Ratio and Proportion: Rs. 3000 is distributed among p, q and r such that p gets 2/3rdof what q and r together get and r gets ½ of what p and q together get. Find q’s share.

Step 1: Let the shares be

P = p

Q = q

R = r Total: p + q + r = 3000

Step 2: Use the first condition

P gets 2/3 of what Q and R together get → p = (2/3)(q + r) Multiply both sides by 3: → 3p = 2q + 2r  (Equation 1)

Step 3: Use the second condition

R gets 1/2 of what P and Q together get → r = (1/2)(p + q) Multiply both sides by 2: → 2r = p + q  (Equation 2)

Step 4: Solve the system

From Equation 1: 3p = 2q + 2r → p = (2q + 2r)/3  (Equation A)

From Equation 2: 2r = p + q → r = (p + q)/2  (Equation B)

Now plug Equation B into Equation A:

p = (2q + 2 × [(p + q)/2]) / 3 → p = (2q + (p + q)) / 3 → p = (3q + p) / 3

Multiply both sides by 3: 3p = 3q + p → 2p = 3q → p = (3/2)q  (Equation C)

Now use Equation 2 again: 2r = p + q → 2r = (3/2)q + q = (5/2)q → r = (5/4)q  (Equation D)

Step 5: Total sum

p + q + r = 3000 → (3/2)q + q + (5/4)q = 3000 → Convert all to same denominator (LCM = 4): → (6/4)q + (4/4)q + (5/4)q = 3000 → (15/4)q = 3000 → q = (3000 × 4) / 15 = 12000 / 15 = 800