Solving Math Puzzles with Fractions: A Comprehensive Guide
In the realm of mathematics, solving puzzles can be both a fun and intellectually stimulating experience. Today, we'll tackle a particular puzzle regarding the distribution of flowers owned by Lira. This puzzle involves fractions, which require a careful approach to solve. Let's dive into the puzzle, explore step-by-step solutions, and understand the algebraic principles behind it.
Introduction to the Puzzle
Lira has a collection of flowers. A portion of these flowers are roses, while the remaining part is divided among sunflowers and tulips. The problem statement initially mentions that 2/9 of the flowers are roses. Further, 3/7 of the remaining flowers are sunflowers. The rest are tulips, and we know Lira has 36 tulips. The task is to determine the number of roses and sunflowers in total.
Step-by-Step Solution
Let's denote the total number of flowers by x.
Step 1: Expressing the number of roses.
[text{Number of roses} frac{2}{9}x]
Step 2: Determining the remaining flowers after accounting for roses.
[text{Remaining flowers} x - frac{2}{9}x frac{7}{9}x]
Step 3: Expressing the number of sunflowers.
[text{Number of sunflowers} frac{3}{7} times frac{7}{9}x frac{3}{9}x frac{1}{3}x]
Step 4: Expressing the number of tulips.
[text{Number of tulips} frac{7}{9}x - frac{1}{3}x frac{7}{9}x - frac{3}{9}x frac{4}{9}x]
Given that Lira has 36 tulips, we can set up the equation:
[frac{4}{9}x 36]
Solving for x.
[x 36 times frac{9}{4} 81]
Step 5: Determining the number of roses and sunflowers.
[text{Number of roses} frac{2}{9} times 81 18]
[text{Number of sunflowers} frac{1}{3} times 81 27]
Step 6: Calculating the total number of roses and sunflowers.
[text{Total} 18 27 45]
Alternative Method: Simplified Algebraic Solution
Another method involves a more direct approach without breaking down the fractions initially:
[1 - left(frac{2}{9} frac{3}{7} times left(1 - frac{2}{9}right)right) frac{4}{9}]
[36 div frac{4}{9} 81]
[81 - 36 45]
Conclusion
Through this detailed breakdown, we have successfully solved the puzzle. Lira has a total of 45 roses and sunflowers combined, with 18 roses and 27 sunflowers.