Question

A man has two types of coffee beans. The first type sells for $5.20/lb, the second for $5.80/lb. How many pounds of the first type must be mixed with 9lb of the second type to produce a blend selling for $5.35/lb?

Answer 1

Hi there!. This is a system of linear equations.

Answer 2

So what you want to do is set up an equation that computes the weighted-average cost of the two types of beans. Do you know what a weighted average is?

Answer 3

No I am not so sure

Answer 4

Thank you for helping me! :)

Answer 5

OK. Well, you probably know what a regular average is, right?

Answer 6

A regular average of two numbers is, for example, (x + y)/2, or (x + y + z)/3. The number in the denominator is the number of items we're averaging over.

Answer 7

Imagine however, that we want an average over three things, but two are the same: x+x+y. The average is (x+x+y)/3, since we still have 3 items, even though two are the same.

We can rewrite this as (2x + y) / 3 -- and now we have a weighted average. It's the sum of each value, times its weight, divided by the total of the weights. Does that make sense? It means x has twice as much effect on the result as y.

Answer 8

Does that make sense so far?

Answer 9

Yes that makes sense!

Answer 10

Great! So, we have two kinds of beans, and they have two different costs. We know the weight -- in this case, literally a weight in pounds -- of one of the kinds, but not the other. We'll call the unknown amount "x". So the weighted average of the two is:

\[{ { (x * 5.20) * (9 * 5.80) } \over {x+9} }= 5.35 \]

Answer 11

x*5.20 is the value of the first bean, 9*5.80 is the value of the second bean, x+9 is the total weight of the beans. Divide them, and you get dollars per pound. We set that equal to what we *want* the dollars per pound to be, or 5.35. Does that make sense?

Answer 12

Oops, that was supposed to be a plus -- sorry, let me write it out again.

\[{ {(x*5.20) + (9*5.80) } \over x+9} = 5.35 \]

Answer 13

Oh that's okay!

Answer 14

In most of math, finding the right equation that represents your problem is the hard part. Solving the equation usually is the easy part. (Computers can usually do that for you, in fact)

Answer 15

Okay so would I simply solve for the equation you typed?? I really appreciate your help!

Answer 16

Yep, that's it! That will give you a value for x, which is the number of pounds of the other kind of bean. To solve the equation, a good first step is to multiply both sides by (x+9).

Answer 17

that looks like a good method. we can also reason as follows:
put x = pounds of first beans used, they cost
\[5.20x\]and 9 pounds of the other beans will cost
\[9\times 5.80=52.50\]for a total of
\[5.20x+52.50\]
if you add x pounds of beans to the 9 pounds of beans you will have
\[x+9\] pounds of beans, and you want this to be worth
\[5.35(x+52.50)\] so set them equal and solve that is write
\[5.20x+52.50=5.35(x+9)\] and solve for x

Answer 18

Thank you both so very much for your help!! Did you get an end result? On the first method I got around 3.65lbs but the second I got almost 30 lbs

Answer 19

i get x = 29

Answer 20

I get 27 pounds

Answer 21

i could have made a mistake

Answer 22

mathematica solved it as 27

Answer 23

oh well let me try again

Answer 24

I might have made a typo in the input though; here's what it says:
In[819]:= Solve[(5.20 x + (9*5.8))/(x + 9) == 5.35, {x}]

Out[819]= {{x -> 27.}}

Answer 25

Oh that makes sense! I just retried it and got 27 as well.! :)

Answer 26

if you start with
\[520x+5250=535(x+9)\] i think we can do it easily

Answer 27

great!

Answer 28

Thank you SO much for all of your time I really appreciate it!!!!

Answer 29

I hope it was helpful :-)

Answer 30

i still get 29
did i make a mistake in the formula?

Answer 31

It definitely was! :)

Answer 32

yeah, 9*5.8 is 5220, not 5250

Answer 33

ahhhhhhhh so it is!