 2 is two whole groups of 8 eighths, which is 16, not a digit written beside the 3. Going back the other way, 19/8 is 2 wholes because 8 × 2 = 16, with 19 - 16 = 3 eighths left, so 23/8.
Use the improper form when you compare eighths. Use the mixed form when the question asks how many whole items are left. 23/8 is two wholes and three eighths. Both writings are the same amount.
Method: 1. A mixed number is whole ones plus a proper fraction. 2. Turn the wholes into the denominator, then add the extra numerator. 3. To go back, split the improper fraction into wholes and a remainder.
Fractions can be ordered only when the pieces are the same size. Give them a common denominator, then compare the numerators. For 3/5, 5/8 and 7/10, a denominator that works is 40, because 5 × 8 = 40, 8 × 5 = 40 and 10 × 4 = 40. You do not have to use the lowest one, but every denominator must divide the one you choose.
Convert each fraction. 3/5 = 24/40, because the bottom was multiplied by 8 and the top must be too. 5/8 = 25/40. 7/10 = 28/40. The numerators in order are 24, 25, 28, so the fractions in order, smallest first, are 3/5, 5/8, 7/10. Write the original fractions in the answer, not the fortieths, unless the question used fortieths.
A larger denominator does not mean a larger fraction. 1/8 is smaller than 1/2, because the whole has been split into more pieces. That shortcut works for unit fractions only. Once the numerators differ, use the common denominator. 5/6 and 7/8 look close: 5/6 = 20/24 and 7/8 = 21/24, so 7/8 is larger by one twenty-fourth.
Method: 1. Give the fractions a common denominator. 2. Compare the numerators once the pieces are the same size. 3. Write the original fractions in that order.
Exam questions often ask you to write one amount as a fraction of another, such as "Write 45 cm as a fraction of 2 m". The first job is to put both amounts in the same unit. Here 2 m = 200 cm, so the fraction is 45/200. The amount you are asked about goes on the top and the amount it is compared with goes on the bottom.
Only then simplify. Divide the top and the bottom by their highest common factor, which for 45 and 200 is 5, giving 9/40. If the question says "in its simplest form", the unsimplified fraction earns the method mark but not the final accuracy mark. Watch the word "of": "Tom as a fraction of Ana" means dfracTomAna, not the other way round.
Method: 1. Underline the two amounts and decide which one is the top and which is the bottom. 2. Write both amounts in the same unit, usually the smaller one. 3. Write the fraction with the top amount over the bottom amount. 4. Divide top and bottom by a common factor until no more division is possible.
Tip: A fraction of two amounts in different units, such as 45 over 2, is an easy way to lose the marks.
Many questions give a table or a short story: "Amir played 24 games. He won 15, drew 3 and lost the rest. What fraction of the games did he lose?" Work out any missing group first, here 24 - 15 - 3 = 6 losses. The bottom of the fraction is always the total, so the fraction is 6/24, which simplifies to 1/4.
Check that your parts add up to the total before you build the fraction. If the fractions for every group are written over the same total, their tops must add up to the total as well. That gives a quick check: 15/24 + 3/24 + 6/24 = 24/24, which is a whole. A question may ask for the fraction of one group, or for the fraction that is not in a group, so read the wording twice.
Method: 1. Find the total number, and work out any group that is missing. 2. Put the group you want over the total. 3. Simplify the fraction fully. 4. Check that all the group fractions together make one whole.
To order numbers that include mixed numbers, compare the whole-number parts first. 31/2 is bigger than 27/8 because 3 is bigger than 2, whatever the fractions are. Only when the wholes are equal do you compare the fractions, using a common denominator. If one number is an improper fraction such as 7/3, turn it into a mixed number first: 7 ÷ 3 = 2 remainder 1, so it is 21/3.
Alternatively, change every number into an improper fraction over a common denominator and compare the tops. Both routes are correct, but pick one and stay with it. Write each converted value beside the original, because the question wants the original numbers in the answer, not your converted versions. A short list of the form 8/24, 9/24, 12/24 is easy to rank and is worth a method mark.
Method: 1. Turn every improper fraction into a mixed number. 2. Compare the whole numbers, then compare the fractions that are left. 3. Use a common denominator for the fractions, such as 24 for halves, thirds and eighths. 4. Write the final order using the original numbers from the question.
Tip: The common denominator does not have to be the smallest. Any shared multiple works, and a larger one is fine.
Some questions ask "Is she right? You must show your working" or "Show that 5/8 is less than 2/3". There is no shortcut that earns the marks: give both fractions the same denominator, write the two equivalent fractions, and then state which is bigger. Here the common denominator is 24, so 5/8 = 15/24 and 2/3 = 16/24.
Finish with a sentence that answers the question: "15/24 is less than 16/24, so 5/8 is less than 2/3". A comparison with 1/2 can be a quick check. If one fraction is under a half and the other is over a half, you know the order without any more work. For example 3/7 is under a half and 4/7 is over a half.
Method: 1. Find a common denominator for the two fractions. 2. Write each as an equivalent fraction over that denominator. 3. Compare the tops and say which fraction is larger. 4. Write a one-line conclusion that answers the question that was asked.
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