algebra Math Problem?

Question

Henry can paint a kitchen in 20 hours and his assistance can do it in 30 hours. How long will it take them together?


please show your work and how things are done step by step

thanks!

Answer 1

Hi,

To solve this you will make a fraction for each person. Let's say they will work together for x hours to do this painting. So your fractions will work like this. Henry could do all the painting in 20 hours, so he could do 1/20 of the job in 1 hour,2/20 in 2 hours, 3/20 in 3 hours, or x/20 in x hours. Likewise his assistant could do all the painting in 30 hours, so he could do 1/30 of the job in 1 hour,2/30 in 2 hours, 3/30 in 3 hours, or x/30 in x hours. So in x hours they could do x/20 and x/30 of the job. Since they finish the painting, they have completed 1 job. So their equation is:

x/20 + x/30 = 1

To solve this, find the least common denominator and multiply every term including the 1 by that number. Since the LCD is 60, here, multiply everything by 60.

60(x/20) + 60(x/30) = 60(1)
3x + 2x = 60
5x = 60
x = 12

They can do the painting in 12 hours working together.

Notice the time working together is always less than any single person's fastest time. It is NOT the average of their times. If Henry can paint the room all by himself in 20 hours, why would having an assistant slow him down so together they'd take 25 hours?

If there were 3 people that could do a job in 15,12 and 10 hours alone, how long would it take all 3 working together?
Your equation would have 3 fractions on the left.

x/15 + x/12 + x/10 = 1

LCD is 60 again so multiply everything including the 1 by 60.

60(x/15) + 60(x/12) + 60(x/10) = 60(1)
4x + 5x + 6x = 60
15x = 60
x = 4

It would take them 4 hours working together.

While these 2 problems came out to "nice" numbers, don't be surprised if you get some ugly fractions or mixed numbers for answers. These don't always work out evenly.

Basically the pattern is

time together___time together
------------------_+_------------------- = 1
time alone A___time alone B

(underlines are only for spacing purposes)


If John can do a job in 10 hours and Mary can do the job in 6 hours, how long will it take them to finish once Mary starts if John starts 1 hour before Mary?

In this case, they didn't both work x hours. John works x+1 hours, so that will be his numerator. Mary still works x hours, so her fraction is normal, with x on top. The equation is

(x+1)/10 + x/6 = 1

LCD is 30, so multiply every term including 1 by 30.

30(x+1)/10 + 30(x/6) = 30(1)

3(x+1) + 5x = 30
3x + 3 + 5x = 30
8x + 3 = 30
8x = 27
x = 27/8 or 3_5/8 hours <== This is their time working together.
John actually worked 4_5/8 hours.

Last example:

If pipe A can fill the swimming pool in 10 hours, pipe B can fill it in 8 hours, but drain C can empty it in 12 hours, find how long it would take to fill the pool completely using both A and B if they mistakenly left drain C open. Since C is not helping to do the work but is actually subtracting from what A and B do, its fraction will be subtracted instead of being added.

x/10 + x/8 - x/12 = 1

LCD is 120 so multiply everthing by 120

120(x/10) + 120(x/8) - 120(x/12) = 120(1)
12x + 15x - 10x = 120
17x = 120
x = 120/17 or 7_1/17 hours

It would be nice if someone figures out to close ALL the pipes and the drain now to keep the water in.

I hope that gives you a clear picture of how to do these.

Answer 2

For these you do 1/x + 1/y = 1/z where z is your answer, is what how I remember.
The logic is that in 1 hour, Henry will have done 1/20 of the kitchen and the assistant will have done 1/30 of the kitchen.
Together they've done 1/20 + 1/30 = 3/60 + 2/60 = 5/60 = 1/12.
If in 1 hour they've done 1/12 of a kitchen, it will take them 12 hours to complete it. (Even if you don't get 1 for the numerator, you still just flip that fraction to get the time).

Answer 3

For Henry:
20 hours = Total work done.
1 hour = 1/ 20 of work done.

For Assistant:
30 hours = Total work done.
1 hour = 1/ 30 of work done.

When two people work together:
1/ 20 + 1/ 30 = [3 + 2]/ 60 ( work done per hour).
1/ 20 + 1/ 30 = 5 / 60
To get all the work done, you must have a full unit.
5 / 60 * 12 = 1. (5 * 12 gives 60).
Answer: 12 hours.

Answer 4

henry rate= 1/20 kitchen per hour
assistance rate= 1/30 kitchen per hour
total rate= 1/20+1/30= 1/12 kitchen per hour
therefore total time taken when both work together=12 hours

Answer 5

Step 1. If Henry Does the job in 20 hours, that means he can do 1/20th part of the whole job during 1 hour. Correspondingly, his assistant can do 1/30th part of the job in an hour.
Step 2. Together they can do (1/20+1/30)th part of the whole job, i.e. in 1/12th part of the job in 1 hour.
Step 3. If they can do the twelfth part of the job in 1 hour, it means they will complete the whole job in 12 hours.

Answer 6

I'm not sure if your teacher will accept this explanation but I can guarantee you the answer is correct.

Henry can work 1.5x faster than his assistant since 30/20=1.5. If we suppose that his assistant works at a rate of 1, the two of the them will work at a rate of 2.5 (because of 1+1.5). If it took his assistant 30 hours to finish his job at a rate of 1, it will take them both 12 hours since 30/2.5=12.

Answer 7

hmmm .. "unit-analysis" might be helpful

[sorta think along teh lines of "miles per hour" only this time we're talking "kitchens/hour"]

let "x" be the number of hours required to paint a kitchen together"
let "k" = kitchen
let "hr" = hour

x(hr) * ( 1k/20hr +1k/30hr) = 1 k

[ multiplying thru and "taking care of the units"
gives us: ]

(x(k hr) / 20hr ) + (x (k hr) / 30hr ) = 1k
xk / 20 + xk / 30 = 1k [factor out the "k"
x / 20 + x / 30 = 1

[ multiply both sides by 20 * 30 = 600 ]

600x / 20 + 600x / 30 = 600
30x + 20x = 600
50x = 600
x = 12

12 hours is your answer ...

I apologize for being extremely verbose, but you asked for the "show your work" stuff so I did in all its gory detail

[yes, you can probly leave out a few of the steps, but in fact this is EXACTLY how it's done]

Answer 8

x/20 +x/30 =1
Multiply by 60
3x+2x = 60
5x = 60
x = 12 hours

These type of problems can also be done by product over sum as follows: 20*30/(20+30) = 600/50 = 12