Understanding the Problem: Numbers in Ratio and LCM

The question provides three numbers that are in a specific ratio, which is 3 : 5 : 7. We are also given that the Least Common Multiple (LCM) of these three numbers is 2415. Our goal is to find the difference between the second number and the first number.

When numbers are in a ratio like 3 : 5 : 7, we can represent them using a common factor. Let this common factor be \(x\). So, the three numbers can be written as:

• First number = \(3x\)
• Second number = \(5x\)
• Third number = \(7x\)

Calculating the LCM of Numbers in Ratio

To find the LCM of the three numbers \(3x\), \(5x\), and \(7x\), we need to consider the LCM of the coefficients (3, 5, and 7) and the common factor \(x\).

Let's find the LCM of 3, 5, and 7. Since 3, 5, and 7 are all prime numbers, their LCM is simply their product.

\(\text{LCM}(3, 5, 7) = 3 \times 5 \times 7 = 105\)

The LCM of the three numbers \(3x\), \(5x\), and \(7x\) will therefore be the LCM of their coefficients multiplied by the common factor \(x\).

\(\text{LCM}(3x, 5x, 7x) = \text{LCM}(3, 5, 7) \times x = 105x\)

Using the Given LCM to Find the Common Factor

We are given that the LCM of the three numbers is 2415. We have also determined that the LCM can be represented as \(105x\).

So, we can set up an equation:

\(105x = 2415\)

To find the value of \(x\), we need to divide 2415 by 105.

\(x = \frac{2415}{105}\)

Let's perform the division:

So, the common factor \(x\) is 23.

Finding the Three Numbers

Now that we have the value of \(x\), we can find the three numbers:

• First number = \(3x = 3 \times 23 = 69\)
• Second number = \(5x = 5 \times 23 = 115\)
• Third number = \(7x = 7 \times 23 = 161\)

Let's quickly verify if the LCM of 69, 115, and 161 is indeed 2415.

• \(69 = 3 \times 23\)
• \(115 = 5 \times 23\)
• \(161 = 7 \times 23\)

The common factor is 23, and the unique factors are 3, 5, and 7. The LCM is the product of the highest powers of all prime factors involved.

\(\text{LCM}(69, 115, 161) = 3 \times 5 \times 7 \times 23 = 105 \times 23 = 2415\)

This matches the given information, so our numbers are correct.

Calculating the Difference

The question asks for the difference between the second number and the first number.

• Second number = 115
• First number = 69

Difference = Second number - First number

Difference = \(115 - 69\)

Difference = 46

Final Answer

The difference between the second number and the first number is 46.

Revision Table: Ratio and LCM Concepts

Additional Information: HCF and LCM Relationship

For two numbers \(a\) and \(b\), the product of their Highest Common Factor (HCF) and LCM is equal to the product of the numbers themselves:

\(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)

For three numbers, there isn't a simple direct formula like the one for two numbers. However, understanding the prime factorization of numbers is key to finding both HCF and LCM for any set of numbers.

In this problem, the numbers are \(3x, 5x, 7x\). Notice that \(x\) is the Highest Common Factor (HCF) of these three numbers because 3, 5, and 7 are coprime (have no common factors other than 1). So, HCF(\(3x, 5x, 7x\)) = \(x\). The LCM is \(\text{LCM}(3, 5, 7) \times x = 105x\).