MathsICSE Class 8 · Rational Numbers

Finding Rational Numbers Between Two Rational Numbers (ICSE Class 8 Maths)

Q: Why is there no "next" rational number after 1/2?

Whatever number you propose, the mean of it and 1/2 is closer, so no smallest gap exists.

Q: Can an answer be an integer?

Yes. Integers are rational, e.g. 0 lies between -\frac{1}{2} and \frac{1}{3}.

Updated 17 October 2026 · Fact-checked

Between any two rational numbers there are infinitely many more. To find one, take their mean $\frac{a + b}{2}$. To find $n$ of them, write both with a common denominator, multiply top and bottom by $(n + 1)$ or more, and list numerators strictly between the two new numerators.

Understand Rational Numbers Between Two Rational Numbers

The CISCE curriculum states that between any two rational numbers there lies another rational number. Repeating the idea shows there are infinitely many, so unlike whole numbers, rational numbers have no "next" neighbour.

Method 1: the mean. For $a < b$ , the number $\frac{a + b}{2}$ lies exactly halfway, so $a < \frac{a + b}{2} < b$ . Apply it again to the new number and either end to get more.

Method 2: common denominator. Write $a$ and $b$ over the same positive denominator. If the numerators are far enough apart, the numerators in between give the answers. If they are adjacent or too close, multiply both fractions top and bottom by a bigger number to make room. To be sure of $n$ numbers, multiply by at least $n + 1$ .

Ordering. To compare or arrange rational numbers, first make all denominators positive, then use the LCM as the common denominator and compare numerators. For negatives, the bigger numerator (closer to $0$ ) is the greater number: $- \frac{27}{36} > - \frac{30}{36}$ .

Number line. On the number line, smaller numbers are to the left. The point at the middle of two points $P$ and $Q$ is their mean. A point one-third of the way from $P$ to $Q$ is $P + \frac{Q - P}{3}$ .

Rational Numbers Between Two Rational Numbers: formulas to remember

Mean of two rationals: $\frac{a + b}{2}$; Lies exactly halfway between $a$ and $b$ .
Making room for n numbers: $multiply numerator and denominator by (n + 1)$; Guarantees at least $n$ numerators in between.
Point one-third of the way: $P + \frac{Q - P}{3}$; Use $\frac{2}{3}$ of the gap for two-thirds.

How to solve Rational Numbers Between Two Rational Numbers questions

Choose the mean for one number and the common-denominator method when several are needed.

1Make both denominators positive.
2Find the LCM of the denominators and rewrite both fractions.
3If you need $n$ numbers, multiply numerator and denominator of both by $(n + 1)$ if there is not already room.
4List numerators strictly between the new numerators.
5Reduce the answers and verify the order by converting to decimals.

Rational Numbers Between Two Rational Numbers in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Marks on a ruler

Suppose a ruler has marks only at $\frac{1}{4}$ cm and $\frac{1}{3}$ cm, and you want to add a finer mark between them.

Find a position for the new mark.

Try it first, then show the solutionSolution

Mean $= \frac{1}{2} (\frac{3}{12} + \frac{4}{12}) = \frac{7}{24}$ .
Decimal check: 0.25 < 0.2917 < 0.3333.

Answer: $\frac{7}{24}$ cm.

Memory hook: No gap is ever too small to split.

Tricks to remember Rational Numbers Between Two Rational Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

Halfway rule
Add the two numbers and halve the sum to land exactly in the middle.
Use it when: When only one number is asked.
Make room first
If numerators are neighbours, scale up both fractions; scaling does not change their value.
Use it when: When many numbers are asked.

Rational Numbers Between Two Rational Numbers: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Pick the number in between · 1 mark · MCQBoard level

Which number lies between $\frac{1}{3}$ and $\frac{1}{2}$ ?
(a) $\frac{2}{5}$ (b) $\frac{3}{5}$ (c) $\frac{1}{6}$ (d) $\frac{5}{6}$

Try it first, then show the solutionSolution

$\frac{1}{3} \approx 0.333$ , $\frac{1}{2} = 0.5$ ; $\frac{2}{5} = 0.4$ lies between. The others are $0.6, 0.167, 0.833$ .

Answer: (a)

Example 2: One number by the mean · 2 marks · Short answerBoard level

Find a rational number between $- \frac{2}{3}$ and $\frac{3}{4}$ .

Try it first, then show the solutionSolution

Common denominator $12$ : $- \frac{8}{12}$ and $\frac{9}{12}$ .
Mean $= \frac{1}{2} \times \frac{- 8 + 9}{12} = \frac{1}{24}$ .

Answer: $\frac{1}{24}$ (check: $- 0.667 < 0.0417 < 0.75$ ).

Example 3: Four numbers between two fractions · 3 marks · Short answerHarder board level

Find four rational numbers between $\frac{3}{5}$ and $\frac{5}{6}$ .

Try it first, then show the solutionSolution

LCM of $5, 6$ is $30$ : $\frac{3}{5} = \frac{18}{30}$ , $\frac{5}{6} = \frac{25}{30}$ .
Numerators from $19$ to $24$ lie between; pick four: $\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}$ .
Reduced: $\frac{19}{30}, \frac{2}{3}, \frac{7}{10}, \frac{11}{15}$ .

Answer: $\frac{19}{30}, \frac{2}{3}, \frac{7}{10}, \frac{11}{15}$ (many answers possible).

Example 4: Making room · 4 marks · Short answerHarder board level

Insert three rational numbers between $- \frac{1}{2}$ and $\frac{1}{3}$ .

Try it first, then show the solutionSolution

Common denominator $6$ : $- \frac{3}{6}$ and $\frac{2}{6}$ . Numerators $- 3$ and $2$ leave $- 2, - 1, 0, 1$ , so there is room.
Choose $- \frac{2}{6}, 0, \frac{1}{6}$ , i.e. $- \frac{1}{3}, 0, \frac{1}{6}$ .
Check: $- 0.5 < - 0.333 < 0 < 0.167 < 0.333$ .

Answer: $- \frac{1}{3}, 0, \frac{1}{6}$ (any three valid numbers).

Example 5: Points on the number line · 4 marks · Long answerHOTS / competency

On a number line $P = - \frac{3}{4}$ and $Q = \frac{5}{8}$ . Find the rational number at (i) the midpoint of $P Q$ , (ii) the point one-third of the way from $P$ to $Q$ .

Try it first, then show the solutionSolution

$Q - P = \frac{5}{8} + \frac{6}{8} = \frac{11}{8}$ .
(i) Midpoint $= \frac{- 6/8 + 5/8}{2} = - \frac{1}{16}$ .
(ii) $P + \frac{1}{3} \times \frac{11}{8} = - \frac{18}{24} + \frac{11}{24} = - \frac{7}{24}$ .
Check: $- 0.75 < - 0.292 < - 0.0625 < 0.625$ .

Answer: (i) $- \frac{1}{16}$ ; (ii) $- \frac{7}{24}$ .

Where students lose marks in Rational Numbers Between Two Rational Numbers

The slips examiners see most often, and what to write instead.

Mistake: Saying no rational number lies between neighbouring fractions
Neighbouring numerators only show that you must scale up.
Wrong working
"Between $\frac{3}{7}$ and $\frac{5}{7}$ there is only $\frac{4}{7}$ ."
Right working
$\frac{6}{14}$ and $\frac{10}{14}$ have $\frac{7}{14}, \frac{8}{14}, \frac{9}{14}$ between them, and so on.
Do this: There are always infinitely many; scale up.
Usually costs: Full marks if the question asks for several numbers.
Mistake: Ordering negative fractions by their size without the sign
For negatives, larger magnitude means smaller number.
Wrong working
$- \frac{3}{4} < - \frac{5}{6}$ because $\frac{3}{4} < \frac{5}{6}$ .
Right working
$- \frac{30}{36} < - \frac{27}{36}$ , so $- \frac{5}{6} < - \frac{3}{4}$ .
Do this: Convert to a common denominator and compare signed numerators.
Usually costs: The ordering mark.

Exam tips

Always verify with decimals: it takes ten seconds and catches slips.
If "n numbers" are asked, show the common denominator step; it carries the method mark.
Reduce final answers to lowest terms unless the question asks for a given denominator.

Practice questions on Rational Numbers Between Two Rational Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Between $\frac{2}{7}$ and $\frac{3}{7}$ , the number of rational numbers is:
(a) zero (b) one (c) six (d) infinite
Hint
Think of repeating the mean.
Answer
(d)
Q2 · 2 marks · Short answerBoard level
Find two rational numbers between $\frac{1}{4}$ and $\frac{1}{3}$ .
Hint
$\frac{3}{12}$ and $\frac{4}{12}$ are neighbours: scale by $4$ .
Answer
$\frac{12}{48}$ and $\frac{16}{48}$ ; choose $\frac{13}{48}$ and $\frac{14}{48} = \frac{7}{24}$ (any two between $0.25$ and $0.333$ ).
Q3 · 3 marks · Short answerHarder board level
Write five rational numbers between $- 2$ and $- 1$ .
Hint
Write $- 2 = - \frac{20}{10}$ and $- 1 = - \frac{10}{10}$ .
Answer
$- \frac{19}{10}, - \frac{17}{10}, - \frac{3}{2}, - \frac{13}{10}, - \frac{11}{10}$ (any five).
Q4 · 3 marks · Short answerHarder board level
Arrange in ascending order: $- \frac{3}{4}, \frac{5}{- 6}, - \frac{7}{9}, \frac{2}{- 3}$ .
Hint
Make denominators positive; LCM is $36$ .
Answer
In 36ths: $- 27, - 30, - 28, - 24$ . Ascending: $- \frac{5}{6}, - \frac{7}{9}, - \frac{3}{4}, - \frac{2}{3}$ .
Q5 · 4 marks · Long answerHOTS / competency
Let $m$ be the mean of $\frac{1}{3}$ and $\frac{1}{2}$ , and $n$ the mean of $\frac{1}{3}$ and $m$ . Find $m$ and $n$ and arrange $\frac{1}{3}, n, m, \frac{1}{2}$ in increasing order.
Hint
Use denominator 12, then 24.
Answer
$m = \frac{4/12 + 6/12}{2}$ ; with $\frac{1}{3} = \frac{4}{12}$ , $\frac{1}{2} = \frac{6}{12}$ the mean is $\frac{5}{12}$ . $n = \frac{1}{2} (\frac{4}{12} + \frac{5}{12}) = \frac{9}{24} = \frac{3}{8}$ . Order: $\frac{1}{3} < \frac{3}{8} < \frac{5}{12} < \frac{1}{2}$ ( $0.333 < 0.375 < 0.417 < 0.5$ ).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the rational number midway between a and b?Show answer
$\frac{a + b}{2}$ .
2.Can two different rationals be so close that nothing fits between them?Show answer
No: infinitely many rationals lie between any two.
3.Which is greater: -3/4 or -5/6?Show answer
-3/4.
4.What do you multiply by to be sure of n numbers between two fractions?Show answer
(n + 1), on top and bottom.

Rational Numbers Between Two Rational Numbers: frequently asked questions

Why is there no "next" rational number after 1/2?

Whatever number you propose, the mean of it and 1/2 is closer, so no smallest gap exists.

Can an answer be an integer?

Yes. Integers are rational, e.g. 0 lies between -\frac{1}{2} and \frac{1}{3}.