What Is a Fraction?

Worked example

1. 1

   $\frac{1}{2}>\frac{1}{3}$ — Cut a whole into 2 and each piece is half of it. Cut it into 3 and each piece is smaller, because the same whole is shared more ways.

2. 2

   $\frac{1}{4}>\frac{1}{6}$ — Same idea: 6 equal pieces are smaller than 4 equal pieces of the same whole. Bigger bottom, smaller piece.

3. 3

   $\frac{4}{4}=1$ — The top equals the bottom: you take all 4 of the 4 pieces, so you have the whole thing.

4. 4

   $\frac{5}{4}=\frac{4}{4}+\frac{1}{4}$ — The top is bigger than the bottom. Split it: 4 quarters make one whole, and 1 quarter is left over.

5. 5

   $\frac{5}{4}=1\frac{1}{4}$ — Swap $\frac{4}{4}$ for 1. So $\frac{5}{4}$ is more than one whole: 1 whole and 1 quarter.

Worked example

1. 1

   $0\text{to}1:5\text{equal gaps}$ — Split the space from 0 to 1 into 5 equal gaps, because the bottom number is 5. Each gap is one fifth.

2. 2

   $\text{count }3\text{ gaps from }0$ — Start at 0 and move 3 gaps, because the top number is 3. Mark the point there.

3. 3

   $3>2.5,\text{so}\frac{3}{5}>\frac{1}{2}$ — Check it: halfway is 2.5 of the 5 gaps. 3 gaps is a little more, so the mark should sit just right of the middle.

4. 4

   $\frac{5}{5}=1$ — After all 5 gaps you reach 1, because five fifths make one whole.

5. 5

   $\frac{7}{5}=1\frac{2}{5}$ — Keep counting past 1: 7 gaps is 5 gaps to reach 1, then 2 more. So $\frac{7}{5}$ sits at $1\frac{2}{5}$.

Worked example

1. 1

   $\text{whole}=12$ — Count every square, shaded or not. That total is the whole, so it goes on the bottom.

2. 2

   $\text{part}=4$ — Count only the shaded squares. That is the part, so it goes on top.

3. 3

   $\frac{4}{12}$ — Part over whole.

4. 4

   $\frac{4÷4}{12÷4}=\frac{1}{3}$ — Divide top and bottom by 4, the biggest number that goes into both. Exams usually want the simplest form.

Worked example

1. 1

   $2\text{hours}=120\text{minutes}$ — Put both amounts in the same unit first. 1 hour is 60 minutes, so 2 × 60 = 120.

2. 2

   $\frac{50}{120}$ — The amount after “of” is the whole, so 120 goes on the bottom. The part, 50, goes on top.

3. 3

   $\frac{50÷10}{120÷10}=\frac{5}{12}$ — Divide top and bottom by 10. 5 and 12 share no other factor, so this is the simplest form.