Question

The number of positive fractions \[{}^{m}/{}_{n}\] such that ${}^{1}/{}_{3}<{}^{m}/{}_{n}<1$ and having the property that the fraction remains the same by adding some positive integer to the numerator and multiplying the denominator by the same positive integer is
A. $1$
B. $3$
C. $6$
D. Infinite

Answer 1

Hint: When we are given unknown fractions, just recall all the properties of the fractions and also the properties of ratios and proportions and apply them accordingly.

Complete step by step solution:
Here, we are going to take one integer K which satisfies the given condition-
${}^{1}/{}_{3}<{}^{m}/{}_{n}<1$
Let K be some positive integer.

As, we know that according to the property that the fraction remains unchanged by adding some positive integer to the numerator and multiplying the denominator by the same positive integer.

Therefore,

$\begin{align}
  & \dfrac{m}{n}=\dfrac{K+m}{kn} \\
 & \\
\end{align}$
By Cross-multiplying-

$\begin{align}
  & knm=n(k+m) \\
 & \Rightarrow knm=nk+mn \\
 & \Rightarrow knm-nk=mn \\
\end{align}$
Taking $n$ common from both the sides,

$\begin{align}
  & \Rightarrow km-k=m \\
 & \Rightarrow k(m-1)=m \\
\end{align}$

The only integers that would satisfy the results are $m=k=2$ ,

Thus, the fraction must be $\dfrac{2}{n}=\dfrac{2+2}{2n}=\dfrac{4}{2n}$

As ${}^{2}/{}_{n}<1$ for $n>2$

Thus substituting the values of n,

The three fractions are -

${}^{2}/{}_{3},{}^{2}/{}_{4},{}^{2}/{}_{5}$

The required answer is ${}^{2}/{}_{3},{}^{2}/{}_{4},{}^{2}/{}_{5}$

Hence, the given option B – 3 is the correct answer.

Note: Sometimes you need to do mental calculations for trial and error methods. Always remember the properties of the fractions which are commutative property, Associative property, identity property, Inverse and multiplicative inverse and distributive properties of multiplication over addition.