Explanation:

Correct option is 1: 126

To solve the problem step by step, let's denote the number of successes as S and the number of failures as F.

Step 1: Set up the initial ratio
From the problem, we know that the success to failure ratio is given as:
S / F=(5 / 2)
This can be expressed as:
S=(5/2)F

Step 2: Express the number of failures in terms of successes
From the equation S=52F, we can also express F in terms of S:
F=(2/5)S

Step 3: Consider the new scenario with increased failures
According to the problem, if the number of failures increases by 14, the new ratio of successes to failures becomes:
S/(F+14)=9/5
This can be rewritten as:
5S=9(F+14)

Step 4: Substitute F in the new ratio equation
Now, substitute F from Step 2 into the new ratio equation:
5S=9[(2/5)S+14}
Expanding this gives:
5S=(18/5)S+126

Step 5: Clear the fraction by multiplying through by 5
To eliminate the fraction, multiply the entire equation by 5:
25S=18S+630

Step 6: Solve for S
Now, isolate S by moving terms involving S to one side:
25S−18S=630
7S=630
S=90

Step 7: Find F
Using the value of S, we can find F:
F=(2/5)S=2/5×90=36

Step 8: Calculate the total number of candidates
The total number of candidates who appeared for the examination is:
Total=S+F=90+36=126

Thus, the total number of candidates who appeared for the examination is 126.