Attetion all Math Geniuses!?

Question

I was working on one of my Algebra 1 worksheets when I encountered a problem that I could not figure out. The problem is below:

Miss Apple can grade her class papers in 3 hours while it takes her assistant 4 1/2 hours. If Miss Apple graded the papers for 1 hour and then left the rest of the job to her assistant to complete, how long will it take her assistant to finish grading the papers?

Thanks so much! Please try to list all the steps so it is easier for me to understand.

Answer 1

Let no. of papers = X

Miss Apple
rate = X / 3 papers / h

Assistant
rate = X / 4.5 papers / h

After 1 h , Miss A marks X / 3 papers
No. remaining = (2/3) X

Assistant
X / 4.5<-> 1 h
2X / 3<-->1 x (2X / 3) / (X / 4.5) h
= (2/3) x 4.5 h
= 3 h

It will take the assistant 3 h to finish.

Answer 2

Well look at it this way. It takes Miss A 3 hours to do all the papers. If she graded for 1 hour, that means that she graded 1/3 of the papers, right? Which means there are 2/3 of the papers left. So you multiply how long it takes the assistant (4.5 hours) times how much is left (2/3) to get the remaining time. That just so happens to be 3 hours.

Answer 3

Miss Apple takes 3 hours to do the entire paper, so she does a third of it every hour. So after one hour she has 2/3 of the paper left to grade. The assistant takes 4.5 hours to do the entire paper, which means that 2/3rds of the paper would take 3 hours to complete
i.e. 2/3 X 4.5 = 3

Answer 4

Mrs Apple completed 1/3 of the job so 2/3 of the job was left for the assistant.
Assistant can do 2/3 of the job in 2/3 of 4 1/2 = 3 hours

Answer 5

assume there are x papers to be graded.
Then Apple can grade x/3 papers per hour.
her assistant can grade x/4.5 papers per hour.

Thus if apple spends one hour grading, she grades x/3
and leaves the assistant with (x-x/3) = 2/3 * x papers left

thus we can see that the time the assistant takes is:

(2/3*x papers) * (4.5/x hours/paper) = 2/3 * 4.5 = 9/3 = 3 hours.


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Answer 6

If Apple can grade all (3/3) of the papers in 3 hours, then she can grade 1/3 of the papers in 1 hour. That leaves 2/3 of the papers left to grade.

Her assistant can grade all the papers in 4.5 hours. So it takes her assistant (2/3)*(4.5) hours to grade 2/3 of the papers.

(2/3)(4.5) = 3 hours

Answer 7

2/3 of the papers r left, so (2/3)*4.5= 3 hours

Answer 8

Say the amount of work required in grading one worksheet is x

If Miss Apple takes 3 hrs in doing x amount of work i.e in marking one full worksheet , than if she works for just 1 hr i.e 1/3 the time required for grading whle sheet ............ so the amount of sheet graded will also be 1/3 of the total sheet

So in one hour , worksheet graded = 1/3 * x = x/3

Amount of worksheet left to be graded = x- x/3 = 2x/3

Now if the assistant takes 4.5 hrs for marking whole sheet ie doing x amount of work.

so how much will he take for doing the work left by miss apple i.e 2x/3 of the worksheet

so if we say from assistants side :

Time required for x units of work = 4.5 hrs
Time required for1 unit of work = 4.5/x hours
Time required for 2x/3 units of work = ( 4.5/x) * (2x/3)
= (4.5 * 2)/ 3
= 9/3 = 3

So time required by the assistant for marking rest of the worksheet = 3 hrs

Answer 9

The ratio is 3hours to 4.5hours for all the papers.

1 hour so it would take the teacher 2 more hours.

2teacher hrs 3teacher hrs
------------------- = --------------------
x assistant hrs 4.5 assistant hrs

3x = 2*4.5 (cross multiply)
3x = 9
x = 3 hours

3 more hours.