# grade 8 fractions math help

##### New member
Jacques takes 3/4h to fill one shelf at the supermarket. Henri can fill the shelves in two-thirds Jacques’ time. There are 15 shelves. Henri and Jacques work together. How long will it take to fill the shelves? Justify your answer.

Thank you so much!

#### Dr.Peterson

##### Elite Member
Jacques takes 3/4h to fill one shelf at the supermarket. Henri can fill the shelves in two-thirds Jacques’ time. There are 15 shelves. Henri and Jacques work together. How long will it take to fill the shelves? Justify your answer.

Thank you so much!
I don't see this as requiring factoring (primarily). Please show us what you have tried, or at least what topics you have recently learned, so we can see where you need help and what sort of help will be useful to you. Did you not read this?

##### New member
I don't see this as requiring factoring (primarily). Please show us what you have tried, or at least what topics you have recently learned, so we can see where you need help and what sort of help will be useful to you. Did you not read this?
Hello! Sorry I didn't see that part in the guidelines. I think I was also trying to say fractions instead of factoring.

This is how I tried to solve it:

Since Jacques takes 3/4 hours to fill one shelf, for 15 shelves, it would take (3/4 hours) x (15 shelves) = 11.25 hours
Then, Henri takes 2/3 of Jacques time, so it would take (11.25 hours) x (2/3) = 7.5 hours
I then did 11.25 - 7.5 = 3.75 hours

Recently, I've been learning about basic calculations with fractions like adding and subtracting, multiplying and dividing. I have a bit of difficulty with word problems though, so I would appreciate the help. Thank you.

#### Dr.Peterson

##### Elite Member
Hello! Sorry I didn't see that part in the guidelines. I think I was also trying to say fractions instead of factoring.

This is how I tried to solve it:

Since Jacques takes 3/4 hours to fill one shelf, for 15 shelves, it would take (3/4 hours) x (15 shelves) = 11.25 hours
Then, Henri takes 2/3 of Jacques time, so it would take (11.25 hours) x (2/3) = 7.5 hours
I then did 11.25 - 7.5 = 3.75 hours

Recently, I've been learning about basic calculations with fractions like adding and subtracting, multiplying and dividing. I have a bit of difficulty with word problems though, so I would appreciate the help. Thank you.
Thanks. You did that part well, up to the subtraction at the end.

My next question is, have you seen any "time to do a job" problems before, perhaps without fractions? This is commonly taught as part of algebra, but doesn't require it.

The basic idea is that this is really a rate problem. You need to find the rate (in shelves per hour, or perhaps supermarkets per hour) for each person, and then add them, because, for example, if I can do something at a rate of 2 tasks per hour, and you can do it at 3 tasks per hour, then together (if we don't interact to help or hinder one another), together we will do 5 tasks per hour.

Does any of that sound familiar? Try doing whatever you can with this idea.

#### Otis

##### Elite Member
Hello PHMTY. We use reciprocals, when dealing with 'combined work rates' type problems. See this page, for some worked examples.

For example, if it takes me 15 minutes to complete a task, then I complete 1/15th of the job per minute. (1/15 is the reciprocal of 15.) If it takes you 10 minutes to complete the same task, then you complete 1/10th of the job per minute. (1/10 is the reciprocal of 10.) We add the reciprocals, to find the fractional amount of the task completed per time unit when we work together.

1/15 + 1/10 = 1/6

In other words, working together, we complete 1/6th of the task per minute. We consider the number 1 to represent 100% of the task. Therefore, 1/6th of the job done per minute means that it takes us 6 minutes, working together. (6 times 1/6 equals 1.)

Go through some worked examples, at the link above. See if that helps. Post your attempt, if you'd like more help.

##### New member
Thanks. You did that part well, up to the subtraction at the end.

My next question is, have you seen any "time to do a job" problems before, perhaps without fractions? This is commonly taught as part of algebra, but doesn't require it.

The basic idea is that this is really a rate problem. You need to find the rate (in shelves per hour, or perhaps supermarkets per hour) for each person, and then add them, because, for example, if I can do something at a rate of 2 tasks per hour, and you can do it at 3 tasks per hour, then together (if we don't interact to help or hinder one another), together we will do 5 tasks per hour.

Does any of that sound familiar? Try doing whatever you can with this idea
Ok that makes a lot more sense. I tried starting all over like this:

For Jacques, it would be 1 shelf per 3/4 hours, so (1)/(3/4) = 1.33 shelves/hour
For Henri, 1 shelf per 1/2 hours, so (1)/(1/2) = 2 shelves/hour
So 1.33 + 2 = 3.33 shelves/hour altogether.
Finally, for 15 shelves it would be 15/3.33 = 4.5 hours?

##### New member
Hello PHMTY. We use reciprocals, when dealing with 'combined work rates' type problems. See this page, for some worked examples.

For example, if it takes me 15 minutes to complete a task, then I complete 1/15th of the job per minute. (1/15 is the reciprocal of 15.) If it takes you 10 minutes to complete the same task, then you complete 1/10th of the job per minute. (1/10 is the reciprocal of 10.) We add the reciprocals, to find the fractional amount of the task completed per time unit when we work together.

1/15 + 1/10 = 1/6

In other words, working together, we complete 1/6th of the task per minute. That means it takes us 6 minutes, working together. (6 is the reciprocal of 1/6.)

Go through some worked examples, at the link above. See if that helps. Post your attempt, if you'd like more help.

Thank you! I'll take a look at the examples later as well.

#### Otis

##### Elite Member
Oh, I forgot to mention the following. Don't convert your fractions to decimal form, as you work. Use only the fractional forms. That way, after adding up the fractional amounts of the job done by each person, you may simply take the reciprocal of the total.

$$\;$$

##### New member
Oh, I forgot to mention the following. Don't convert your fractions to decimal form, as you work. Use only the fractional forms. That way, after adding up the fractional amounts of the job done by each person, you may simply take the reciprocal of the total.

$$\;$$
ohh ok thanks I'll keep that in mind

#### Otis

##### Elite Member
(1)/(3/4) = 1.33

15/3.33 = 4.5 hours?
4.5 hours is a correct answer. You method is okay. But, if you're expected to use exact arithmetic (instead of calculator approximations), there might be an issue with your work on a class assignment.

1/(3/4) is not 1.33 (it's actually 4/3).

1.33 is only a decimal approximation for 4/3.

1.33 is exactly 133/100.

Likewise, 15/3.33 is not 4.5

15/3.33 is actually 500/111.

##### New member
4.5 hours is the correct answer. You method is okay. But, if you're expected to use exact arithmetic (instead of calculator approximations), there might be an issue with your work on a class assignment.

1/(3/4) is not 1.33 (it's actually 4/3).

1.33 is a decimal approximation for 4/3.

15/3.33 is not 4.5 (it's actually 45/4).
ah so for the calculations, instead of changing them to decimal form, I would keep the fractions and go like this

4/3 + 2 = 10/3 shelves per hour
then for 15 shelves: (15) / (10/3) = 45/10 hours

#### Otis

##### Elite Member
Very good. Here's another way to go.

Mr. J fills a shelf in 3/4 hour.

15 × 3/4 = 45/4

Mr. J completes the job in 45/4 hour.

Mr. H requires 2/3 the time Mr. J does, to complete the job.

2/3 × 45/4 = 15/2

Therefore, Mr. J completes 4/45ths of the job per hour, and Mr. H completes 2/15ths of the job per hour. We combine those reciprocals.

4/45 + 2/15 = 2/9

Working together, they complete 2/9ths of the job per hour, so it takes them 9/2 hour to finish.

9/2 = 4.5

##### New member
Very good. Here's another way to go.

Mr. J fills a shelf in 3/4 hour.

15 × 3/4 = 45/4

Mr. J completes the job in 45/4 hour.

Mr. H requires 2/3 the time Mr. J does, to complete the job.

2/3 × 45/4 = 15/2

Therefore, Mr. J completes 4/45ths of the job per hour, and Mr. H completes 2/15ths of the job per hour. We combine those reciprocals.

4/45 + 2/15 = 2/9

Working together, they complete 2/9ths of the job per hour, so it takes them 9/2 hour to finish.

9/2 = 4.5

haha ok got it! thank you so much it took me so long to figure this out

#### jonah2.0

##### Full Member
Beer induced filibuster follows.
haha ok got it! thank you so much it took me so long to figure this out
Hard earned gains are not easily forgotten. Basic principle of no pain no gain mantra. One has a tendency to remember and value them permanently.

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#### HallsofIvy

##### Elite Member
Jacques takes 3/4h to fill one shelf at the supermarket. Henri can fill the shelves in two-thirds Jacques’ time. There are 15 shelves. Henri and Jacques work together. How long will it take to fill the shelves? Justify your answer.

Thank you so much!
2/3 of 3/4 is, of course, 2/4= 1/2 hour. Jacques, filling one shelf in 3/4 hour is working at a rate of 4/3 shelves per hour. Henri, filling one shelf in 1/2 hour, is workingt at a rate of 2 shelves per hour. When people work together, their rates add. So Jacques and Henri, working together fill 4/3+ 2= 4/3+ 6/3= 10/3 shelves per hour.

Together they can fill 15 shelves in 15/(10/3)= 15(3/10)= 9/2 or 4 and a half hours.

#### eddy2017

##### Elite Member
this problem drew my attention.
i tried to solve it using another way and i'm bringin' it before you to approve of it or destroy it.
I did exactly what the poster did until i found how much hours each of them work.
Jason 3/4 * 15 =45/4=11.25 hrs
Henry 2/3 * 11.25 = 7.5 hrs
here i veered direction and went this way, (though might be the wrong way!)

i set up this formula to find out combined rate of work
t/A + t/B =1

where t = the amount of time working together to accomplish the task
i let Jason be A
I let Henry be B
So, again, here's the formula
t/A + t/B =1
i'll plug in what i have
t/11.25 + t/7.5 =1
i will round up and down to make finding a common denominator easier
t/11 + t/8 =1
common denominator of 8 and 11 =88

___ + _____
88 88

now i will have to multiply both fractions for a number that yields 88.
t/11 (8/8)= 8t/88

t/8 (11/11) =11t/88

so now i can add this fractions

8t/88 + 11t/88 = 19t/88

19t/88=1
solvin' for t
88( 19t/88) = 1 * 88
19t=88
19t/19 =88/19
=4.6 which is pretty close to the result he got.
if Jason and Henri work together they will fill the shelves in approximately 4.6 hours.

#### Dr.Peterson

##### Elite Member
i'll plug in what i have
t/11.25 + t/7.5 =1
i will round up and down to make finding a common denominator easier
t/11 + t/8 =1
No. Rounding means you'll be solving a different problem. It's pure luck that your answer is fairly close to the right answer.

There are several ways to make it easier without rounding.

One is to just multiply both sides of the equation by 11.25*7.5, which is the actual LCM. You get 7.5t + 11.25t = 84.375. Add and divide, and you get t = 84.375/18.75 = 4.5 hours.

Another way is not to use decimals at all; the two times are 11 1/4 = 45/4 and 7 1/2 = 15/2, so the equation is (4/45)t + (2/15)t = 1. The LCD is 45, so we multiply by that and get 4t + 6t = 45, so that 10t = 45 and t = 45/10 = 4.5 again.

#### eddy2017

##### Elite Member
awfully good!.. thank you for rectifying, Doc.

t/11.25 + t/7.5 =1
multiply both sied by 11.25*7.5

11.25*7.5( t/11.25)+t/7.5 = 11.25*7.5(1)
7.5t+ 11.25 t= 84.375
18.5t=84.375
18.5t/18.5 =84.375/18.5
t=4.56081081
t=4.5
wow!

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#### Dr.Peterson

##### Elite Member
7.5t+ 11.25 t= 84.375
18.5t=84.375
Please check that. What is 7.5 + 11.25?

t=4.56081081
t=4.5
I showed you the correct work, completely. It gives the exact answer, not one you have to round (incorrectly, even!) at the end.