Solving the Fruit Pricing Puzzle: A Mathematical Journey

Imagine you have a challenge: for 1 Rs, you can get 4 apples, or 1 banana, and 1 mango costs 5 Rs. How can you buy 100 fruits with 100 Rs? This intriguing puzzle requires both strategic thinking and mathematical precision. Let's break it down and solve it step-by-step.

Defining the Problem

To tackle this problem, we first need to define variables for each fruit:

Let a be the number of apples. Let b be the number of bananas. Let m be the number of mangoes.

Setting Up the Equations

We know the cost of each fruit and the total number of fruits and their cost:

Cost of Apples: 4 apples for 1 Rs, so each apple costs 1/4 Rs. Cost of Bananas: 1 Rs for 1 banana. Cost of Mangoes: 1 Rs for 1 mango.

The total number of fruits and their cost equations are:

Total number of fruits: a b m 100 Total cost: (1/4)a b 5m 100

Solving the Equations

Now we have a system of two equations:

a b m 100 (1/4)a b 5m 100

From the first equation, we can express b in terms of a and m as follows:

b 100 - a - m

Substituting this into the second equation:

(1/4)a (100 - a - m) 5m 100

Simplifying the equation:

(1/4)a 100 - a - m 5m 100

(1/4)a - a 4m 0

-3a/4 4m 0

4m 3a/4

m 3a/16

Substituting m back into the first equation:

a b 3a/16 100

16a 16b 3a 1600

19a 16b 1600

Solving for b in terms of a:

b 100 - 19a/16

The equation for b is always true, indicating that there are infinitely many solutions. However, we need to find integer solutions for the number of apples, bananas, and mangoes.

Finding Integer Solutions

To find integer solutions, m must be an integer. Since m 3a/16, a must be a multiple of 16. Let a 16k where k is a non-negative integer. Then:

b 100 - 19(16k)/16 100 - 19k m 3(16k)/16 3k

For b to be non-negative:

100 - 19k ≥ 0

19k ≤ 100

k ≤ 100/19 ≈ 5.26

Hence, k can take the values 0, 1, 2, 3, 4, 5.

Summary of Solutions

k 0: a 0, b 100, m 0 k 1: a 16, b 81, m 3 k 2: a 32, b 62, m 6 k 3: a 48, b 43, m 9 k 4: a 64, b 24, m 12 k 5: a 80, b 5, m 15

These are the possible combinations of apples, bananas, and mangoes that satisfy the conditions of buying 100 fruits for 100 Rs.

Conclusion

By considering the constraints and solving the equations, we have found multiple solutions to the puzzle. Depending on your preference, you can choose from these combinations:

0 apples, 100 bananas, and 0 mangoes 16 apples, 81 bananas, and 3 mangoes 32 apples, 62 bananas, and 6 mangoes 48 apples, 43 bananas, and 9 mangoes 64 apples, 24 bananas, and 12 mangoes 80 apples, 5 bananas, and 15 mangoes

Enjoy exploring the different combinations and solving this fascinating mathematical puzzle!