Question

HELP!! ALGEBRA FRACTIONS

Answer 1

\[\frac{ 1 }{ 8} + \frac{ 1 }{ 6 } + \frac{ 1 }{ x } = \frac{ 1 }{ y }\]

Answer 2

this is the original problem. i set the equation where x is for joseph and y is for the total of time they all take

Hayden is a manager at a landscaping company. He has 3 workers to landscape an entire park, Cody, Kaitlyn, and Joseph. Cody can complete the project in 8 hours. Kaitlyn can complete the project in 6 hours. Joseph is new, so no one knows how long it will take him.

Hayden assigns all of them to complete the park together. Explain to Hayden how this project can tell him how long it would take Joseph to complete the project if he worked by himself. Use complete sentences.

Answer 3

I would take the reciprocal of both sides and just get rid of the fractions

Answer 4

to make all numerator the same right?

Answer 5

LCD would be 24xy am i right?

Answer 6

or LCM im not sure which one is it lol

Answer 7

You don't have to do that in this case since all the numerators are 1.

Answer 8

yes but dont i need the denominators to be equal in order to solve?

Answer 9

\[(\frac{ 1 }{ 8} + \frac{ 1 }{ 6 } + \frac{ 1 }{ x } )^{-1}= (\frac{ 1 }{ y })^{-1} \rightarrow 8 +6+x=y\]

Answer 10

You only have one equation and twon unknowns, so it can't be solved. How did you come up with the equation? Was it given to you or is that what you think it needs to be?

Answer 11

thats what i came up with.. so it cant be solved?

Answer 12

the original problem is a word problem

Answer 13

I think the question can be solved, but not by your equation. We need to come up with a different one. The word prolem is weird, bc I feel like the answer woud be just let the guy do it himself and time him

Answer 14

i know... i honestly think this problem needs two equations

Answer 15

This is worded terribly. We have no base line to compare Joseph. We don't know his work ethic so we can't write an equation for him. Is that the entire problem you posted?

Answer 16

i think it it will be 1/8 + 1/6 = 1/y then when we find y we add 1/8 + 1/6 + 1/x = 1/y. we substitue y for the answer we got before and we solve for x to find josephs timing?

Answer 17

i don't know if i explained myself right

Answer 18

That wont work because if two terms equal 1/y then three terms won't equal 1/y

Answer 19

when y is not the total of all of them but of 2 of them

Answer 20

I think you have to set up ratios. I'd write an equation for Cody and Kaitlyn first and relate one to the other

Answer 21

FOr instance, Kaitlyn does it 3/4 as fast as Cody.

Answer 22

mmm is confusing huh. a real math problem

Answer 23

i know i was thinking about it too but we don't know that that would be like a guess

Answer 24

i think i got it mm let me write it down

Answer 25

no nvm...

Answer 26

\[C+K+J=t\]Cody, Kaitlyn, Joseph's time = total project time. we know \[{3 \over 4}C=K\]so the equation becomes \[C + {3 \over 4}C+J=t\]\[{7 \over 4}C+J=t\]

Answer 27

where do the 7/4 comes from?

Answer 28

I added C to 3/4 C. So 4/4 + 3/4 = 7/4

Answer 29

I'm thinking on where to go from here still. This is kinda tricky

Answer 30

ohh i see i see and ummm it i tricky. is complicated

Answer 31

idk man... this is breaking my head. backward i think it would be even more complicated

Answer 32

question is it poissible to add 1/8 + 1/6 + 1/x?

if so x would be the total, then we would set up an equation 1/x - 1/8 - 1/6 = 1/J substituting x for the answer we got before

Answer 33

idk that what i came up with. but idk if its possible to add 1/8 + 1/6 + 1/x

Answer 34

x would be the total of three of them and J the total of joseph..?

Answer 35

I see what you're thinking. But why are you putting 1 over everything

Answer 36

because they need to complete 1 project and 8, 6, and x is the amount of time you know? 1/8 i project in 8hrs 1/6 is 1 project in 6hrs we need to find how long does joseph takes to get done 1 poject 1/x

Answer 37

I would let the rates at which they work be denoted \(c, k, j\).

Answer 38

We know \[
8c = 6k
\]

Answer 39

We assume that they work at constant rates, and when they work together the combined rate is just the sum of their rates (so they don't get in each other's way). It's not completely realistic but for this problem it will do.

Answer 40

Suppose it takes \(t\) hours for them to finish when working together.
This means we have: \[
t(c+k+j) = 8c=6k
\]

Answer 41

but then what would J be? if we were to substitue values we woudlnt eb able to substitue J

Answer 42

We don't actually want \(j\). We want \(n\) (number of hours) such that: \[
nj = 8c=6k
\]

Answer 43

And weirdly enough, we're supposed to get \(n\) in terms of \(t\).

Answer 44

n in terms of t? that doesn't sound right

Answer 45

Well, the only information that needs to be disclosed, according to the question, is \(t\) the time it takes for all three of them to complete the landscaping.

Answer 46

\[
k = \frac{4}{3}c
\]And \[
j = \frac{8}{n}c
\]

Answer 47

So we have: \[
t\left(\frac{8}{n}c+\frac{4}{3}c+c\right) = 8c
\]

Answer 48

i see i see/ so next step would be what..? distributive property?

Answer 49

so you understand what I've done so far?

Answer 50

yes kinda. i just dont get why it all equals 8c in your equation

Answer 51

Okay, so let's introduce units then...

Answer 52

Let \(L\) be the unit representing landscaping projects. Also \(h\) is hours.
The speed of people's work is measured at a rate of \(L/h\).

Answer 53

yes thats better

Answer 54

It takes Cathy \(8h\) to complete \(1L\) so she works at a rate of \((1/8) L/h\)

Answer 55

We could say \(k = 1/8\).

Answer 56

yes i understand that part and it takes cody 6hr so its 1/6 right? and im guessing joseph is 1/x

Answer 57

Yeah, for her it is \(1/6\)

Answer 58

or 1/h

Answer 59

We say Joseph is \(1/x\)

Answer 60

ok 1/x. what about all of them together?

Answer 61

However, if we want an actual equation... we would say this: \[
1L = k(6h) = c(8h) =jx
\]

Answer 62

ohh now i get it now i get it

Answer 63

And the final equation is that if it takes \(t\) time for all of them to complete together, then we say: \[
t(k+c+j) = 1L = (8h)c
\]

Answer 64

Now, from here on out we can not include units since we don't need them.

Answer 65

ok so what would be th enext step int he eqaution?

Answer 66

\[
t\left(\frac{8}{n}c+\frac{4}{3}c+c\right) = 8c
\]

Answer 67

Simplify

Answer 68

okok hold on i got a notebook with me

Answer 69

wait. im confuse now. how do we simplify that?

Answer 70

Add like terms.

Answer 71

What is \(3a + 6a\)?

Answer 72

9a

Answer 73

so i add whats in the parenthesis?

Answer 74

Yes

Answer 75

\[\frac{ 12 }{ 3n }c\]
Maybe?

Answer 76

There should be \(n\) in the numerator as well.

Answer 77

ohh okok hold on

Answer 78

so \[\frac{ 28n }{ 3n }c\]

Answer 79

??*

Answer 80

Nope, it would help to see your work.

Answer 81

\[\frac{ 8 }{ n}c (\frac{ 3 }{ 3 }) + \frac{ 4 }{ 3 } (\frac{ n }{ n }) + c\]

Answer 82

thats the parenthesis. i multiplied by the reciprocal.. ?

Answer 83

@wio

Answer 84

Okay first thing is first...

Answer 85

\[
\frac 8n+\frac 43
\]

Answer 86

reciprocal?

Answer 87

No, just add them

Answer 88

okok 12/3n

Answer 89

\[
\frac ab +\frac cd = \frac{ad+bc}{bd}
\]

Answer 90

\[\frac{ 24 + 4n }{ 3n }\]

Answer 91

nono sorry got confuse

Answer 92

is it like that?

Answer 93

@wio ?

Answer 94

sorry if im bugging, is the last question for a test and ive been for 2hrs on it already..

Answer 95

let me see

Answer 96

Now you must add \(1\) to the fraction.

Answer 97

add 1 to the fraction? i dont understand

Answer 98

\[\frac{ 24 + 4n }{ 3n } + 1c\] ?