Typo in one of the Question Bank Question (2/7 or 5.7)

[anabbasi]

On January 1, 2076, Lake Loser contains x liters of water. By Dec 31 of that same year, 2/7 of the x liters have evaporated. This pattern continues such that by the end of each subsequent year the lake has lost 2/7 of the water that it contained at the beginning of that year. During which year will the water in the lake be reduced to less than 1/4 of the original x liters?


2077
2078
2079
2080
2081


Solution Says

This fraction problem contains an "unspecified" total (the x liters of water in the lake). Pick an easy "smart" number to make this problem easier. Usually, the smart number is the lowest common denominator of all the fractions in the problem. However, if you pick 28, you will quickly see that this yields some unwieldy computation.

The easiest number to work with in this problem is the number 4. Let's say there are 4 liters of water originally in the lake. The question then becomes: During which year is the lake reduced to less than 1 liter of water?

At the end of 2076, there are 4 × (5/7) or 20/7 liters of water in the lake. This is not less than 1.

At the end of 2077, there are (20/7) × (5/7) or 100/49 liters of water in the lake. This is not less than 1.

At the end of 2078, there are (100/49) × (5/7) or 500/343 liters of water in the lake. This is not less than 1.

At the end of 2079, there are (500/343) × (5/7) or 2500/2401 liters of water in the lake. This is not less than 1.


At the end of 2080, there are (2500/2401) × (5/7) or 12500/16807 liters of water in the lake. This is less than 1.

Notice that picking the number 4 is essential to minimizing the computation involved, since it is very easy to see when a fraction falls below 1 (when the numerator becomes less than the denominator.) The only moderately difficult computation involved is multiplying the denominator by 7 for each new year.

The correct answer is D

[StaceyKoprince]

Nope, not a typo. The lake loses 2/7 each year, which means 5/7 remains at the end of each year. The calculation uses 5/7 because it is calculating how much water remains at the end of each year.