Question

How many multiples of 3 are between 23 and 82.
$\left( A \right)$ 19
$\left( B \right)$20
$\left( C \right)$ 21
$\left( D \right)$ 22
$\left( E \right)$ 23

Answer 1

Hint – In this question first find out the first, second and last multiple of 3 between 23 and 82, then check out which series it will make (i.e. multiples of three) whether it is A.P, G.P or H.P, then apply the formula of nth term of the series and calculated the number of terms in the series so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
Multiples of 3 between 23 and 82.
The set of three is given as {3, 6, 9, 12, 15, 18, 21, 24, 27,......, 81, 84.............. up to so on}.
So after 23 the first multiple of 3 is 24 and after that it is 27 and before 82 the multiple of three is 81.
So the multiple of 3 between 23 and 81 are
24, 27, 30, ........., 81.
So as we see this will follow the rule of A.P (arithmetic progression) with first term (a) = 24, common difference (d) = (27 – 24) = (30 – 27) = 3 and last term (an) is = 81.
Let n be the number of terms of this series.
Now as we know the nth term of an A.P is given as
$ \Rightarrow {a_n} = a + \left( {n - 1} \right)d$, where symbols have their usual meaning.
Now substitute the values in the above equation we have,
$ \Rightarrow {a_n} = a + \left( {n - 1} \right)d$
$ \Rightarrow 81 = 24 + \left( {n - 1} \right)3$
Now simplify this we have,
$ \Rightarrow 81 - 24 = \left( {n - 1} \right)3$
$ \Rightarrow 57 = \left( {n - 1} \right)3$
$ \Rightarrow \dfrac{{57}}{3} = 19 = \left( {n - 1} \right)$
$ \Rightarrow n = 19 + 1 = 20$
So there are 20 multiples of 3 between 23 and 82.
Hence option (B) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that the formula of the nth terms of an arithmetic progression which is written above then using this formula substitute the values of the first term, common difference and nth term of the series and simplify as above we will simply get the number of terms in the series.