Question

Write four more rational numbers in each of the following pattern:
$\dfrac{-3}{5},\dfrac{-6}{10},\dfrac{-9}{15},\dfrac{-12}{20},....$

Answer 1

Hint: Here, the first number of the series $\dfrac{-3}{5}$, can be used to find the four more rational numbers. The numerator and denominator of all the numbers given in the series are multiples of the numerator and denominator of $\dfrac{-3}{5}$.

Complete Step-by-Step solution:
Here, we have to find the four more rational numbers of the pattern:
$\dfrac{-3}{5},\dfrac{-6}{10},\dfrac{-9}{15},\dfrac{-12}{20},....$
 A rational is a number of the form $\dfrac{p}{q}$ where, $p$ and $q$ are integers and $q\ne 0$.
Here, we are given with the series of rational numbers $\dfrac{-3}{5},\dfrac{-6}{10},\dfrac{-9}{15},\dfrac{-12}{20},....$
 The first number is $\dfrac{-3}{5}$. Now, the second number $\dfrac{-6}{10}$ can be expressed in terms of $\dfrac{-3}{5}$.That is,
$\dfrac{-6}{10}=\dfrac{-3}{5}\times \dfrac{2}{2}$
Similarly, all the other numbers in the series can also be expressed in terms of $\dfrac{-3}{5}$
Now, consider the number $\dfrac{-9}{15}$, the numerator and denominator of $\dfrac{-9}{15}$ is 3 times of the numerator and denominator of $\dfrac{-3}{5}$ . Hence, we can write:
$\dfrac{-9}{15}=\dfrac{-3}{5}\times \dfrac{3}{3}$
Consider the next number, $\dfrac{-12}{20}$, the numerator and denominator of $\dfrac{-12}{20}$ is 4 times the numerator and denominator of $\dfrac{-3}{5}$. That is,
$\dfrac{-12}{20}=\dfrac{-3}{5}\times \dfrac{4}{4}$
Now, according to the series then the next number’s numerator and denominator will be 5 times the numerator and denominator of $\dfrac{-3}{5}$. The number will be:
$\dfrac{-3}{5}\times \dfrac{5}{5}=\dfrac{-15}{25}$
Hence, the next number will be $\dfrac{-15}{25}$.
Similarly, the next number of the series can be calculated by multiplying the numerator and denominator of $\dfrac{-3}{5}$by 6. So the number is:
$\dfrac{-3}{5}\times \dfrac{6}{6}=\dfrac{-18}{30}$
So, the next number is $\dfrac{-18}{30}$.
The other two numbers of the series can also be calculated by multiplying the numerator and denominator of $\dfrac{-3}{5}$ by 7 and 8 respectively. Hence, we will obtain:
$\begin{align}
  & \dfrac{-3}{5}\times \dfrac{7}{7}=\dfrac{21}{35} \\
 & \dfrac{-3}{5}\times \dfrac{8}{8}=\dfrac{24}{40} \\
\end{align}$
Therefore, the other two numbers are $\dfrac{21}{35}$ and $\dfrac{24}{40}$.
Hence we can say that the next four numbers of the series are $\dfrac{-15}{25},\dfrac{-18}{30},\dfrac{21}{35}$ and $\dfrac{24}{40}$.
Now, we will get the series as $\dfrac{-3}{5},\dfrac{-6}{10},\dfrac{-9}{15},\dfrac{-12}{20},\dfrac{-15}{25},\dfrac{-18}{30},\dfrac{21}{35},\dfrac{24}{40},...$

Note: Here, you have to check the connection of numbers with each other especially with the first number. It is neither an AP nor GP, since the difference or the ratios are not the same. In such cases you have to understand the pattern in which the series goes.