A floor-finishing machine uses bars of wax. Once the bar is worn down to 1/2 its original size, it must be replaced with a new bar. Only full-size bars can be put in the machine. The pieces are melted and recycled to make full-size bars. If you buy 24 bars and recycle all the pieces, how many full-size bars can you use? How much wax will be left over? Your goal is to have as little wasted wax as possible.
1st cycle:
equivalent of 12 bars used, 24 half-bars left over
recycled to make 12 new bars
2nd cycle:
equivalent of 6 bars used, 12 half-bars left over
recycled to make 6 new bars
3rd cycle:
equivalent of 3 bars used, 6 half-bars left over
recycled to make 3 new bars
4th cycle:
equivalent of 1.5 bars used, 3 half-bars left over
recycled to make 1.5 new bars
since only full-size bars can be used, only one more bar can be used
total used: 12+6+3+1.5+1 = 23.5
So 1/2 bar is left over.
thanks for your help.
To solve this problem, let's break it down step by step.
1. Calculate the number of usable bars:
Since the bar must be replaced when it is worn down to 1/2 its original size, each bar will be used until it reaches the size of 1/2. Therefore, each full-size bar will be used twice (the original size and when it wears down to 1/2 size). So, for 24 bars, we can use a total of 24 x 2 = 48 times.
2. Calculate the number of full-size bars needed to reach 48 uses:
Since each full-size bar can be used twice, we need a total of 48/2 = 24 full-size bars to achieve the 48 uses.
3. Calculate the leftover wax:
To determine the amount of leftover wax, we need to consider the size of the wasted wax when a bar is worn down to 1/2 its size. Since the wasted wax will also be recycled to make new bars, we need to calculate the total amount of wasted wax.
When a bar is worn down to 1/2 its size, it means there is only 1/2 of the wax remaining. Therefore, for each bar that is used, it will result in (1 - 1/2) = 1/2 amount of wasted wax.
So, for 48 uses, we will have 48 x 1/2 = 24 bars worth of wasted wax.
4. Calculate the number of full-size bars we can make from the wasted wax:
Since each full-size bar requires the equivalent of 1 full bar of wasted wax, we can make a total of 24 full-size bars from the 24 bars worth of wasted wax.
Therefore, the final result is 24 full-size bars used and no wax left over.
To summarize:
- You can use a total of 24 full-size bars.
- No wax will be left over.