Introduction

This article provides a detailed explanation on how to find the price ratio of grapefruits to oranges when given the ratios of oranges to lemons and grapefruits to lemons. Understanding ratios and their applications is a fundamental skill in mathematics and economics, particularly when dealing with price comparisons of different goods.

Step-by-Step Explanation

The problem at hand involves finding the price ratio of grapefruits to oranges using the given ratios of oranges to lemons and grapefruits to lemons. The given information is as follows:

The price ratio of oranges to lemons is 2:3. The price ratio of grapefruits to lemons is 3:5.

First, let's establish the price of each fruit using a common factor. Let:

Price of oranges be ( 2x ), where ( x ) is the common factor. Price of lemons be ( 3x ) since the ratio of oranges to lemons is ( 2:3 ). Price of grapefruits be ( 3y ), where ( y ) is another common factor. Price of lemons be ( 5y ) since the ratio of grapefruits to lemons is ( 3:5 ).

From the second point, we can derive:

3x 5y

Next, we solve for ( y ) in terms of ( x ):

y (frac{3x}{5})

Substituting ( y ) back into the price of grapefruits:

Price of grapefruits ( 3y 3 left( frac{3x}{5} right) frac{9x}{5} )

Now, the prices are:

Price of oranges ( 2x ) Price of grapefruits ( frac{9x}{5} )

To find the ratio of grapefruits to oranges, we use the formula:

(text{Ratio of grapefruits to oranges} frac{text{Price of grapefruits}}{text{Price of oranges}} frac{frac{9x}{5}}{2x})

Simplifying this expression:

(frac{9x}{5} cdot frac{1}{2x} frac{9}{10})

Therefore, the ratio of grapefruits to oranges by price is:

(boxed{9:10})

Alternative Method

An alternative, simpler method to solve this problem involves recognizing that the required ratio (G/O) can be derived by inverting the first ratio (O/L) and multiplying it with the second ratio (G/L). This method directly leverages the given ratios and avoids the need for complex algebraic manipulations.

Conclusion

This problem demonstrates the practical application of ratios in solving real-world economic and mathematical problems. Understanding and correctly applying the principles of ratios can simplify complex problems and provide clear, efficient solutions.