Find three distinct rational numbers that lie strictly between and .

Question

Homi · Accepted Answer

Understand the Question

To find rational numbers strictly between $-\dfrac{1}{2}$ and $\dfrac{1}{4}$ , first convert both fractions to a common denominator.
If there are not enough intermediate integers between the numerators, multiply the numerator and denominator by a suitable integer to create a larger common denominator.
Any three distinct values lying strictly between the resulting endpoints can be chosen.

Step 1 · Find a Common Denominator

Given rational numbers are $-\dfrac{1}{2}$ and $\dfrac{1}{4}$ . Diagram 1

The LCM of denominators $2$ and $4$ is $4$ .

$-\dfrac{1}{2} = -\dfrac{1 \times 2}{2 \times 2} = -\dfrac{2}{4}$

The two numbers are $-\dfrac{2}{4}$ and $\dfrac{1}{4}$ .

Step 2 · Convert to a Larger Denominator and Choose Values

The integers strictly between the numerators $-2$ and $1$ are $-1$ and $0$ , which provides only two rational numbers ( $-\dfrac{1}{4}$ and $0$ ).

Multiply numerator and denominator of both fractions by $2$ to obtain denominator $8$ :

\begin{aligned} -\dfrac{2}{4} &= -\dfrac{2 \times 2}{4 \times 2} = -\dfrac{4}{8} \\[0.6em] \dfrac{1}{4} &= \dfrac{1 \times 2}{4 \times 2} = \dfrac{2}{8} \end{aligned}

The integers strictly between $-4$ and $2$ are $-3, -2, -1, 0, 1$ .

Rational numbers between $-\dfrac{4}{8}$ and $\dfrac{2}{8}$ are: $-\dfrac{3}{8}, \quad -\dfrac{2}{8} = -\dfrac{1}{4}, \quad -\dfrac{1}{8}, \quad \dfrac{0}{8} = 0, \quad \dfrac{1}{8}$

Selecting any three distinct numbers: $-\dfrac{1}{4}, \quad 0, \quad \dfrac{1}{8}$

Answer

$-\dfrac{1}{4}, \quad 0, \quad \dfrac{1}{8}$

Common Mistakes

Including Endpoints: Numbers must lie strictly between $-\dfrac{1}{2}$ and $\dfrac{1}{4}$ ; the endpoint values themselves cannot be included.
Negative Number Ordering: Remember that on a number line, $-4 < -3 < -2 < -1 < 0$ . Do not reverse the order for negative numbers.
Not Expanding Denominator: Denominator $4$ gives only two intermediate fractions ( $-\dfrac{1}{4}$ and $0$ ). To find three or more, scale up the denominator by multiplying both numerator and denominator by an integer like $2$ or $10$ .

Question 2

More questions in Exercise 3.4