Solving the Marble Conundrum: A Unique Mathematical Puzzle

Today, we delve into a mathematical puzzle involving marbles, a classic example that can be solved using basic algebra and the quadratic formula. The problem revolves around two individuals, Rahul and Rohan, who together have 45 marbles, and after losing 5 marbles each, find that the product of their remaining marbles is 124. Let's explore this step by step.

Understanding the Problem

Let the number of marbles Rahul initially had be R and the number of marbles Rohan initially had be H. According to the problem, we have the following two equations:

R H 45 (R - 5)(H - 5) 124

We start by expressing H in terms of R from the first equation:

H 45 - R

Solving the Equations

We next substitute H 45 - R into the second equation:

(R - 5)(45 - R - 5) 124

Expanding this, we get:

R(40 - R) - 5(40 - R) 124

Simplifying further:

R 40 - R2 - 5(40 - R) 124

Further simplification:

40R - R2 - 200 - 5R 124

Combining like terms:

-R2 35R - 324 0

Multiplying through by -1 to get the standard form:

R2 - 45R 324 0

Solving the Quadratic Equation

We can use the quadratic formula to solve for R:

R (-b plusmn; radic;b2 - 4ac) / 2a

Where:

a 1 b -45 c 324

Plugging in the values:

R (45 plusmn; radic;2025 - 1296) / 2

Further simplifying the expression:

R (45 plusmn; radic;729) / 2

R (45 plusmn; 27) / 2

Finding the Solutions

Calculating the two possible values for R:

R (72 / 2) 36 R (18 / 2) 9

Now, we find H for both values of R:

If R 36, then H 45 - 36 9 If R 9, then H 45 - 9 36

Conclusion

Thus, the initial number of marbles Rahul and Rohan had were:

Rahul: 36 marbles, Rohan: 9 marbles OR Rahul: 9 marbles, Rohan: 36 marbles

To summarize, Rahul had 36 marbles and Rohan had 9 marbles, or vice versa.