Finding the Number of Marbles Per Box (Value of x)

The problem asks us to find the number of marbles in each box, denoted by 'x', such that Meena can pack 70 red marbles and 105 blue marbles into boxes without mixing colors. This means all red marbles must be packed into boxes of size 'x', and all blue marbles must also be packed into boxes of size 'x'.

For this to be possible, the number of marbles in each box ('x') must be a number that can divide both the total number of red marbles (70) and the total number of blue marbles (105) exactly. In mathematical terms, 'x' must be a common divisor of 70 and 105.

While any common divisor would allow packing (resulting in a different number of boxes), the question implies we are looking for the largest possible number of marbles per box. This is a common scenario in such problems to maximize efficiency. Therefore, we need to find the greatest common divisor (GCD) of 70 and 105.

Calculating the Greatest Common Divisor (GCD) of 70 and 105

We can find the GCD by listing the factors of each number or by using prime factorization.

Method 1: Prime Factorization

Let's find the prime factors of 70 and 105:

• Prime factorization of 70: $70 = 2 \times 35 = 2 \times 5 \times 7$
• Prime factorization of 105: $105 = 3 \times 35 = 3 \times 5 \times 7$

Now, we identify the common prime factors. The prime factors common to both 70 and 105 are 5 and 7.

To find the GCD, we multiply the common prime factors:

$\text{GCD}(70, 105) = 5 \times 7 = 35$

Method 2: Listing Factors

Let's list the factors of 70 and 105:

• Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
• Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

The common factors are the numbers that appear in both lists: 1, 5, 7, and 35. The greatest among these common factors is 35.

So, $\text{GCD}(70, 105) = 35$.

Conclusion for the Value of x

The greatest common divisor of 70 and 105 is 35. Therefore, the value of x, the number of marbles in each box, is 35.

With x = 35:

• Number of boxes for red marbles: $70 \div 35 = 2$ boxes
• Number of boxes for blue marbles: $105 \div 35 = 3$ boxes

Each box will contain 35 marbles, and all marbles of each color can be packed without mixing.

Revision Table: Marbles Packing Problem

Additional Information: Understanding GCD in Real Life

The concept of the Greatest Common Divisor (GCD) is useful in many real-life situations, not just packing marbles. Here are a few examples:

• Cutting Fabric/Wire: If you have two pieces of fabric or wire of different lengths and want to cut them into the longest possible pieces of equal length without any waste, you would find the GCD of their lengths.
• Arranging Items: When arranging items (like students for a parade or books on shelves) into equal rows or groups from different total numbers, the GCD helps find the largest possible group size.
• Scheduling: Finding when events that repeat at different intervals will occur simultaneously involves concepts related to both GCD and Least Common Multiple (LCM).

In the marble packing problem, the GCD represents the largest size of the boxes that allows all marbles of both colors to be packed perfectly into boxes of that size without mixing and without any leftovers.