Question

How many multiple of 9 lie between 10 and 300?

Answer 1

Hint: First project the given condition into a type of arithmetic progression. So, use the formula of \[{{n}^{th}}\] term of arithmetic progression to find the n. you know the first term. As the question is on 9 multiples, we know the common difference must be 9. So, find the least multiple greater than 10 that will be first term and greatest multiple less than 300 that will be last term. From this find the number n which is last term. That will be the number of multiples.

Complete step-by-step answer:
A sequence of numbers such that the difference of any two consecutive numbers is a constant is called an arithmetic progression. For example, the sequence 1, 2, 3, 4……… is an arithmetic progression with common difference = 2-1=1. Now, we need to find \[{{n}^{th}}\] terms of such a sequence. Let us have a sequence with first term a and common difference d. We know the difference between succession terms is d. So, second term= first term + d= a + d, third term = second term + d = a+ 2d, and so on calculating, we get,
${{n}^{th}}={{\left( n-1 \right)}^{th}}term+d=a+\left( n-2 \right)d+d=a+\left( n-1 \right)d$
Given the range in the question, the condition on that range is: number of multiples of 9 between 10 and 300. So, the least multiple greater than 10 will be 18. The greatest multiple less than 300 will be 297. Now take an arithmetic sequence from 18 with common difference 9 till 297, we get the sequence as:
18, 27, 36, ………………………., 297.
By looking at this sequence we can say all the terms are multiple of 9. If we have first term as a and common difference d, then \[{{n}^{th}}\] term t of an arithmetic progression, given by:
$t=a+\left( n-1 \right)d$
If we find a position of 297 indirectly, we get the number of multiples between 10 and 300.
So, let the 297 term be at \[{{k}^{th}}\] position. By formula, we have a= 18, d=9, t=297, n=k. by substituting all these values, we get the equation as:
$297=18+\left( k-1 \right)9$
To remove brackets multiple the constant 9 inside, we get it as:
$297=18+9k-9$
By simplifying the right-hand side of equation, we get it as:
$297=9+9k$
By subtracting 9 on both sides of the equation, we get it as:
$288=9k$
By dividing with 9 on both sides of the equation, we get:
$\dfrac{288}{9}=\dfrac{9k}{9}$
By simplifying the above equation, we get the value of k as:
k=32
So, the value of 297 is at 32nd position. We can also say 297 comes after 31 multiples greater than 10. So, total multiples are 32 in between 10 and 300.

Note: The idea of projecting it as arithmetic progression and then getting a sequence of multiples is very crucial. If not, you must take a range of 10 numbers and find the number and count all numbers which are multiples by writing in a list. This method avoids you to not write all multiples of 9 in the range of 10 to 300, which is not tricky at all.