Find 5 rational numbers between 1/3 and 1/2

Question

Find 5 rational numbers between $\dfrac{1}{3}$ and $\dfrac{1}{2}$ .

Solution — Step by Step

To find rational numbers between two fractions, we need both fractions to have the same denominator — or at least a common denominator large enough to give us at least 5 integers between them. There are two methods: the LCM method and the multiplication method. We’ll use the multiplication method first.

We need at least 6 integers between the numerators (to pick 5 from). The easiest way: multiply both fractions by $\dfrac{(n+1)}{(n+1)}$ where $n = 5$ .

Multiply both fractions by $\dfrac{6}{6}$ :

\frac{1}{3} = \frac{1 \times 6}{3 \times 6} = \frac{6}{18}

\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}

That didn’t unify denominators — let’s take the LCM approach.

LCM of 3 and 2 is 6.

\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}

Between $\dfrac{2}{6}$ and $\dfrac{3}{6}$ , there’s only one integer numerator (none between 2 and 3). So we need to scale up further. Multiply both by $\dfrac{7}{7}$ (or any number ≥ 6):

\frac{2}{6} = \frac{14}{42}, \quad \frac{3}{6} = \frac{21}{42}

Now the integers between 14 and 21 are: 15, 16, 17, 18, 19, 20. That gives us 6 choices — more than enough.

Any 5 fractions with denominator 42 and numerators between 14 and 21:

\frac{15}{42},\ \frac{16}{42},\ \frac{17}{42},\ \frac{18}{42},\ \frac{19}{42}

These can also be simplified where possible. For example, $\dfrac{18}{42} = \dfrac{3}{7}$ .

Check: $\dfrac{1}{3} = \dfrac{14}{42}$ and $\dfrac{1}{2} = \dfrac{21}{42}$ .

Since $14 < 15 < 16 < 17 < 18 < 19 < 21$ , all five fractions lie strictly between $\dfrac{1}{3}$ and $\dfrac{1}{2}$ .

Answer: $\dfrac{15}{42},\ \dfrac{16}{42},\ \dfrac{17}{42},\ \dfrac{18}{42},\ \dfrac{19}{42}$ (or their simplified equivalents)

Why This Works

Rational numbers are dense — between any two distinct rational numbers, there are infinitely many more. This means no matter how close two fractions are, we can always find more by scaling up the denominators. There is no “next” rational number after any given one.

The scaling trick works because multiplying numerator and denominator by the same number doesn’t change the fraction’s value — it just reveals more integer positions on the number line between the two fractions.

Alternative Method

Use the mean method (or averaging method) repeatedly:

Mean of $\dfrac{1}{3}$ and $\dfrac{1}{2}$ : $\dfrac{\frac{1}{3} + \frac{1}{2}}{2} = \dfrac{\frac{5}{6}}{2} = \dfrac{5}{12}$

Now find the mean between $\dfrac{1}{3}$ and $\dfrac{5}{12}$ , and between $\dfrac{5}{12}$ and $\dfrac{1}{2}$ , and so on. This generates rational numbers in between, though the computation gets more tedious.

For board exams, the LCM + scaling method is faster and easier to present. Write the 5 fractions clearly and verify that all lie between the original two — examiners specifically check this.

Common Mistake

A very common error is to stop at LCM = 6 and write $\dfrac{2}{6}$ and $\dfrac{3}{6}$ , then claim there are no rationals between them. There are infinitely many — you just need to scale both fractions further. Never assume adjacent-looking fractions have no numbers between them.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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