Between any two numbers there are infinitely many numbers. This is because the number system is continuous so it has no gaps between numbers. This also applies to fractions, we can find a number between any two fractions no matter how close the fractions seem to be. Let’s look at how we can find a number between two fractions.
Worked example 1
Find a fraction between $\frac{2}{5}$25 and $\frac{4}{5}$45.
Think: Since the fractions have the same denominator, we can use a fraction with the same denominator but a numerator between $2$2 and $4$4, which is $3$3.
Do: So a fraction between $\frac{2}{5}$25 and $\frac{4}{5}$45 is $\frac{3}{5}$35.
Reflect: We can picture this on a number line as finding a missing number between between $\frac{2}{5}$25 and $\frac{4}{5}$45 as:
Worked example 2
Find a fraction between $\frac{1}{4}$14 and $\frac{2}{4}$24.
Think: The fractions have the same denominator, so we could try to use a fraction with the same denominator with a numerator between $1$1 and $2$2, but there is no integer between these numbers. So, we can multiply the numerator and denominator by a whole number to make the numerators and denominators larger, and therefore the gap between the numerators larger.
Do: Let's try multiplying the numerators and denominators by $2$2 since it is a small number:
| $\frac{1}{4}\times\frac{2}{2}$14×22 | $=$= | $\frac{2}{8}$28 |
| $\frac{2}{4}\times\frac{2}{2}$24×22 | $=$= | $\frac{4}{8}$48 |
Now we are trying to find a fraction between $\frac{2}{8}$28 and $\frac{4}{8}$48 which could be $\frac{3}{8}$38.
Worked example 3
Find a fraction between $\frac{2}{5}$25 and $\frac{2}{6}$26.
Think: To easily compare, we need to convert these two fractions to equivalent fractions with the same denominator.
Do: We can do this by multiplying the numerator and denominator of each fraction by the other fraction's denominator:
| $\frac{2}{5}\times\frac{6}{6}$25×66 | $=$= | $\frac{12}{30}$1230 |
| $\frac{2}{6}\times\frac{5}{5}$26×55 | $=$= | $\frac{10}{30}$1030 |
Now we are trying to find a fraction between $\frac{10}{30}$1030 and $\frac{12}{30}$1230 which could be $\frac{11}{30}$1130.
Reflect: You can find other fractions that are in between the two original fractions by making the denominators even larger.
Worked example 4
Find the fraction exactly half-way between $\frac{1}{4}$14 and $\frac{7}{12}$712.
Think: To find a fraction that is exactly half-way between two fractions, we need to add them and divide them by two. To do this we must first convert these two fractions to equivalent fractions with the same denominator.
Do: For this example we can just multiply the numerator and denominator of the first fraction by three so that both fractions have a denominator of twelve:
| $\frac{1}{4}\times\frac{3}{3}$14×33 | $=$= | $\frac{3}{12}$312 |
Now we must add the fractions:
| $\frac{3}{12}+\frac{7}{12}$312+712 | $=$= | $\frac{10}{12}$1012 |
Then divide the sum by two:
| $\frac{10}{12}\div2$1012÷2 | $=$= | $\frac{10}{12}\times\frac{1}{2}$1012×12 |
| $=$= | $\frac{5}{6}\times\frac{1}{2}$56×12 | |
| $=$= | $\frac{5}{12}$512 |
So the fraction half-way between $\frac{1}{4}$14 and $\frac{7}{12}$712 is $\frac{5}{12}$512.