Question

A property was to be divided among P, Q, R, S and T in the ratio of \(1 : 1.5 : 2 : 2.5 : 3\). If instead, it was divided in the duplicate ratio, then who among the four would be benefitted most?

• A. Q
• B. T
• C. R
• D. S
• E. P

Hint:

Convert both ratios to a common total (20 for the original, 90 for the duplicate ratio) and compare each person's share as a fraction of the property before and after.

Answer 1

The Correct Option is B

Step 1: Write the original ratio in whole numbers.
The shares of P, Q, R, S, T are in the ratio \(1 : 1.5 : 2 : 2.5 : 3\). Multiplying every term by 2 to remove the decimals gives the equivalent ratio \(2 : 3 : 4 : 5 : 6\). The sum of parts is \(2+3+4+5+6=20\), so treating the whole property as 20 units, the original shares are P = 2, Q = 3, R = 4, S = 5, T = 6 units.

Step 2: Form the duplicate ratio.
The duplicate ratio of \(2:3:4:5:6\) is formed by squaring every term: \(4 : 9 : 16 : 25 : 36\). The sum of these parts is \(4+9+16+25+36=90\).

Step 3: Express both sets of shares as a fraction of the total property.
Original fraction of property: P = \(2/20=0.1000\), Q = \(3/20=0.1500\), R = \(4/20=0.2000\), S = \(5/20=0.2500\), T = \(6/20=0.3000\).
New fraction of property (duplicate ratio): P = \(4/90=0.0444\), Q = \(9/90=0.1000\), R = \(16/90=0.1778\), S = \(25/90=0.2778\), T = \(36/90=0.4000\).

Step 4: Compare the change for every person.
P: \(0.0444-0.1000=-0.0556\) (a loss)
Q: \(0.1000-0.1500=-0.0500\) (a loss)
R: \(0.1778-0.2000=-0.0222\) (a loss)
S: \(0.2778-0.2500=+0.0278\) (a gain)
T: \(0.4000-0.3000=+0.1000\) (the biggest gain, 10 percentage points of the whole property)

Step 5: Conclude.
Only S and T end up with a bigger share than before, and T's increase of 0.10 is by far the largest of the two, and the largest change overall. So T is benefitted the most. The correct option is (b) T.