For the first one, think about how much they can do in one hour and together. If it takes, for example, 4 hours to do a job, then in one hour you can do 1/4 of the job, right?
So, in one hour, Jake can do 1/10th of the job. Betty can do 1/x of the job (i.e., she can do the whole job in x hours.) If they work together they can do 1/6 of the job. Another way of representing the hourly amount they can get done when working together is simply to add 1/10 + 1/x.
This gives . Using 30 as a common denominator:
, so x = 15.
Let's say he uses x pounds of the 20% wool fiber. Then the amount of wool fiber from that type of cloth or carpet or what-have-you is (.20)*x.
Now, if he used x pounds of the 20%, he must use 500-x of the 50% fiber (since he used 500 pounds total). The wool from the 50%-type can be expressed, then, as (.50)(500-x).
We also know that the total wool in the final, 500 pounds can be written as (.35)*500.
If we take the wool from the 20%, add it to the wool from the 50%, it should equal the wool in the 35% mixture.
.2x + 250 -.5x = 175.
Solving for x gives 250. So he uses 250 of one fiber, 250 of the other.
This makes sense since 35% is exactly halfway between 20 and 50. Thus, he should use half of each.