Question

Find four rational numbers equivalent to the following.
$\dfrac{7}{-15}$

Answer 1

Hint: We need to find the four rational numbers that are equal to $\dfrac{7}{-15}$ . The equivalent fraction can be found out by multiplying and dividing the given rational number with any number other than zero. We multiply the given rational number with 2, 3, 4, 5 to get the desired result.

Complete step by step solution:
In mathematics, a rational number is a number that can be expressed as a fraction. The general representation of a rational number is given as follows,
$\Rightarrow \dfrac{p}{q}$
Here,
p, q are integers and $q\ne 0$
 We are given a rational number and are asked to find four equivalent rational numbers to the given rational number. We multiply the given rational number with numbers 2, 3, 4, 5 to find the equivalent rational numbers for the given rational number.
Two rational numbers are the same if their lowest forms are the same. Equivalent rational numbers have the same value but are represented in a different form.
 If $\dfrac{p}{q}$ is a rational number and $r$ is any non-zero integer. The equivalent rational number to $\dfrac{p}{q}$ can be found out as follows,
$\Rightarrow \dfrac{p\times r}{q\times r}$
According to the question, we need to find the equivalent rational numbers to $\dfrac{7}{-15}$
The first equivalent rational number to $\dfrac{7}{-15}$ can be found by multiplying and dividing the rational number by the number 2.
Multiplying and dividing $\dfrac{7}{-15}$ with the number 2, we get,
$\Rightarrow ~\dfrac{7\times 2}{-15\times 2}$
Simplifying the above expression, we get,
$\Rightarrow \dfrac{14}{-30}$
The second equivalent rational number to $\dfrac{7}{-15}$ can be found by multiplying and dividing the rational number by the number 3.
Multiplying and dividing $\dfrac{7}{-15}$ with the number 3, we get,
$\Rightarrow ~\dfrac{7\times 3}{-15\times 3}$
Simplifying the above expression, we get,
$\Rightarrow \dfrac{21}{-45}$
The third equivalent rational number to $\dfrac{7}{-15}$ can be found by multiplying and dividing the rational number by the number 4.
Multiplying and dividing $\dfrac{7}{-15}$ with the number 4, we get,
$\Rightarrow ~\dfrac{7\times 4}{-15\times 4}$
Simplifying the above expression, we get,
$\Rightarrow \dfrac{28}{-60}$
The fourth equivalent rational number to $\dfrac{7}{-15}$ can be found by multiplying and dividing the rational number by the number 5.
Multiplying and dividing $\dfrac{7}{-15}$ with the number 5, we get,
$\Rightarrow ~\dfrac{7\times 5}{-15\times 5}$
Simplifying the above expression, we get,
$\Rightarrow \dfrac{35}{-75}$
$\therefore$ The four equivalent rational numbers to $\dfrac{7}{-15}$ are $\dfrac{14}{-30},\dfrac{21}{-45},\dfrac{28}{-60},\text{ and }\dfrac{35}{-75}$ respectively.

Note: The equivalent rational numbers to $\dfrac{7}{-15}$ are correct only if they give $\dfrac{7}{-15}$ on simplifying them to their lowest forms by cancelling out the common factors.
$\Rightarrow \dfrac{14}{-30}=\dfrac{7\times 2}{-15\times 2}=\dfrac{7}{-15}$
$\Rightarrow \dfrac{21}{-45}=\dfrac{7\times 3}{-15\times 3}=\dfrac{7}{-15}$
$\Rightarrow \dfrac{28}{-60}=\dfrac{7\times 4}{-15\times 4}=\dfrac{7}{-15}$
$\Rightarrow \dfrac{35}{-75}=\dfrac{7\times 5}{-15\times 5}=\dfrac{7}{-15}$
All the rational numbers $\dfrac{14}{-30},\dfrac{21}{-45},\dfrac{28}{-60},\text{ and }\dfrac{35}{-75}$ give $\dfrac{7}{-15}$ when simplified to their lowest forms. The result attained is correct.