Number cards are arranged with numbers in ascending order. The difference between any two consecutive cards is the same. Which of the following best replaces the question mark?

$\frac{1}{5}$

?

$\frac{1}{4}$

Given :

Number cards are arranged with numbers in ascending order.

The first number card is $\frac{1}{5}$

The fourth number card is $\frac{1}{4}$

To find :

We have to find the number which replaces the question mark

Solution :  

Let the difference between any two number cards be x.

Therefore,

$\frac{1}{4} =3x+\frac{1}{5}$

$3x=\frac{1}{4} -\frac{1}{5}$

$3x=\frac{1\times 5-1\times 4}{20}$                 [ LCM of 4 and 5 is 20]

$3x=\frac{5-4}{20}$

$3x=\frac{1}{20}$

$x=\frac{1}{3\times 20}$

$x=\frac{1}{60}$

Therefore, The third number card( ?) $ =2x+\frac{1}{5}$

$=2\left(\frac{1}{60}\right) +\frac{1}{5}$

$ =\frac{1}{30} +\frac{1}{5}$

$ =\frac{1\times 1+1\times 6}{30} $       [ LCM of 30 and 5 is 30]

$ =\frac{1+6}{30}$

$=\frac{7}{30}$ 

The number is $\frac{7}{30}$