Solving a Fruit Stall Problem Using Algebra and Bar Models

Fruit stalls are a common sight in many parts of the world, offering a variety of fresh produce. In this article, we will solve a problem involving a fruit stall to determine the initial number of apples and oranges using algebraic methods and bar models. This problem is not only educational but also provides a real-world application of mathematical concepts.

Problem Statement

At a fruit stall, 5/13 of the fruits were apples, and the rest were oranges. After 1/2 of the apples and 3/8 of the oranges were sold, there were a total of 120 apples and oranges left at the stall. Determine the initial number of apples and oranges in the stall.

Solution Approach

Let the total number of fruits at the stall be x. We will use algebraic equations to find the initial number of apples and oranges, and then visualize the problem using a bar model.

Step 1: Setting Up the Equations

n - The number of apples is 5/13x.

n - The number of oranges is x - 5/13x 8/13x.

Step 2: Calculating the Number of Fruits Sold

n - Apples sold: 1/2 * 5/13x 5/26x

n - Oranges sold: 3/8 * 8/13x 3/13x

Step 3: Calculating the Remaining Fruits

n - Remaining apples: 5/13x - 5/26x 10/26x - 5/26x 5/26x

n - Remaining oranges: 8/13x - 3/13x 5/13x

Total remaining fruits:

5/26x 5/13x 5/26x 10/26x 15/26x

Set up the equation:

15/26x 120

Solve for x:

x 120 * 26/15 208

Step 4: Determining the Initial Number of Apples and Oranges

n - Number of apples: 5/13 * 208 80

n - Number of oranges: 8/13 * 208 128

Final Answer

The initial number of apples is 80, and the initial number of oranges is 128.

Visual Aid: Bar Models

To further illustrate the problem, let's use a bar model:

13 units - total fruits

5 units - apples (A)

8 units - oranges (O)

Half the apples sold is 2.5 units, so remaining apples are 2.5 units.

3/8 of the oranges sold is 3.75 units, so remaining oranges are 4.25 units.

Total remaining units: 2.5 4.25 6.75 units

Hence each unit is 120 / 6.75 16 fruits.

Apples sold: 2.5 * 16 40

Total fruits initially: 13 * 16 208

Conclusion

Using algebraic equations and bar models helps in solving complex problems involving fractions and ratios. This method not only ensures accuracy but also provides a clear understanding of the problem. Understanding such applications can be extremely helpful in real-world scenarios, such as managing inventory in retail or grocery stores.