math - Andy, Thursday, October 25, 2007 at 7:37pm
can you please help me with this math problem? If you can here is the question:
Seventy-five years ago, pharmacists weighed medicine on balances like the ones you have been using. The mass pieces were very expensive, so a pharmacist would buy as few mass pieces as possible. If a pharmacist had 1-g, 3-g, and 9-g mass pieces, he or she could weigh out any number of grams from 1g to 13g. Show how you could measure all of the masses from ig to 13g using only the three mass pieces given
for 1 g, use the 1g piece on one side top balance the material bing measured.
For 2 g, put 3g on one side and 1 on the other. Add material to the side with the 1 g mass.
For 3 g, use the 3g piece to balance.
For 4 g, put the 1 g and 3 g together on one side
For 5 g, Put 9 g on one side and 3 + 1 on the other
Similarly, 6 = 9 - 3
7 = 9 + 1 - 3
8 = 9 - 1
9 = 9
10 = 9 + 1
11 = 9 + 3 - 1
12= 9 + 3
13 = 9+3+1 (all on the same side)
To solve this problem, we need to find a combination of the 1-g, 3-g, and 9-g mass pieces that can be used to measure all masses from 1g to 13g. Let's go step by step:
1. Start with 1g. Since we have a 1-g mass piece, we can directly measure this mass.
2. Now, let's see how we can measure 2g. We don't have a 2-g mass piece, but we can use the 1-g mass piece twice. So, we use the 1-g mass piece twice to measure 2g.
3. Similarly, to measure 3g, we can use the 1-g mass piece three times.
4. Now, we need to find a way to measure 4g. Since we only have 1-g, 3-g, and 9-g mass pieces, we cannot directly measure 4g. However, we can add 1g to the 3g we already have to get 4g. So, we use the 1-g and 3-g mass pieces together to measure 4g.
5. Moving on, to measure 5g, we can use the 1-g mass piece and the 4g mass we previously obtained together.
6. For 6g, we can use the 3-g mass piece and the 3g mass we previously obtained.
7. To measure 7g, we can use the 3-g and 4g masses together.
8. Next, for 8g, we can use the 1-g, 3-g, and 4g masses together.
9. For 9g, we already have a 9-g mass piece, so we can directly measure this mass.
10. To measure 10g, we can use the 1-g and 9g masses together.
11. For 11g, we can use the 3-g and 9g masses together.
12. For 12g, we can use the 1-g, 3-g, and 9g masses together.
13. Finally, for 13g, we can use the 4-g, 9-g masses together.
So, by using different combinations of the 1-g, 3-g, and 9-g mass pieces, we can measure all masses from 1g to 13g.