A rabbit on a controlled diet is fed daily 300 grams of a...

[I_need_a_700plus]

Can someone help me solve the following? I originally chose Answer choice C, but the correct answer is B - 140.

A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food x and food y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit's diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture?

A) 100
B) 140
C) 150
D) 160
E) 200

Thanks!

[vrajesh.dave]

I used process of elimination for this answer

-- I started with C - 150

if X is 150 g -- then >> 0.10 * 100 = 15 g of protein

That leave us with Y = 150 g =>> 0.15 * 150 = 22.5g of protein

hence we have 15 + 22.5 = 32.5g

So I know we need to have less of X

B - 140g >> 140 *0.10 = 14 g of protein

hence Y = 160 g >> 160 * 0.15 = 24 g of protein

I used 10% of 160 = 16
+
5 % of 160 = 8 ==>> 16 + 8 = 24 g [to do the math quickly]

24 + 14 = 38 g of protein.

================================

You can also use this approach
-----
X + Y = 300 [since the total food is 300 g]

0.10X + 0.15Y = 38 [since the total protein is 38 g]

And you can substitute Y = 300 - X

0.10X + 0.15(300 - X) = 38
solving it you will get
X = 14

which is 10% of 300 [but 14g of protein]

SO we get 140 g of X

This 2nd approach is length and takes longer. So I would use the first one

[RonPurewal]

the post above is great.

below the line is an ADVANCED SHORTCUT TECHNIQUE for solving this question.

--

ADVANCED TECHNIQUE: NUMBER LINE SHORTCUT FOR WEIGHTED AVERAGES

if you have a WEIGHTED AVERAGE of two quantities, then you can FIGURE OUT THE RELATIVE AMOUNTS OF EACH by doing the following:
* plot the individual averages on a number line
* plot the weighted average ("WA") on the same number line
* find the distances between the individual averages and the WA
* REVERSE the ratio of these distances
that reversed ratio will be the ratio of the amounts.

in this problem:
* food X contains 10% protein
* food Y contains 15% protein
* the mixture contains 38/300 = 12 2/3 % protein

plot on a number line:
<--------X=10%------------WA=12 2/3 %----------Y=15%------->

so the distances are 2 2/3 (for X) and 2 1/3 (for Y)
the reverse ratio is X:Y = 2 1/3 : 2 2/3
which equals 7:8 (note that you can multiply by 3)

therefore the food is 7/15 "X" and 8/15 "Y".
7/15 of 300 is 140.

true, this shortcut isn't much of a shortcut in this problem, but it can solve other very difficult problems IMMEDIATELY.
such as this one:
official-guide-240-t1212.html

--

finally, you can ESTIMATE

you can notice that the final mixture is 12 2/3 % protein, which is closer to Y than to X - but BARELY so (it's almost right in the middle).

therefore, the mixture should be EVER SO SLIGHTLY MORE THAN 1/2 Y and ever so slightly LESS than 1/2 X.
the only choice that satisfies this condition is 140.

[georgepa]

I'm posting a solution using the same number line shortcut that Ron has posted above but using values instead of percents.

Calculate the values for gms of protein
gms of protein in X = 0.10 (300) = 30
gms of protein in Y = 0.15 (300) = 45
gms of protein in M = 38 (given)

Draw The Weighted Number Line
Note the following:

1. The number line is broken in 15ths (45-30 = 15). That is each unit in the number line is 1/15.


2. If the gms protein in M was 37.5 - then both X and Y would have contributed equally


3. Since the gms of protein in M is 38, Y contributed more to the average than X


4. Distance of M from X = 8 (1/15) = 8/15


5. Distance of M from Y = 7 (1/15) = 7/15


6. So Y contributed 8/15 to the average (pulled the Mean 8/15 units from X to Y)


7. And X contributed 7/15 to the average (pulled the mean 7/15 units from Y to X)

Code: Select All Code

X                        M                  Y
|                        |                   |
|<-- -- -- 8/15 -- -- -->|< -- -- 7/15 -- -->|
|                        |                   |
30 -- -- -- -- -- -- -- 38 -- -- -- -- -- -- 45


Direct Calculation
X contributed 300 (7/15) = 140

Indirect Calculation

1. Each unit is 1/15


2. Regular average is 37.5 - which means 150 gms of protein from X and Y each


3. Weighted average is 38 - 37.5 = 0.5 units or 1/2 unit away from mean


4. 1/2 unit is [1/15]/[2] = 1/[15*2] = 1/30


5. Therefore 0.5 units of 300 grams = 300/30 = 10 gms


6. So X contributed 150 - 10 = 140 gms


7. and Y contributed 150 + 10 = 160 gms

Estimation
What Ron posted above


Thanks to Ron's explanation above and Stacey's post here:
https://www.manhattangmat.com/blog/inde ... roblems-2/

[RonPurewal]

George"”just curious: why are you posting this on a thread that is (a) already answered, and, more importantly, (b) almost five years old?

[georgepa]

I was searching for an explanation for this problem after doing a GMATPrep exam. I found Stacey's blog post extremely helpful. And since I was having trouble with weighted averages I was trying all different ways to solve the problem. I figured - other people who are just browsing may also benefit.

[RonPurewal]

Ok.

This kind of posting (just adding to the discussion) is fine, I suppose. But please don't answer other posters' questions"”that's our job.

Thanks.