p% of Bobby's income is equal to q% of Ron's income. If Bobby's income was Rs y less than what it is and Ron's income was Rs z more than what it is, the ratio of the incomes of Bobby and Ron would have been m : n. The actual combined income of the duo is Rs x.

Which one of the following is correct in respect of the Question and the Statements given below?

Statement 1: The values of p, q, y, z, m, n and x are respectively 40, 56, 3200, 2000, 6, 5 and 67200.

Statement 2: The values of p, q, y, z, m, n and x are respectively 20, 32, 1200, 5500, 4, 3 and 83200.

Correct option is D

Given:

P% of Bobby's income = Q% of Ron's income.

If Bobby's income decreases by Rs. y and Ron's income increases by Rs. z, the ratio of their incomes becomes m : n

The actual combined income of Bobby and Ron is Rs. x

Two statements are provided with specific values for P, Q, y, z, m, n, and x.

Concept Used:

Percentage Relationship:

​$$\frac{P}{100} \times \text{Bobby's Income} = \frac{Q}{100} \times \text{Ron's Income}$$​​

Modified Income Ratio:

​$$\frac{\text{Bobby's Income} - y}{\text{Ron's Income} + z} = \frac{m}{n}$$​​

Total Income Constraint:

Bobby's Income + Ron's Income = x

Solution:

Let Bobby's actual income = B

Let Ron's actual income = R

Given: B + R = x

Applying Percentage Condition

​$$P \times B = Q \times R \implies B = \frac{Q}{P} \times R$$​​

Applying Modified Income Ratio

After changes:

Bobby's new income = B - y

Ron's new income = R + z

Given ratio:

​$$\frac{B - y}{R + z} = \frac{m}{n}$$​​

Substituting B = $$\frac{Q}{P} \times R:$$​​

​$$\frac{\frac{Q}{P} \times R - y}{R + z} = \frac{m}{n}$$​​

​$$n \left(\frac{Q}{P} \times R - y\right) = m (R + z)$$​​

​$$\frac{nQ}{P} \times R - n y = m R + m z$$​​

R = $$\frac{m z + n y}{\frac{nQ}{P} - m}$$​​

Statement 1:

P = 40 , Q = 56 , y = 3200, z = 2000 , m = 6 , n = 5, x = 67200

Now,

R = $$\frac{6 \times 2000 + 5 \times 3200}{\frac{5 \times 56}{40} - 6} = \frac{12000 + 16000}{7 - 6} = 28000$$​​

Calculating B :

B = $$\frac{56}{40} \times 28000 = 39200$$​​

total income:

B + R = 39200 + 28000 = 67200

Statement 1 is Feasible.

Statement 2:

P = 20 , Q = 32, y = 1200 , z = 5500, m = 4, n = 3, x = 83200

Calculating R:

R = $$\frac{4 \times 5500 + 3 \times 1200}{\frac{3 \times 32}{20} - 4} = \frac{22000 + 3600}{4.8 - 4} = 32000$$​​

Calculating B:

B = $$\frac{32}{20} \times 32000 = 51200$$​​

total income:

B + R = 51200 + 32000 = 83200 

Statement 2 is Feasible.

Both Statement 1 and Statement 2 are feasible.