1. First let’s find Hannah’s and Destiny’s speed of work. Since Hannah can paint a room in 12 hours, she can paint 1/12 of the room in 1 hour. Destiny can paint the same room in 15 hours, so in 1 hour she can paint 1/15 of it. If they are working together, then in 1 hour they can cover 1/12 + 1/15 of the room. Let’s say they need t hours to paint 1 whole room. Notice that the full job is equal to 1, one whole thing. Then the equation will look like:
t (1/12 + 1/15) = 1 (It can be written as t/12 + t/15 = 1, which is same thing).
To solve it we will do 1/12 + 1/15 = 3/20 first.
Now we have 3/20 t = 1. Multiplying both sides by 20 and dividing by 3 we get time t=20/3 hours. It is equal to 6 and 2/3 hours or 6 hours 40 minutes.
2. Let’s label s the time it would take Seth to paint the room by himself. Since it takes Ted twice as long, his time would be 2s. So in 1 hour Seth would cover 1/s of the room, and Ted would do 1/2s of it. Together they would cover 1/s+1/2s in 1 hour. We know that they can do the whole job, 1 whole room, in 3 hours, so we can write it down as
3(1/s + 1/2s)=1 <-- one whole room.
To add the fractions we should get them to the common denominator, which is 2s. So, 1/s=2/2s.
Now 1/s+1/2s becomes 2/2s+1/2s=3/2s.
So we get 3(3/2s)=1
9/2s=1 Multiplying both parts of the equation by s we get s=9/2 hours. It would take Seth 4.5 hours to do the job on his own.
3. In 1 hour inlet pipe can fill 1/6 of the pool, and drain pipe can empty 1/15 of the pool. Let’s say it will take t hours to fill 1 whole pool if they both are open. Then we can produce an equation
t(1/6 – 1/15)=1 Notice that since the second pipe removes water, we subtract 1/15.
1/10 t=1. Multiplying both parts of the equation by 10, we get t=10.
4. Let’s label Mikhail’s time M. So in 1 hour Alejandra weeds 1/15, and Mikhail covers 1/M of the garden.
10(1/15+1/M)=1
Divide both sides by 10: 1/15+1/M=1/10
Subtract 1/15 from both sides: 1/M=1/30
M=30 hours.
