Question

If $ \dfrac{{37}}{{13}} = 2 + \dfrac{1}{{x + \dfrac{1}{{y + \dfrac{1}{z}}}}} $ where $ x $ $ y $ $ z $ are natural numbers, then find $ x + y + z $ .
(a) $ 6 $
(b) $ 7 $
(c) $ 8 $
(d) $ 9 $

Answer 1

Hint: As we know that above question has algebraic expression and we use algebraic formulas and basic identities to solve this question. In this kind of question where a lot of algebraic variables are given in the form of $ 1 $ by or $ 1 $ upon , then we have to simplify this form in the simpler part. Here we well equate both the left hand side and the right hand side of the equation and solve them in the simpler form.

Complete step-by-step answer:
As per the question we have
 $ \dfrac{{37}}{{13}} = 2 + \dfrac{1}{{x + \dfrac{1}{{y + \dfrac{1}{z}}}}} $ ,
and the variables $ x,y,z $ are natural numbers and we have to find it.
We will first transfer $ 2 $ in the left hand side and then subtract it to the number:
$ \dfrac{{37}}{{13}} - 2 = \dfrac{1}{{x + \dfrac{1}{{y + \dfrac{1}{z}}}}} \\
\Rightarrow \dfrac{{11}}{{13}} = \dfrac{1}{{x + \dfrac{1}{{y + \dfrac{1}{z}}}}} $ .
Now after removing $ 2 $ from the right hand side we can now reciprocate both the left hand side and the right hand side. We will now reciprocate both the sides and we get:
$ \dfrac{{13}}{{11}} = x + \dfrac{1}{{y + \dfrac{1}{z}}} $ .
Here we will have to bring the left hand side in a suitable form i.e. in the form of right hand side to get the value of $ x $ . Also $ \dfrac{{13}}{{11}} $ can be written as $ 1 + \dfrac{2}{{11}} $ , so by putting this value on the right hand side, we get:
 $ 1 + \dfrac{2}{{11}} = x + \dfrac{1}{{y + \dfrac{1}{z}}} $ .
On comparing both the equations we can say that $ x = 1 $ . Now we are left with
$ \dfrac{2}{{11}} = \dfrac{1}{{y + \dfrac{1}{z}}} $ ,
On reciprocating it again we have:
$ y + \dfrac{1}{z} = \dfrac{{11}}{2} $ .
Similarly we can write the value of the right hand side in the form of the left hand side i.e.
 $ \dfrac{{11}}{2} = 5 + \dfrac{1}{2} $ .
Comparing the equation again we have
$ y + \dfrac{1}{z} = 5 + \dfrac{1}{2} $ .
Therefore we can say
$ y = 5 $ and $ \dfrac{1}{z} = \dfrac{1}{2} $ i.e. $ z = 2 $ .
Since we have all the values of $ x,y,z $ , now we will add the variables $ x + y + z = 1 + 5 + 2 = 8 $
 Hence the correct option of $ x + y + z = 8 $ is (c).
So, the correct answer is “Option C”.

Note: Before solving this kind of question we need to have full knowledge of algebraic expressions, their reciprocals and how to solve them. Also we should be careful while solving as we can make calculation mistakes. Expansion of algebraic expressions must be correct and signs must be checked as it may lead to wrong answers.