Ratios

Let the required ratio be 1 : x

Now, as per the given fact, we can make the following equation:

(2+3x)/(1+x)=2.75; simplifying, this we get; 0.25x=0.75
Thus, x=3

A

Thrills4ever wrote:Hey Guys,

So this problem is pretty straightforward but i wanted to know if there's an easier way to solve it?

A mixture of red beans and black beans is to be prepared. The price of red beans is $2 per pound, and the price of black beans is 3$ per pound. What is the ratio of red beans to black beans if the mixture is to be sold for $2.75 per pound?

(A) 1:3
(B) 1:2
(C) 2:3
(D) 1:1
(E) 3:1

All help is really appreciated

For most mixture and weighted average problems, I use alligation.
Alligation is used to determine how much weight must be given to each element in a mixture of 2 elements:

The proportion of each element in the mixture = the distance between the value associated with the mixture and the value associated with the other element.

Proportion of red beans in the mixture = the distance between the black bean price and the mixture price = 3.00-2.75 = .25.
Proportion of black beans in the mixture = the distance between the red bean price and the mixture price = 2.75-2 = .75.

The results above give the ratio of red beans to black beans in the mixture:
Red beans:black beans = (.25): (.75) = 1:3.

The correct answer is A.

The problem above also could be solved with some clever reasoning.
The answer choices represent the ratio of red beans to black beans in the mixture.

The $2.75 selling price of the mixture is closer to the price of the black beans ($3).
Thus, the black beans must be the higher value in the ratio.
Eliminate D and E.

The values in the correct ratio represent how much of each kind of bean is in the mixture.
The sum of the values in the correct ratio represent the total weight of the mixture.
Since the prices of the beans are integer values, when the total weight and the price of the mixture are multiplied, the product should be an integer.

Of the remaining answers, only answer choice A (ratio = 1:3) works:
1+3 = 4.
4*(2.75) = 11.

To confirm that red:black = 1:3 is the correct ratio:
Price of 1 pound of red beans = 1*2 = 2.
Price of 3 pounds of black beans = 3*3 = 9.
Total price = 2+9 = 11.
Total weight = 1+3 = 4 pounds.
Price per pound = 11/4 = 2.75.

The correct answer is A.