Question

The sum of a number and its reciprocal is \[5\dfrac{1}{5}.\] Then the required equation is:
\[\left( a \right){{y}^{2}}+\dfrac{1}{y}=\dfrac{26}{5}\]
\[\left( b \right)5{{y}^{2}}-26y+5=0\]
\[\left( c \right){{y}^{2}}+\dfrac{1}{y}+\dfrac{26}{5}=0\]
\[\left( d \right)5{{y}^{2}}+26y+5=0\]

Answer 1

Hint: To solve the given question, firstly we will convert the given mixed fraction into an improper fraction. Then we will assume that the number is y. So, the other number will be \[\dfrac{1}{y}.\] Then we will add both of them and we will equate it to the mixed fraction obtained. Then, we will convert the given equation into the quadratic equation by multiplying the whole equation with y.

Complete step by step answer:
Before starting the question, we will convert the given mixed fraction into an improper fraction. So, if a mixed fraction \[x\dfrac{y}{z}\] is given, then its improper form will be \[\dfrac{xz+y}{z}.\] In our question, the mixed fraction given to us is \[5\dfrac{1}{5}.\] To convert it into an improper fraction, we will do the following:
\[5\dfrac{1}{5}=\dfrac{5\times 5+1}{5}=\dfrac{25+1}{5}=\dfrac{26}{5}\]
Now, we will assume that one of the two numbers given is y. It is given in the question that the other number will be the reciprocal of the first number. So the other number will be \[\dfrac{1}{y}.\] Now, it is given in the question that the sum of the numbers is \[5\dfrac{1}{5}.\] So, we have the following equation.
\[y+\dfrac{1}{y}=5\dfrac{1}{5}\]
\[\Rightarrow y+\dfrac{1}{y}=\dfrac{26}{5}\]
Now, we will make the above equation quadratic by multiplying the whole equation with y. Thus, we will get the following equation.
\[\Rightarrow y\left( y \right)+\dfrac{1}{y}\times y=\dfrac{26}{5}y\]
\[\Rightarrow {{y}^{2}}+1=\dfrac{26y}{5}\]
\[\Rightarrow {{y}^{2}}-\dfrac{26y}{5}+1=0\]
Now, we will multiply the whole equation with 5. Thus, we will get the following equation.
\[5{{y}^{2}}-26 y+5=0\]

So, the correct answer is “Option B”.

Note:
We can also solve this question alternatively by the following method. The mixed fraction can also be written as a sum of an integer and a proper fraction. Thus, we can say that,
\[5\dfrac{1}{5}=5+\dfrac{1}{5}\]
Here, we can see that 5 and \[\dfrac{1}{5}\] are the two inverse numbers whose sum is \[5\dfrac{1}{5}.\] So, y = 5 and \[y=\dfrac{1}{5}.\] Hence, we can say that,
\[\left( y-5 \right)\left( y-\dfrac{1}{5} \right)=0\]
\[\Rightarrow {{y}^{2}}-5y-\dfrac{1}{5}y+1=0\]
\[\Rightarrow {{y}^{2}}-\dfrac{26}{5}y+1=0\]
\[\Rightarrow 5{{y}^{2}}-26y+5=0\]