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Fractions: what a fraction is, and where to start

A fraction is a part of a whole that was cut into equal pieces. The bottom number says how many pieces the whole became, and the top number says how many of them you hold.

Fractions look like two numbers stacked on top of each other, and most of the trouble starts when you read them that way. They are one number: some of a thing that was cut into equal pieces. Below you get the picture that makes it click, the unit fraction every other fraction is built from, where fractions sit on a number line, and the slip that costs marks in tests. More practice is in the numerical practice hub.

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What is a fraction? One number, read as a count of equal pieces

A fraction describes some of a whole that was cut into equal pieces. The bottom number, the denominator, says how many pieces the whole became. The top number, the numerator, says how many of them you hold. Pieces of different sizes do not make a fraction, however neat the drawing looks.

Formula

Fraction = (Top) divided by (Bottom)

  • Top is how many pieces you hold.
  • Bottom is how many equal pieces make the whole: a team, a budget, a shift.
  • A fraction is also a division left undone: the top divided by the bottom.

Worked example

A sales team has 8 people, and 3 of them hit quota in September. What fraction of the team hit quota, and what fraction missed?

Writing 3/5, the hitters over the misses, is the usual slip. The bottom number is always the whole team, never the other group.

  1. Missed quota: 8 − 3 = 5 people
  2. Check the pieces: 3 + 5 = 8, so the two fractions fill the whole team
  3. Read it as a division: 3 ÷ 8 = 0.375, a little under half

3/8 of the team hit quota and 5/8 missed.

Check it: the two fractions have the same bottom number, and their tops add back up to the whole.

Unit fraction: why is 1/8 smaller than 1/5?

A unit fraction has one on top: a single piece of a whole cut into equal parts. Every other fraction is a unit fraction counted several times. The more pieces the whole is cut into, the smaller each piece is, so a bigger bottom number means a smaller fraction.

Formulas

Unit = (1) divided by (Bottom)

Fraction = Top × Unit

  • Unit is the unit fraction, one piece. Top is how many pieces you hold.

Worked example

A bonus pot of 240 euros is shared equally. Team A cuts it into 5 shares and team B into 8. Whose single share is bigger, and by how much?

Reading 8 as bigger than 5, so 1/8 as bigger than 1/5, treats the bottom number as the size of the piece. It counts the pieces, and more pieces means thinner ones.

  1. Team A: 240 ÷ 5 = 48 euros
  2. Team B: 240 ÷ 8 = 30 euros
  3. The gap: 48 − 30 = 18 euros

Team A's share is 48 euros and team B's is 30 euros, so A's is bigger by 18 euros.

Check it: 5 shares of 48 and 8 shares of 30 both rebuild the 240.

Fractions on a number line: where does 3/8 sit?

Draw a line from zero to one, and cut it into as many equal steps as the bottom number says. Then walk as many steps as the top number says. Where you stop is the fraction, and you can see whether it is more or less than a half before you calculate anything.

  • The size of one step. Eighths cut the line into 8 equal steps, so one step is 1 ÷ 8 = 0.125 long.
  • Walk the top number. 3/8 is 3 steps: 3 × 0.125 = 0.375, a little short of the middle.
  • The halfway test. 4 steps of eighths land exactly on the middle: 4 × 0.125 = 0.5. Fewer steps than that is under a half, and more is over.
  • A top bigger than the bottom. 10/8 walks off the end of the line: 10 × 0.125 = 1.25, which is more than one whole.

How to work out fractions: name the whole, then shrink the numbers

Start every fraction question by saying what the whole is and how many pieces it has. Write the fraction from those two counts, then look for a whole number that goes into both the top and the bottom. Dividing both by it gives a smaller fraction of the same size.

Formula

New = (Top ÷ n) divided by (Bottom ÷ n)

  • n is any whole number that goes into both the top and the bottom.

Worked example

A screening table shows 45 of 60 applicants passed. Write that as a fraction with the smallest numbers you can.

Dividing only the top by 15 gives 3/60, a far smaller fraction. The same number has to act on both, or the size changes.

  1. Divide the top: 45 ÷ 15 = 3
  2. Divide the bottom: 60 ÷ 15 = 4
  3. Check it as a division: 3 ÷ 4 = 0.75

3/4 of the applicants passed.

Check it: 45 ÷ 60 is also 0.75, so the new fraction is the same size as the old one.

Which job is the question asking you to do?

Most fraction questions in a test ask for one of a few jobs, and naming the job is the first move. Full lessons on each job are still to come; here is how to tell them apart.

  • Same size, smaller numbers. Cutting every piece in two doubles both numbers: 1/2 is 2/4, and 2 × 2 = 4.
  • Putting pieces together. Pieces add only when they are the same size: 3 eighths and 2 eighths make 3 + 2 = 5 eighths.
  • A share of an amount. The bottom number divides and the top number counts: a fifth of 240 is 240 ÷ 5 = 48.
  • A fraction of a fraction. Half of three quarters is 3/8, since 2 × 4 = 8 pieces to the whole and 1 × 3 = 3 of them held.
  • A decimal or a percentage. 3/8 is 3 ÷ 8 = 0.375, which is 0.375 × 100 = 37.5 percent.

Fraction questions in a test: which whole is each fraction of?

In a timed table the sum is rarely the hard part. Marks go when a second fraction is taken of the original whole and not of the smaller group the first one left behind.

Each fraction acts on the amount it names, and that amount changes as you go. Find the first part, then treat that part as the new whole for the next fraction.

Formulas

Piece = (Whole) divided by (Bottom)

Part = Piece × Top

  • Whole is the amount this fraction was taken from, and it can change from one step to the next.

Worked example

Of 80 candidates, 3/4 passed the numerical test. Of those who passed, 1/3 also passed the verbal test. How many passed both?

Taking a third of all 80 gives 26.67, which is not a whole person. The second fraction belongs to the 60 who passed, not to everyone.

  1. One quarter of 80: 80 ÷ 4 = 20
  2. Three quarters: 20 × 3 = 60 passed the numerical test
  3. A third of those 60: 60 ÷ 3 = 20

20 candidates passed both tests.

Check it: 20 of 80 is one quarter, and a quarter is what you get by taking a third of three quarters.

Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.

Start numerical practice

Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.

Start numerical practice

How we write, check and mark the practice questions is set out on the Methodology page.

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