The sum of a number and its reciprocal is 5 (1/5) . Find the number.
I found the answer to be:
x+(1/x)=5(1/5)
(x^2 + 1)/x= 26/5 and so on...
Now, I am trying to figure out how a number multiplied by a fraction could lead to 26 and that would only happen by adding a whole number to the fraction 5+(1/5).
But that is not the case here, right? I don't understand why 5(1/5) is suddenly calculated as 5+(1/5)? What am I missing out on? Thank you in advance for any clarification.
On
$$5\frac12=5+\frac12=\frac{11}2=5.5\neq(5)(\frac12)\left(\text{which is}=5\times0.5=\frac52=2.5\right)$$
Basically, mixed fraction is just a way to write $\frac{11}2$ informing that we need $5$ wholes and a half.
And given sum of number and it's reciprocal as $5\frac15$ then it's quite obvious that one solution is $x=5$ (and the other is $x=\frac 15$). Seen simply by comparing $x+\frac1x=5+\frac15 (\text{or }\frac15 + 5)$.
If I understand correctly, the question probably looked like this:
When writing a positive rational number, one way of doing so is called a "mixed number," where we write the integer part, followed by the fractional part. In this case, while the numbers $5$ and $\frac{1}{5}$ are written next to each other, this is not intended to indicate multiplication, as it would if one or both of them were a variable, but rather, addition. So, $5\frac{1}{5}$ signifies $5+\frac{1}{5}.$
Interestingly, this gives us one solution to the problem immediately, because we need to solve the equation $x+\frac{1}{x}=5+\frac{1}{5},$ which clearly has $x=5$ as a solution. On the other hand, we can observe that the substitution $x=\frac{1}{t}$ yields $\frac{1}{t}+t=5+\frac{1}{5},$ or equivalently, $t+\frac{1}{t}=5+\frac{1}{5},$ so $t=5$ is a solution, and so $x=\frac{1}{5}$ is the other solution to the initial equation!