The problem I picked was on Logs because it's just a build on algebra but a new topic that I enjoy learning.
(x^log9^1/3 *y^log25^625)^log16^1/32 all over x^log10^100 *y^log81^9
I chose this problem because I struggled with how to solve this at first because I only had a basic understanding of logs. But like all algebra, if you follow the rules you can solve even the problems with the most crazy exponents and letters. The logs are like exponents but written differently so if you were to have Log3^81=4 you would say the exponent of 3 that gets 81 is 4.
1)So if I were to start solving the bottom I would say the exponent of 10 to the power of ? is 100. ?=2 so it would be x^2.
2)The next step on the bottom is similar, 81^?=9. ?=1/2 so it would be y^1/2
3) The bottom would now look like x^2*y^1/2
4)Now to the top, 16^?=1/32. ?=-5/4
5)Inside the parenthesis 9^?=1/3. ?=-1/2 so it would be x^-1/2
6)Next 25^?=625. ?=2 so it would be y^2
7) The new equation would be (x^-1/2*y^2)^-5/4 all over x^2*y^1/2
8)Simplify it and you would have x^5/8*y^-5/2 all over x^2*y^1/2
9)Then you would simplify common variables, x^5/8 over x^2 would turn into 1 over x^-11/8
10) Also y would simplify from y^-5/2 over y^1/2 to 1 over y^-3
The final answer is 1 over x^11/8*y^-3
(x^log9^1/3 *y^log25^625)^log16^1/32 all over x^log10^100 *y^log81^9
I chose this problem because I struggled with how to solve this at first because I only had a basic understanding of logs. But like all algebra, if you follow the rules you can solve even the problems with the most crazy exponents and letters. The logs are like exponents but written differently so if you were to have Log3^81=4 you would say the exponent of 3 that gets 81 is 4.
1)So if I were to start solving the bottom I would say the exponent of 10 to the power of ? is 100. ?=2 so it would be x^2.
2)The next step on the bottom is similar, 81^?=9. ?=1/2 so it would be y^1/2
3) The bottom would now look like x^2*y^1/2
4)Now to the top, 16^?=1/32. ?=-5/4
5)Inside the parenthesis 9^?=1/3. ?=-1/2 so it would be x^-1/2
6)Next 25^?=625. ?=2 so it would be y^2
7) The new equation would be (x^-1/2*y^2)^-5/4 all over x^2*y^1/2
8)Simplify it and you would have x^5/8*y^-5/2 all over x^2*y^1/2
9)Then you would simplify common variables, x^5/8 over x^2 would turn into 1 over x^-11/8
10) Also y would simplify from y^-5/2 over y^1/2 to 1 over y^-3
The final answer is 1 over x^11/8*y^-3
Teamwork Problem
Team A can complete a job in 8 days while Team B can complete the job in 12 days. If Team A works alone for 3 days then has Team B join, how long will it take for the job to be complete?
Step 1: I started with seeing how much can be done in a single day.
Team A:
1/8 +1/8+1/8+1/8+1/8+1/8+1/8+1/8=8/8
1=the task
8=total number of days
1/8=how much work in a day
Team B:
1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12=12/12
1=the task
12=total number of days
1/12=how much work in a day
Step 2: Then I found a common denominator so I could see how quickly Team A and B could work together.
Team A 1 day: 1/8=3/24
Team B 1 day: 1/12=2/24
Team A and Team B 1 day: 2/24+3/24=5/24
Step 3: The next step was to add up the new fractions. I started with just A because they worked alone for three days.
Day 1: Team A 3/24=Total 3/24
Day 2: Team A 3/24=Total 6/24
Day 3: Team A 3/24=Total 9/24
Day 4: Team A and B 5/24=Total 16/24
Day 5: Team A and B 5/24=Total 19/24
Day 6: Team A and B 5/24=Total 24/24
Step 4: I also did a visual representation of this. There are 24 square and I'm using the same fractions for each day, 3/24 for Team A, 2/24 for Team B and 5/24 for Team A and Team B.
Team A can complete a job in 8 days while Team B can complete the job in 12 days. If Team A works alone for 3 days then has Team B join, how long will it take for the job to be complete?
Step 1: I started with seeing how much can be done in a single day.
Team A:
1/8 +1/8+1/8+1/8+1/8+1/8+1/8+1/8=8/8
1=the task
8=total number of days
1/8=how much work in a day
Team B:
1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12+1/12=12/12
1=the task
12=total number of days
1/12=how much work in a day
Step 2: Then I found a common denominator so I could see how quickly Team A and B could work together.
Team A 1 day: 1/8=3/24
Team B 1 day: 1/12=2/24
Team A and Team B 1 day: 2/24+3/24=5/24
Step 3: The next step was to add up the new fractions. I started with just A because they worked alone for three days.
Day 1: Team A 3/24=Total 3/24
Day 2: Team A 3/24=Total 6/24
Day 3: Team A 3/24=Total 9/24
Day 4: Team A and B 5/24=Total 16/24
Day 5: Team A and B 5/24=Total 19/24
Day 6: Team A and B 5/24=Total 24/24
Step 4: I also did a visual representation of this. There are 24 square and I'm using the same fractions for each day, 3/24 for Team A, 2/24 for Team B and 5/24 for Team A and Team B.
Reflection
I chose this problem because when I originally solved this problem I struggled with trying to figure out how to find how much each worked every day. I started trying to add 8 and 12 but that doesn't make sense because that will get me how many days if they worked separately on different jobs. As I worked it out with a partner, we used each others tecniques to understand how to get to the answer. Once I figured out that I needed a common denominator it was very simple. This can be used in multiple real life scenarios because it uses two different rates of people or things and tries to figure out the total time they will take to complete a job together
I chose this problem because when I originally solved this problem I struggled with trying to figure out how to find how much each worked every day. I started trying to add 8 and 12 but that doesn't make sense because that will get me how many days if they worked separately on different jobs. As I worked it out with a partner, we used each others tecniques to understand how to get to the answer. Once I figured out that I needed a common denominator it was very simple. This can be used in multiple real life scenarios because it uses two different rates of people or things and tries to figure out the total time they will take to complete a job together