Solving the Marble Problem: A Comprehensive Guide

In this guide, we will solve a classic marble problem that involves algebraic equations. This problem is not only entertaining but also a great way to practice your problem-solving skills. We will provide a detailed solution and explore different methods to reach the correct answer.

Understanding the Problem

Peter had 15 more marbles than Andrew and 10 more than Ben. They have 86 marbles altogether. How many marbles does Ben have?

Algebraic Approach

We will start by setting up the problem using algebra:

Let the number of marbles Andrew has be $A$ the number of marbles Ben has be $B$ and the number of marbles Peter has be $P$.

The given relationships are:

$P A 15$ $P B 10$ $A B P 86$

We will express $P$ in terms of $A$ and $B$:

Step 1: Express $P$ in terms of $A$

Substitute $P$ into the total equation:

$2A B - 15 86$

$2A B 71$

Step 2: Express $P$ in terms of $B$

Substitute $P$ into the total equation, then:

$A 2B - 10 86$

$A 2B 76$

Step 3: Solve the System of Equations

Now we have the system of equations:

$2A B 71$ $A 2B 76$

Solving for $B$ from the first equation:

$B 71 - 2A$

Substitute $B$ into the second equation:

$A 2(71 - 2A) 76$

$A 142 - 4A 76$

$-3A 76 - 142$

$-3A -66$

$A 22$

Substitute $A 22$ back into the expression for $B$:

$B 71 - 22 49 - 15 27$

Finally, find the number of marbles Peter has:

$P A 15 22 15 37$

So, Ben has 27 marbles.

Alternative Methods

Let's explore some alternative methods to solve the same problem:

Direct Substitution Method

Using the same variables as before:

Express $P$ in terms of $B$: $P B 10$ Substitute $P$ into the total equation: $A B (B 10) 86$ Simplify: $A 2B 10 86$ Subtract 10 from both sides: $A 2B 76$ Another equation is: $A B 5$ because $P A 15$ Substitute: $B 5 2B 76$ Simplify: $3B 5 76$ Solve: $3B 71$ Divide by 3: $B 27$ Therefore, Ben has 27 marbles.

Systematic Approach

Using a systematic approach:

Let $x$ be the number of marbles Peter has. Then Andrew has $x - 15$ marbles. Ben has $x - 10$ marbles. The total is: $x (x - 15) (x - 10) 86$ Combine like terms: $3x - 25 86$ Add 25 to both sides: $3x 111$ Divide by 3: $x 37$ Peter has $37$ marbles. Andrew has $37 - 15 22$ marbles. Ben has $37 - 10 27$ marbles.

Conclusion

In this article, we explored various methods to solve the marble problem. Whether you use the algebraic approach, direct substitution, or systematic substitution, the solution remains the same. Ben has 27 marbles. This problem not only tests your algebraic skills but also enhances your logical reasoning skills.