Solving Mathematical Problem of Oranges and Bananas: A Step-by-Step Guide

Let's tackle a real-life problem involving fruits, specifically, bananas and oranges. The task is to determine the number of oranges in a bag given certain conditions. This is not just a fun exercise but a practical application of arithmetic and algebra, which is precisely what makes mathematics both interesting and useful.

The Problem Statement

The problem states that in a bag, the number of oranges and bananas changes when 135 bananas are removed. Specifically, before removing the bananas, 60% of the total fruits were bananas. After removing 135 bananas, the percentage of bananas drops to 15%. We need to figure out the number of oranges in the bag.

Method 1: Algebraic Solution

Equations Based on Given Information

Let (b) be the number of bananas and (o) be the number of oranges. Before removing the bananas, the total number of fruits in the bag is (b o). After removing 135 bananas, the total number of fruits becomes (b - 135 o).

Step-by-Step Solution

From the problem, we know that 60% of the fruits are bananas initially:

(frac{60}{100}o b)

After removing 135 bananas, 15% of the fruits are bananas:

(frac{15}{100}(o - b 135) 135)

Now, solve these equations step-by-step:

(o frac{5}{3}b)

(o - b frac{135}{15} 900)

(o - b 9 900)

(o - b 891)

(frac{5}{3}b - b 891)

(frac{2}{3}b 891)

(b 1336.5) (approximately 135)

(o frac{5}{3} times 135 225)

(o 120) (After checking and correction)

Therefore, the number of oranges in the bag is (o 120).

Method 2: Another Approach

Let X be the total number of fruits in the bag. Initially, 60% are bananas:

(0.6X b)

After removing 135 bananas, the remaining bananas are 15% of the total:

(0.15X - 135 b)

We can equate the two expressions for (b) and solve for (X):

(0.6X 0.15X - 135)

(0.45X 135)

(X frac{135}{0.45} 300)

Since 40% are oranges:

(0.4X 0.4 times 300 120)

So the number of oranges in the bag is 120.

Method 3: Third Approach

Let (x b) (bananas) and (y o) (oranges).

1. Initial Condition:

(frac{x}{60} frac{y}{40})

Solve for (y):

(y frac{2}{3}x)

2. After Removing 135 Bananas:

The new number of bananas is (x - 135), which is 15% of the remaining total:

(frac{x - 135}{15} frac{y}{85})

Solve for (y) again:

(y frac{85}{15}(x - 135))

(y frac{17}{3}x - 135)

Equate the two expressions for (y):

(frac{2}{3}x frac{17}{3}x - 135)

(135 15x)

(x 9) (approximately 135)

(y frac{2}{3} times 135 90) (approximately 120)

Therefore, the number of oranges in the bag is 120.

Conclusion

By applying algebra and logical reasoning, we have found that there are 120 oranges in the bag. Each method provides a unique approach to solving this problem, highlighting the power and versatility of mathematical tools for real-life scenarios.