Mixture questions with pounds involved

[jess.scalfano]

How do you solve questions such as I am going to mix one type of flour that costs 2.70 per pound with another type of flour that costs 1.95 per pound in order to make a 40 pound mixture worth $2.25 per pound. How many pounds of each type should I use?


I don't understand how to combine this information. Is it two equations? It seems there are too many variables to solve for the question asked. This is just an example problem of the same type of question I see over and over and can't solve.

[sen.sreeroop]

2.7 1.95
\ /
2.25
/ ? ?

Leaves you with the below -

2.7 1.95
\ /
2.25
/ 0.30 0.45

0.30 is derived from (2.25-1.95) and 0.45 is derived from (2.7-2.25).
Hence the ratio is the mix would be 0.30 : 0.45 => 2:3
So, they will be mixed as 2/5 * 45 = 18 lbs and 3/5 * 45 = 27 lbs resp.

[mschwrtz]

sen.sreeroop, I'm afraid that the formatting options might have let you down here. Your reasoning is correct, except that you assume 45 lbs of flour when you should read 40, but I'm not sure how clear you were able to make your diagram.

sen.sreeroop is approaching this mixture problem as a weighted average, often the most efficient approach if you have a good intuitive understanding. Well, what feels like intuition is really a well-developed sense of weighted averages.

You can usefully represent sen.sreeroop's reasoning on something like a number line or line segment (see page 107 of the Manhattan GMAT Word Translation Strategy Guide for examples).

1.95----------2.25---------------2.70

(The gap between 1.95 and 2.25)/(the gap between 1.95 and 2.70)=the portion of the mixture due to 2.70 flour. So the mixture is 30/75 1.95 flour, or 2/5 1.95 flour.

Why? The more 2.70 flour you have, the closer the weighted average will be to 2.70, and the more 1.95 flour you have, the closer the weighted average will be to 1.95.

We could begin this same approach with a more algebraic account:

(The gap between 1.95 and 2.25)x(the number of lbs of 1.95 flour)=(The gap between 2.70 and 2.25)x(the number of lbs of 2.70 flour)

[jinkala2]

total weight of the flour should be 40 pounds so

X1+X2 = 40 ---- equation 1

cost of first flour per pound 2.70
cost of second flour per pound 1.95

2.70(x1)+1.95(x2)=40*2.22 -----equation 2


solve both equations you will get the values of x1 and x2

[mschwrtz]

Yep, you can use a similar pair of equations for every mixture problem.