Solving a Ratio and Average Problem: Finding the Number of Lilies in a Garden

When dealing with problem-solving in mathematics, particularly in subjects like algebra and geometry, it's common to encounter questions involving ratios and averages. In this article, we dive into a specific problem that deals with the ratio of roses to lilies and their average number in a garden. We'll explore the problem-solving process step-by-step, making sure to break down the calculations and explain each step in detail. This will not only provide a solution but also enhance your understanding of the underlying mathematical principles.

Problem Description

The problem at hand is as follows: In a garden, the ratio of roses to lilies is 3:2. The average number of roses and lilies in the garden is 180. The question is, how many lilies are there in the garden?

Step-by-Step Solution

Given:

The ratio of roses to lilies is 3:2. The average number of roses and lilies is 180.

Step 1: Define Variables

Let x be the common multiplier for the ratio. Then, the number of roses can be represented as 3x and the number of lilies as 2x.

Step 2: Set Up the Equation for the Average

The average number of roses and lilies is given by:

Average (Number of roses Number of lilies) / 2

Substituting the values from our variables:

180 (3x 2x) / 2

Multiplying both sides by 2 to clear the denominator:

360 5x

Step 3: Solve for x

Dividing both sides by 5:

x 360 / 5 72

Step 4: Find the Number of Lilies

The number of lilies is given by 2x:

Number of lilies 2 * 72 144

Verification

To ensure our solution is correct, let's check the conditions given in the problem:

The ratio of roses to lilies is 3:2. We have 3 * 72 216 roses and 2 * 72 144 lilies, which satisfies the ratio 3:2. The average number of roses and lilies is 180. The average of 216 and 144 is (216 144) / 2 360 / 2 180, which matches the given average.

Conclusion

Therefore, the number of lilies in the garden is 144.

Understanding step-by-step problem solving in mathematics, such as this ratio and average problem, can significantly enhance your problem-solving skills and prepare you for more complex mathematical challenges. With practice and a clear understanding of the principles involved, you can tackle a wide range of similar problems with confidence.

Keywords: ratio problem, average problem, garden flowers