Question

Carl can paint a room 3 hours faster than Jennifer can. If they work together, they can complete the job in 2 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jennifer to complete this job on her own.

Answer 1

Okay I think I know the answer, I just know that my teacher will not accept the answer without the formula and steps.

Answer 2

which is confusing me

Answer 3

so u just don't know the formula

Answer 4

you could say that

Answer 5

I would use the formula
rate * time = work done
in this case, the work done is 1 room
so
rate_carl * t1 = 1 (room)
and
rate_jen * t2 = 1
where t1 is the time it takes carl and t2 the time for jen
carl is 3 hours faster than jen which means t1 is 3 hours shorter than t2
in other words t1= t2-3

Answer 6

Here is an example similar to the question. Maybe this will ring a bell.

Answer 7

i included the steps

Answer 8

They are using some "short cut" formula that is not easy to follow, but does work.
If we ignore why it works (which is not something I usually do), and just use it,
we write
\[ \frac{1}{t_1}+ \frac{1}{t_2}= \frac{1}{t_{total}}\]
where t1 is the time it takes for one person to do the job (working alone)
t2 the amount of time for the other person to do the job working alone
and t_total is the amount of time to do the job when working together

Answer 9

okay so I would plug 3 and 2 as the first two denominators?

Answer 10

They say
If they work together, they can complete the job in 2 hours
that means t_total is 2
we can put that into the formula
\[ \frac{1}{t_1}+ \frac{1}{t_2}= \frac{1}{t_{total}} \\
\frac{1}{t_1}+ \frac{1}{t_2}= \frac{1}{2}\]

Answer 11

they want to find Jen's time. that is obviously unknown, so let's assume she is t1, but we will use "x" for unknown. Put x in for t1
\[ \frac{1}{x}+ \frac{1}{t_2}= \frac{1}{2} \]

Answer 12

Now you have to figure out what this means
Carl can paint a room 3 hours faster than Jennifer can

what is the time t2 (Carl's time) ?

we are using x for jen's time, so we can say
t2 (carl's time) is 3 hours less than x

Answer 13

do you know how to write 3 less than x
in algebra?

Answer 14

3>x ?

Answer 15

3<x sorry

Answer 16

good guess, but that is not what they mean

what is 3 less than 10 ?

Answer 17

7

Answer 18

and 3 less than 8 ?

Answer 19

5

Answer 20

notice 3 less than 10 you figured out by doing 10-3
and for 3 less than 8 you did 8-3

what is 3 less than x?

Answer 21

-3x ?

Answer 22

use the same idea as when you do
3 less than 5
you write
5
then a - sign
then 3
5-3 =2
3 less than 5 is 2

now do 3 less than x

Answer 23

x-3

Answer 24

yes. it is a way of "thinking" that is useful.

so if jen takes x hours, carl takes x-3 hours (3 hours less than jen's x)

Answer 25

replace t2 with x-3 in the equation
\[ \frac{1}{x}+ \frac{1}{x-3}= \frac{1}{2} \]

Answer 26

do you have to solve this ? you get a quadratic equation that you have to factor.

Answer 27

ok i understand so I solve x+x-3=2

Answer 28

no, it's messier.
one way to proceed is to multiply both sides (and all terms) by x(x-3)
can you do that?

Answer 29

i think so let me try

Answer 30

sorry for being slow. if I was to multiply 2 by x(x-3) would i plug in 2 for the xs

Answer 31

sorry i dont know how to multiply each term

Answer 32

you write x(x-3) next to each term

Answer 33

notice you can simplify the first term because x divided by x "cancels"
and in the 2nd term (x-3)/(x-3) also cancels.

Answer 34

yes I see that, would the same apply for (x-3) over (x-3)

Answer 35

\[ \frac{x(x-3)}{x}+ \frac{x(x-3)}{(x-3)}= \frac{x(x-3)}{2} \]
or after simplifying
\[ (x-3) + x = \frac{x(x-3)}{2} \]

Answer 36

gotcha

Answer 37

can you simplify the left side x-3+x

Answer 38

2x-3?

Answer 39

yes, so you have
\[ 2x-3 = \frac{x(x-3)}{2} \]
if we multiply both sides by 2 we can get rid of the fraction on the right side
\[ 2(2x-3) = \cancel{2}\cdot \frac{x(x-3)}{\cancel{2}} \]
\[ 2(2x-3)= x(x-3) \]

I would "distribute the 2" on the left side
can you do that ?

Answer 40

4x-12

Answer 41

ok except 2*3 is 6 (not 12)

Answer 42

i feel so stupid sorry

Answer 43

\[ 4x -6 = x(x-3) \]
it's easy to make simple mistakes. you have to go slower and more carefully
(remember, you have to learn math and it's painful)

now distribute the x on the right side

Answer 44

4x-6 = x^2 -3x

Answer 45

now "move" the terms on the left side to the right side:
add -4x+6 to both sides

Answer 46

x^2 -7x +6 ?

Answer 47

ok, but it's an equation
\[ 0= x^2 -7x +6 \]
or
\[ x^2 -7x +6 =0 \]

Answer 48

okay so do i factor?

Answer 49

yes , and it does factor

Answer 50

(x-6)(x-1)

Answer 51

yes
but you should write the whole equation
(x-6)(x-1)= 0

you get
x=6 or x=1

x is the time it takes Jen to do the job.
carl is 3 hours faster. So , even though the math has a solution x=1 (as a possibility) we can rule it out, because carl's time can't be 1-3 = -2 hours
that leaves jen's time as 6 hours
and carl's time as 3 hours

Answer 52

so that is the solution?

Answer 53

yes

Answer 54

Okay thank you very much for dealing with me and helping.