Problem Solving

anastasios1984 wrote: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 much of food X (in grams) is in the mixture?
(A) 100
(B) 140
(C) 150
(D) 160
(E) 180

Here's how to solve with ALLIGATION.

Percentage of protein in X = 10%.
Percentage of protein in Y = 15%.
Percentage of protein in the mixture = 38/300 * 100 = 38/3.

Step 1: Plot the 3 percentages on a number line, with the two starting percentages (10% and 15%) on the ends and the goal percentage (38/3%) in the middle.
X(10%)------------------------38/3-----------Y(15%)

Step 2: Calculate the distances between the percentages.
X(10%)----------8/3---------38/3----7/3----Y(15%)

Step 3: Determine the ratio in the mixture.
The ratio of X to Y in the mixture is the RECIPROCAL of the distances in red.
X : Y = 7/3 : 8/3 = 7:8.

Since 7+8 = 15, of every 15 grams, 7 grams must be X and 8 grams must be Y.
Thus, the amount of X is equal to 7/15 of the total:
(7/15)300 = 140.

The correct answer is B.

Another problem solved with alligation:

Two tins A and B contain mixtures of wheat and rice. In A, the weights of wheat and rice are in
the ratio 2 : 3 and in B they are in the ratio 3 : 7. What quantities must be taken from A and B to
form a mixture containing 5 kg of wheat and 11 kg of rice?

1. 3kgs from A, 13 kg from B
2. 8kgs from A, 8 kg from B
3. 2kgs from A, 14 kg from B
4. 6kgs from A, 10 kg from B

Percentage of wheat in A = 2/5 = 40%
Percentage of wheat in B = 3/10 = 30%.
Percentage of wheat in the mixture = 5/16 = 31.25%.

The following approach is called alligation.
It's a very good way to handle MIXTURE PROBLEMS.

Step 1: Plot the 3 percentages on a number line, with the two starting percentages (40% and 30%) on the ends and the goal percentage (31.25%) in the middle.
A(40%)-------------------------31.25%--------------B(30%)

Step 2: Calculate the distances between the percentages.
A(40%)----------8.75---------31.25%----1.25----B(30%)

Step 3: Determine the ratio in the mixture.
The ratio of A to B in the mixture is the RECIPROCAL of the distances in red.
A : B = 1.25 : 8.75 = 1:7.

Only answer choice C has the same ratio:
2:14 = 1:7.

The correct answer is C.

One more:

Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75 percent fescue. If a mixture of X and Y contains 30 percent ryegrass, what percent of the weight of the mixture is X ?

(A) 10%
(B) 33.1/3%
(C) 40%
(D) 50%
(E) 66.2/3%

X = 40% ryegrass, Y = 25% ryegrass, the mixture = 30% ryegrass.

X(40%)--------10--------30%-----5-----Y(25%)

X:Y = 5:10 = 1:2.
Since 1+2 = 3, of every 3 liters, 1 liter must be X and 2 liters must be Y, implying that the fraction of X in the mixture = 1/3 = 33 1/3%.

The correct answer is B.