Solve Ratio Word Problem: Add 1.6667 lbs Barley to Mixture

[LearningMath]

Homework Statement

A 5 pound mixture of barley and oats is 4/5 oats by weight.

How much barley in pounds must be added to the mixture to make it 3/5 oats by weight?

Homework Equations

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The Attempt at a Solution

I honestly cannot figure this problem out. The book gives the answer and writes out the steps, but it does not help me one bit. The book does not explain how it arrived at the variables that it uses and, more importantly, I feel like I couldn't take its solution and understand it enough to use on any related type of problem. Here is the book's answer and basic steps:

A: 1 and 2/3 pounds [1.66667 lbs].

First it converts the mixes into percents - original = 80% oats/20% barley; desired = 60% oats/40% barley. Then,

i. (.8)(5)/5+b = 60%
ii. (.8)(5)=.6(5+b)
iii. 4 = 3 + .6b
iv. b= 1/.6
v. b = 1 & 2/3 [1.6667] pounds of barley must be added. Please explain how to solve this type of problem. I can see the book's work and understand all the variables, but it doesn't make any sense to me. That is, I couldn't set up the solution to a new, similar problem.

Thanks for the help!

 

[symbolipoint]

The addition of barley will dilute the fraction of the oats. You know that you want a result of 3/5 oats by weight.

Consider these expressions:

Amount of oats = 5 pounds x (4/5)
Amount of resulting mixture = 5 pounds + w
w = unknown pounds of barley to add.

The rest I withhold so that you can try to continue.

 

[LearningMath]

Ok, I am not being facetious or lazy, but I really don't know how to solve this one. I've spent a while on it and it's just not clicking. When I tried setting up your equation I got -1 (4=5+w)!, which is clearly not right.

What (I think) I know is that there is an 80/20 split between oats and barley (4 oats/ 1 barley = 5 lbs mix) and we're looking for a 60/40 division (4 lbs oats / x pounds barley) via dilution.

I just don't know how to set up an equation to solve for that. Please advise.

 

[HallsofIvy]

You are almost there! Since you want to find out how many pounds to add, instead of letting "x" equal the number of pounds of barley in the new mixture, let x be the number of pounds added to that original 1 pound of barley so that your second formula is (4 lbs oats/1+ x pounds barley)= 4+ 1+ x= 5+ x. 40% of 5+ x is .4(5+ x)= 2+ .4x so you want 1+ x= 2+ .4x. Solve that equation for x.

 

[symbolipoint]

Your work (post #1), mostly looks good and you found the correct value but understand these:

You mishandled grouping symbols and displayed order of operations (you seem to know what you are doing and must have used the correct form on paper, but some doubt). This WILL become a problem if you do not learn to manage it correctly. First error is found in your first step.

You could do your example using simple ordinary fractions; no need in this case to convert to decimals or percents.

 

[LearningMath]

Hi all!

You are almost there!

Thanks for the encouragement - math frustrates me sometimes.

Anyway, sorry for the delayed response and thanks for the help! I've come up with an equation that produces the right answer, but it doesn't look exactly like the book's nor your's, Halls.

Since you want to find out how many pounds to add, instead of letting "x" equal the number of pounds of barley in the new mixture, let x be the number of pounds added to that original 1 pound of barley

This has made a lot clearer in terms of how everyone's arrived at these variables. Here's my equation, please check it as I'm still not sure if I'm only right coincidentally:

i) (3/2)=(4/[1+x])
ii) cross multiply and get -> 3+3x=8 or 3x = 5
iii) 3x=5 = x =1.6667 (which is the answer the book gives!)​

Ok, I think that works but could someone confirm that this is a way to set it up?

Once I finally got that X = # of lbs added to barley, like Halls said (and everyone else was hinting at), and not X = total # of lbs, that has helped. Hopefully this way works...

Thanks a lot! I asked my a couple friends and they all told me to just stop trying to be a farmer...

EDIT: It's probably obvious to yall, but I put 3/2 in step i b/c it's the final ratio (oats:barley) the problem asks for...