There are two vessels, one containing 12 litres of water, the other 6 litres of wine. If a litre be taken out of each and poured into the other, and if this be repeated four times, find how much wine will be in container one?
Solution
Correct Answer: Option B
Initial volumes of first and second vessel = 12 and 6 litres
Formula to be used,
Volume of wine after exchange in the first vessel = Volume of wine in first vessel before this exchange –(1 × fractional volume of wine in the first vessel during the exchange) + (1 × fractional volume of wine in the second vessel during the exchange)
Volume of wine after exchange in the second bottle = Volume of wine in second vessel before this exchange – (1 × fractional volume of wine in the second vessel during the exchange) + (1 × fractional volume of wine in the first vessel during the exchange)
After first exchange,
volume of wine in the first vessel = 0 – 0 + 1 × 1 = 1litre
∴fractional volume of wine in the first vessel = 1/12
Volume of wine remaining in the second vessel = 6 – 1 × 1 + 0 = 5litres
∴fractional volume of wine in the second vessel = 5 / 6
After 2nd exchange,
Volume of wine in the first vessel = 1 – 1 × 1 /12 + 1 × 5 / 6 = 21 / 12 litres
∴fractional volume of wine in the first vessel = (21 / 12)/12 = 21 / 144
Volume of wine in the second vessel = 5 – 1 × 5 / 6 + 1 × 1 / 12 = 51 / 12 litres
∴fractional volume of wine in second vessel = ( 51 / 12 ) / 6 = 51 / 72
After 3rd exchange,
Volume of wine in the first vessel = 21 / 12 – 1 × 21 /144 + 1 × 51 /72 = 37 / 16 litres
∴fractional volume of wine in first vessel = (37 / 16) /12 = 37 / 192
Volume of wine in the second vessel = 51 / 12 – 1 × 51 / 72 + 1 × 21 / 144 = 59 /16
∴fractional volume of wine in second vessel = ( 59 / 16 ) / 6 = 59 / 96
After 4thexchange,
Volume of wine in the first vessel = 37 / 16 – 1 × 37 / 192 + 1 × 59 / 96 = 175 / 64 = 2.734
∴volume of wine in the first vessel after 4 exchanges = 2.734
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