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

Answer: (B)

Let k be some positive integer. Then

$\frac{m}{n}=\frac{k+m}{kn}$

$\Rightarrow kmn=n(k+m)$

$\Rightarrow kmn=kn+mn$

$\Rightarrow km=k+m$

$\Rightarrow m=km-k$

$\Rightarrow m=K(m-1)$

The only integers that would satisfy this result are $m=k=2.$

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

As$2/n<1$for $n>2$

Thus n=3,4,5,

then three fraction are $2/3,2/4,2/5$