Question

What is the $9^{th}$ multiple of $15$?

Answer 1

Hint: We know that a multiple is a number which is obtained by multiplying the number with a series of numbers. For example, if the number is 4, then we have 4, 8, 12, 16 … as its multiples. So, we have obtained this by multiplying 4 with 1, 2, 3, 4, …. Similarly, we will multiply 15 by 9 to obtain the answer.

Complete step by step solution:
Multiples of any number can be found by multiplying the number by the series of natural numbers. Natural numbers are the positive integers numbers starting from $1$ till infinity and it does not include zero $\left( 0 \right)$ .They are represented by the letter ‘N’ and they form a part of real numbers but exclude negative integers. Thus, natural numbers are given as $\left\{ 1,2,3,4,5,6,7,8,9..... \right\}$.
We have to find the $9^{th}$ multiple of $15$ .
We know that multiples are found by multiplying the number with natural numbers i.e. $\left\{ 1,2,3,4,5,6,7,8,9..... \right\}$.
Hence we can find the multiples of $15$ as shown below:
$\begin{align}
  & 15\times 1=15 \\
 & 15\times 2=30 \\
 & 15\times 3=45 \\
 & 15\times 4=60 \\
 & 15\times 5=75 \\
 & 15\times 6=90 \\
 & 15\times 7=105 \\
 & 15\times 8=120 \\
 & 15\times 9=135 \\
\end{align}$
Hence representing the above data in the form of set we get:
Multiples of 15 are:
$\left\{ 15,30,45,60,75,90,105,120,135,.. \right\}$

Hence the $9^{th}$ multiple of $15$ is $135$.

Note: We can get the solution using another method:
The $9^{th}$ multiple of $15$ can be written as $\left( 9\times 15 \right)$.
Here we can use the fact that it is easy to multiply the numbers by 10.
Thus, we can write $15=10+5$
$9\times \left( 10+5 \right)$
$9\times 10+9\times 5$ …(1)
We also know that,
 $\begin{align}
  & 5\times 2=10 \\
 & 5=\dfrac{1}{2}\times 10 \\
\end{align}$
Thus equation (1) can be written as:
$\begin{align}
  & 9\times 10+\left( 9\times \dfrac{1}{2}\times 10 \right) \\
 & =\left( 9\times 10 \right)+\dfrac{1}{2}\left( 9\times 10 \right) \\
\end{align}$
Multiply the terms term by term we have,
$\begin{align}
  & =90+45 \\
 & =135 \\
\end{align}$
Hence in this case also the answer obtained is similar to that shown above, hence you can use any method which you may find easier to solve the question in order to evaluate the value of $9^{th}$ multiple of $15$.