Question

Find ten rational numbers between $\dfrac{3}{5}$ and $\dfrac{3}{4}$.

Answer 1

Hint: For finding rational numbers between two given numbers, the first step is to make the denominator of two numbers same.

Complete step-by-step answer:
We first convert $\dfrac{3}{5}$ and $\dfrac{3}{4}$ to rational numbers with the same denominator.
$\dfrac{3}{5} \times \dfrac{4}{4} = \dfrac{{12}}{{20}}$
$\dfrac{3}{4} \times \dfrac{5}{5} = \dfrac{{15}}{{20}}$
Since, we need to find $10$ rational numbers, so we multiply and divide by $11$.
$\dfrac{{12}}{{20}} \times \dfrac{{11}}{{11}} = \dfrac{{132}}{{220}}$
$\dfrac{{15}}{{20}} \times \dfrac{{11}}{{11}} = \dfrac{{165}}{{220}}$
$\therefore 10$ rational numbers between $\dfrac{3}{5}$ and $\dfrac{3}{4}$ are:
$\dfrac{{133}}{{220}},\dfrac{{134}}{{220}},\dfrac{{135}}{{220}},\dfrac{{136}}{{220}},\dfrac{{137}}{{220}},\dfrac{{138}}{{220}},\dfrac{{139}}{{220}},\dfrac{{140}}{{220}},\dfrac{{141}}{{220}},\dfrac{{142}}{{220}}$

Additional Information: Rational Number Definition: A rational number can be defined as any number which can be represented in the form of $\dfrac{p}{q}$ where $q \ne 0$ . It is also a type of real number. Any fraction with non-zero denominators is a rational number. Hence, we can say that $'0'$ is also a rational number, as we can represent it in many forms such as $\dfrac{0}{1},\dfrac{0}{2},\dfrac{0}{3}$etc. But, $\dfrac{1}{0},\dfrac{2}{0},\dfrac{3}{0}$are not rational, since they give us infinite values.
Irrational number Definition: Irrational numbers are the real numbers that cannot be represented as a simple fraction. It can’t be expressed in the form of a ratio, such as$\dfrac{p}{q}$ , where $q \ne 0$ . It is a contradiction of rational numbers. An irrational number can be represented with a decimal. It has endless non-repeating digits after the decimal point.

Note: The simplest method to find a rational number between two rational numbers $x$and $y$ is to divide their sum by $2$. For ex- The rational no. between $3$ and $4$ is $\dfrac{{3 + 4}}{2} = \dfrac{7}{2}$.
For finding rational numbers between two rational numbers with different denominators, we first find their equivalent fraction with the same denominator and then find the rational number between them.