Q1.
Ariba and Arun have some marbles. Ariba says, “The number of marbles with me is 120% of the marbles Arun has”. What would be an appropriate statement Arun could make comparing the number of marbles he has with Ariba’s?
Answer
Arun could say: “The number of marbles with me is 83⅓% of the marbles Ariba has” — or, equivalently, “I have 16⅔% fewer marbles than Ariba.”
Let Arun have n marbles. Then Ariba has 1.2n.
Arun as a fraction of Ariba = n / 1.2n = 1/1.2 = 10/12 = 5/6
5/6 × 100 = 500/6 = 83.33%
Shortfall = 100% – 83.33% = 16.67% (exactly 16⅔%)
Arun as a fraction of Ariba = n / 1.2n = 1/1.2 = 10/12 = 5/6
5/6 × 100 = 500/6 = 83.33%
Shortfall = 100% – 83.33% = 16.67% (exactly 16⅔%)
A quick check with numbers: if Arun has 50 marbles, Ariba has 60. Then 50/60 = 83.33%, and Arun has 10 fewer, which is 10/60 = 16.67% of Ariba's.
Why it happens: The tempting answer, “Arun has 20% fewer marbles than Ariba”, is wrong — and the reason is the base. Ariba's extra 10 marbles are 20% of Arun's 50, but only 16.67% of Ariba's 60. The same gap, measured against two different wholes, gives two different percentages. Reversing a percentage comparison is never as simple as changing the sign: if A is (1 + r) times B, then B is 1/(1 + r) times A, and 1/1.2 is not 0.8.