Problem Solving

[shashwats]

Q:Each of the products produced yesterday was checked by x or y. 0.5% of the products checked by x are defective and 0.8% of the products checked by y are defective. If the total defective rate of all the products checked by x and y is 0.75%, what fraction of the products was checked by y?

[sairakarim07]

Q:Each of the products produced yesterday was checked by x or y. 0.5% of the products checked by x are defective and 0.8% of the products checked by y are defective. If the total defective rate of all the products checked by x and y is 0.75%, what fraction of the products was checked by y?

This is a problem of weighted average so we can use the following shortcut similar to alligation method (or based on inverse proportion method)
X/Y =(.8-.75)/(.75-.50) [here we need the difference of individual percentage and final percentage, so do not feel bothered about + or - sign)
or, X/Y = 1/5
the fraction done by Y = [spoiler]5/6[/spoiler]
Ans [spoiler]5/6[/spoiler]

[shashwats]

What about ?
0.5% of x + 0.8% of y= 0.75% of (x+y)
solving we get y= 5x.
hence y proportion is 5/6

[Jim@StratusPrep]

(75 - 50) / (80 - 50) = 5/6

This is a quick way to do the calculation. It will work on all weighted average problems.

[apoorva.rattan]

Jim - can you explain the formula you used or point me to a post which talks about the formula used ?

[GMATGuruNY]

Q:Each of the products produced yesterday was checked by x or y. 0.5% of the products checked by x are defective and 0.8% of the products checked by y are defective. If the total defective rate of all the products checked by x and y is 0.75%, what fraction of the products was checked by y?

Defective rate of X: 0.5%.
Defective rate of Y: 0.8%.
Defective rate of the MIXTURE of X and Y: 0.75%.

The following approach is called ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.

Step 1: Plot the 3 percentages on a number line, with X and Y's percentages (0.5% and 0.8%) on the ends and the mixture's percentage (0.75%) in the middle.
X 0.5%------------0.75%-----------0.8% Y

Step 2: Calculate the distances between the percentages.
X 0.5%----.25-----0.75%----.05-----0.8% Y

Step 3: Determine the ratio of X to Y in the mixture.
The required rate of X to Y is the RECIPROCAL of the distances in red.
X:Y = .05 : .25 = 5:25 = 1:5.

Since X:Y = 1:5, of every 6 products, 1 was checked by X and 5 were checked by Y.
Thus:
Y/total = 5/6.

For two similar problems, check here:

https://www.beatthegmat.com/ratios-fract ... 15365.html

[Jim@StratusPrep]

The explanation above should cover it...

The key is that the difference between to weighted average and the two figures you are averging have a distinct relationship.