Question

Rate Problem: Jeff and Lucy have been asked to wash their mom’s minivan. It takes Jeff 2 hours to wash the van by himself, and it takes Lucy 1.5 hours to wash the van by herself. How long will it take Jeff and Lucy to wash the van if they work together?
a.What equation is used to solve this problem? What does each variable represent?
b.Identify the given information.
c.Enter the given information into the equation.
d.Solve the equation.
e.State the solution.

Answer 1

Does anyone know?

Answer 2

Ah, the classic work-rate problem . . .

Answer 3

Yes. :)

Answer 4

There are different ways to set up the equation, so the 'right' one will depend on what your teacher prefers.
I like something like this

\(\large \frac{1}{A}+\frac{1}{B}=\frac{1}{T}\)

Where A is the time one takes to do a job, B is the time the other takes, and T is the total time of them working together.

Answer 5

(It's similar to finding the equivalent resistance in a parallel electrical circuit, ... but you'll learn that later.)

Answer 6

Hmmm...okay! let me try it, and you can check it it, ok?

Answer 7

Would you do Jeff=1 van/2 hrs, lucy=1 van/1.5 hours?

Answer 8

Yes.

Answer 9

Would you combine it together?

Answer 10

Yep. If they are working together, then you add their rates together to get the total rate.
Remember that the rate is in units of vans-per-hour, and you're looking for the amount of time, so be careful.

Answer 11

Would you multiply 2 and 1.5 hours together? And you get 3, Then you would set it up like...3t/2, 3t/1.5?

Answer 12

Hmmm... Not sure what you mean. Are you just trying to add the fractions?

Answer 13

Yeah, isn't that what you do?

Answer 14

Yes.

Answer 15

So..would you get 6/7 hours for the answer? Because that is what I got?

Answer 16

That is exactly right, good job!
That works out to be about 51 and a half minutes. That makes sense given that One can do it in 1.5 hours and has help to get it done quicker than that.

Answer 17

Ok, thx so much! :D

Answer 18

what about this one?

Answer 19

Mixture Problem: Addison is mixing hummingbird food to fill her feeders. Ideally, the hummingbird food should contain 25% sugar. Addison remembers that she has 8 ounces of leftover hummingbird food that contains 35% sugar. She decides to mix some diluted food to add to the leftover food. If the diluted food contains 10% sugar, how much should she add to the food with 35% sugar to get the ideal hummingbird food with 25% sugar?
a.What equation is used to solve this problem?
b.Identify the given information. How can you represent each food mixture with an expression? What amount will you represent with a variable?
c.Enter the given information into the equation.
d.Solve the equation.
e.State the solution.

Answer 20

Do you know?

Answer 21

Yeah, these are pretty easy too once you get used to them.

Answer 22

Yeah, I don't understand how to solve some of these.

Answer 23

A couple things to think about to get this right:
1) The total amount of sugar on the left side of your equation has to be equal to the amount of sugar on the right side.
2) The amount of sugar = ounces of food × %concentration
3) The total volume of food on both sides of the equals sign also must be the same.

Answer 24

Okay...hmm.

Answer 25

The equation will basically look like this:

Ax+By=Cz
A=concentration of first mixture
x=volume of first mixture
B=concentration of second mixture
y=volume of second mixture
C=concentration of final total mixture
z=volume of total mixture.

Since the volumes on both side have to be the same, z = x+y

Answer 26

Let me try it, and I will get back to you.

Answer 27

Would you do it somewhat luike thuis?
8(0.35) and x(0.10)

Answer 28

Yes, I would. :-)

Answer 29

So, then tyou wpould what?

Answer 30

Add those together and set it equal to what you want the final total mixture to be.

Answer 31

so would you get 0.45?

Answer 32

No, don't add the concentrations, add the amounts of sugar.
8(0.35) oz is the amount of sugar in one mix, x(0.10) is the amount of sugar in the other.
Add 8(0.35)+x(0.10) = total amount of sugar.

Answer 33

So how ould you soilve for x?

Answer 34

x also appears on the other side of the equals sign. Remember that the total volume has to be the same. You're starting with 8 oz and adding x oz, so the total volume is (8+x).

Answer 35

Oh, then what?

Answer 36

Given that you want the final concentration to be 25%, what is the final total amount of sugar?

Answer 37

Umm, 25/8 is 3.125. 8+3.125=11.125?

Answer 38

How did you get the 25/8?

Answer 39

I thought we had to divde? or do we do like this 25+8

Answer 40

Remember this:

\(\large Ax+By=Cz\)

A=concentration of first mixture = 0.10
x=volume of first mixture = ?
B=concentration of second mixture = 0.35
y=volume of second mixture = 8 oz
C=concentration of final total mixture = 0.25
z=volume of total mixture = x+y = x+8.

Altogether, that is \(\large (0.10)x + (0.35)(8 oz) = (0.25)(x+8 oz)\)

Answer 41

Oh, okay! A little more help?

Answer 42

I think you can take it from here. Simplify and solve normally.

I'm going to take a break for dinner.

Answer 43

Ok, thx for your help!