Solving Maria's Mango Puzzle: A Comprehensive Guide
Dealing with fractions and ratios can sometimes feel like a complex puzzle. Let's break down the intriguing problem of Maria and her mangoes for better understanding. Maria sold and gave away some of her mangoes, and it's interesting to figure out the exact number. We'll walk through the step-by-step process to solve this mathematical puzzle using both word problems and mathematical equations.
Introduction
Maria faced a situation where she sold some fraction of her mangoes and gave away the same amount. In the end, she kept a third of the mangoes she initially had. We aim to find out how many more mangoes Maria sold compared to what she kept for herself.
Mathematical Approach
Let's begin by defining variables that will make the problem easier to solve:
Step 1: Setting Up the Problem
Assume that the number of mangoes Maria initially had is N.
Step 2: Expressing Given Information in Terms of N
She sold ( frac{4}{9} ) of her mangoes. She gave away 42 mangoes. She kept ( frac{1}{3} ) of her mangoes.Step 3: Calculating the Total Number of Mangoes
To find the total number of mangoes Maria initially had, we can use the information that she gave away 42 mangoes, which is ( frac{2}{9} ) of the total mangoes.
Mathematically, we can express this as:
Let 2/9 of the mangoes be 42. So, ( frac{2}{9}N 42 ). Multiplying both sides by ( frac{9}{2} ) to isolate N gives us: N 189.Step 4: Calculating the Number of Mangoes Sold and Kept
She sold ( frac{4}{9} times 189 84 ) mangoes. She kept ( frac{1}{3} times 189 63 ) mangoes.Step 5: Finding the Difference
We need to find how many more mangoes Maria sold than she kept. This can be calculated as:
Number of mangoes sold – Number of mangoes kept 84 – 63 21.
Therefore, Maria sold 21 more mangoes than she kept for herself.
Visual Representation Using a Bar Model
A visual representation can help us understand the problem more clearly. Let's assume each unit in the bar model represents 21 mangoes (since ( frac{2}{9} ) of the total is 42 mangoes).
1. Maria had 9 units of mangoes in total (9X). 2. She sold 4 units (4X). 3. She gave away 2 units (2X), which equals 42 mangoes. 4. She kept 3 units (3X), which is 63 mangoes.
As we can see, the difference between the units she sold and the units she kept is 1 unit, which equals 21 mangoes.
Alternative Mathematical Approach Using Equations
We can also solve the problem using algebraic representation:
Step 1: Express the Total Number of Mangoes
Let the total number of mangoes Maria initially had be ( x ).
Step 2: Write Equations Based on Given Information
Through the problem, we know:
She sold ( frac{4}{9}x ) mangoes and gave away 42 mangoes.
She kept ( frac{1}{3}x ).
So, we can write the equation:
[ x - frac{4}{9}x - 42 frac{1}{3}x ]
Multiplying through by 9 to clear the denominators, we get:
[ 9x - 4x - 366 3x ]
Simplifying, we get:
[ 2x 366 ]
[ x 189 ]
Now, we can calculate how many mangoes she sold and kept:
'She sold ( frac{4}{9} times 189 84 ) mangoes.
She kept ( frac{1}{3} times 189 63 ) mangoes.
Thus, she sold 21 more mangoes than she kept.
Conclusion
Understanding fraction arithmetic is crucial for solving such problems. By breaking down the problem into steps and using visual models, we can easily identify the number of mangoes Maria sold and kept. Whether through word problems or mathematical equations, the process remains the same, making the solution clearer and more accessible.