A group of hardy trekkers started out on a challenging trek. Their number was
less than 100. At night they found a shelter which could just about
accommodate all the trekkers. However, in the shelter there were three-tiered
bunk beds. Only one trekker could sleep on the lowest bunk. The middle tier
had room for double that, and the topmost could accommodate double the
number of sleepers on the middle bunk. Before retiring for the night they
played a game after dividing themselves into teams of five members each. The
enthusiasm of 12 trekkers waned on the second day and they turned back to go
home. That left more than two thirds of the original group and they soldiered
ahead. How many trekkers started from home?
To solve this problem, let's break down the information provided step by step.
1. The shelter has three-tiered bunk beds, where the bottom tier can accommodate 1 trekker, the middle tier can accommodate double that, and the top tier can accommodate double the number on the middle tier.
2. The group of trekkers was less than 100, but we don't know the exact number.
3. They divided themselves into teams of five members each before playing a game.
4. 12 trekkers decided to turn back and go home, leaving more than two-thirds of the original group to continue the trek.
Now, let's work through the problem:
Let's assume the number of trekkers started from home is 'x'.
1. Since the shelter can accommodate all the trekkers, the total number of trekkers must be a multiple of 5 (team size) and fit the bunk bed criteria.
2. Since the bottom tier can accommodate only one trekker, we know that the total number of trekkers must be divisible by 1.
3. As the middle tier can accommodate double the number of the bottom tier, the total number of trekkers must also be divisible by 2.
4. Similarly, the top tier can accommodate double the number of the middle tier, so the total number of trekkers must also be divisible by 2 again.
To summarize, the total number of trekkers must be divisible by 1, 2, and 2.
Let's find the smallest number that fits this criterion: 1 * 2 * 2 = 4.
Therefore, the total number of trekkers must be a multiple of 4.
Now, let's find the number of trekkers who continued the trek after 12 members turned back:
Remaining trekkers = Total trekkers - Trekkers who turned back
Remaining trekkers = x - 12
According to the problem, more than two-thirds of the original group continued the trek:
Remaining trekkers > (2/3) * Total trekkers
x - 12 > (2/3) * x
Multiplying both sides of the equation by 3 to get rid of the fraction gives us:
3*(x - 12) > 2*x
Expanding the equation: 3*x - 36 > 2*x
Now, let's solve the inequality:
3*x - 36 > 2*x
3*x - 2*x > 36
x > 36
Therefore, the number of trekkers who started from home must be greater than 36.
However, we don't have enough information to determine the exact number of trekkers who started from home. The only thing we know is that it is greater than 36.