Solving Word Problems on Money Distribution - A Case with Danny and Eira

Algebra can be a powerful tool in solving complex word problems, particularly those involving money distribution. In this article, we will delve into a delightful puzzle shared by Danny and Eira, illustrating the step-by-step process of solving such problems through algebraic methods. This example will not only help in understanding how to approach these types of word problems but also provide insights into how Google might index and rank content relevant to such mathematical exercises.

Introduction to the Problem

In the scenario presented by Danny and Eira, they share a sum of 52 dollars between themselves. After Danny spends 3/5 of his money and Eira spends 3/4 of her money, they end up with an equal amount. This problem can be approached using algebraic equations, which serve as powerful tools to solve such distribution problems. Let's dive into the details.

Step-by-Step Solution

Step 1: Define the Variables
Let Danny have ( D ) dollars.
Let Eira have ( E ) dollars.

Step 2: Set Up the Equation
From the problem, we know that ( D E 52 ).
Danny spends ( frac{3}{5}D ), so he has ( frac{2}{5}D ) left.
Eira spends ( frac{3}{4}E ), so she has ( frac{1}{4}E ) left.
Since they end up with an equal amount, the equation becomes:
[ frac{2}{5}D frac{1}{4}E ]

Step 3: Solve for One Variable in Terms of the Other
From the equation ( frac{2}{5}D frac{1}{4}E ), we can solve for ( E ) in terms of ( D ):
[ E frac{8}{5}D ]

Step 4: Substitute and Solve the System of Equations
Substitute ( E frac{8}{5}D ) into ( D E 52 ):
[ D frac{8}{5}D 52 ]

Multiplying through by 5 to clear the fraction gives:
[ 5D 8D 260 ]

Simplifying, we get:
[ 13D 260 ]

Thus:
[ D 20 ]

Substituting ( D 20 ) back into ( E frac{8}{5}D ):
[ E frac{8}{5} times 20 32 ]

Step 5: Verify the Solution
If Danny has 20 dollars and Eira has 32 dollars, then after spending: Danny has ( frac{2}{5} times 20 8 ) dollars.
Eira has ( frac{1}{4} times 32 8 ) dollars.
Both have the same amount, confirming our solution.

Conclusion

To summarize, Danny initially had 20 dollars, and Eira had 32 dollars. Therefore, Eira had 12 dollars more than Danny at the beginning.

Additional Insights

This problem not only showcases the application of algebra but also highlights the importance of systematic problem-solving in mathematics. By breaking down the problem into manageable steps and using algebraic tools, we can efficiently solve complex word problems involving money distribution.

For SEO purposes, including keywords such as word problems, algebra, money distribution, equations, and inequality ensures that the content is optimized for search engines and relevant to students, educators, and anyone interested in mathematical problem-solving.