Question: Instead of dividing Rs. 133 among A, B and C in the ratio \displaystyle \frac{1}{2} : \displaystyle\frac{1}{4} : \displaystyle\frac{1}{6} , by mistake it was divided in the ratio of 2:4:6. Who gained in the transaction?
(a) Only A
(b) Only B
(c) Only C
(d) Both B and C
Ans: (d) Both B and C
Explanation:
Original:
A : B : C = \displaystyle \frac{1}{2} : \displaystyle\frac{1}{4} : \displaystyle\frac{1}{6} = 6 : 3 : 2
A’s share = \displaystyle\frac{6}{11}
B’s share = \displaystyle\frac{3}{11}
C’s share = \displaystyle\frac{2}{11}
By Mistake:
A : B : C = 2 : 4 : 6 = 1 : 2 : 3
A’s share = \displaystyle\frac{1}{6}
B’s share = \displaystyle\frac{2}{6}
C’s share = \displaystyle\frac{3}{6}
We just need to find out which fraction increases by mistake.
Hence both B and C gains in the transaction.
Note: In solving the problem our sole concern was who gains/loses, not the exact amount of gain/loss. If you want to find out the gain/loss amount we have to use 133 as multiplier to each fraction and get the difference.
Distributing using the Original ratio 6 : 3 : 2
A’s share = \displaystyle\frac{6}{11}\times 133 = 72.55
B’s share = \displaystyle\frac{3}{11}\times 133 = 36.27
C’s share = \displaystyle\frac{2}{11} \times 133 = 24.18
Distributing Using mistaken ratio 1 : 2 : 3
A’s share = \displaystyle\frac{1}{6}\times 133 = 22.17
B’s share = \displaystyle\frac{2}{6}\times 133 = 44.23
C’s share = \displaystyle\frac{3}{6}\times 133 = 66.50
| Person | Original share | share by mistake | Differences |
| A | 72.55 | 22.17 | -50.38 (loss) |
| B | 36.27 | 44.33 | 8.06 (gain) |
| C | 24.18 | 66.50 | 42.32 (gain) |
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How did you compare the fraction?
Say, for A, how did you get into 36, 11 from 6/11, 1/6
LCM of denominator is multiplied to each fraction.
LCM(11, 6) = 66
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