Understanding the Problem: LCM and Ratio of Two Numbers

The question provides two key pieces of information about two numbers: their Least Common Multiple (LCM) is 48, and their ratio is 2:3. We need to find the sum of these two numbers.

Using the Ratio to Represent the Numbers

When the ratio of two numbers is given as 2:3, it means the numbers can be represented as multiples of some common factor. Let this common factor be \(x\). Therefore, the two numbers can be written as:

• First Number = \(2x\)
• Second Number = \(3x\)

Here, \(x\) is also the Highest Common Factor (HCF) of the two numbers, \(2x\) and \(3x\).

Calculating the LCM from the Ratio

The LCM of two numbers \(a\) and \(b\) is given by the formula: \(\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}\). In our case, the numbers are \(2x\) and \(3x\), and their HCF is \(x\).

So, the LCM of \(2x\) and \(3x\) is:

\[ \text{LCM}(2x, 3x) = \frac{(2x) \times (3x)}{x} = \frac{6x^2}{x} = 6x \]

Alternatively, we can think of the LCM of \(2x\) and \(3x\) as the product of their HCF and the remaining unique factors. The HCF is \(x\), the remaining factor from \(2x\) is 2, and the remaining factor from \(3x\) is 3. So, \(\text{LCM} = x \times 2 \times 3 = 6x\).

Solving for the Common Factor \(x\)

We are given that the LCM of the two numbers is 48. We have calculated the LCM in terms of \(x\) as \(6x\). Now, we can set up an equation:

\[ 6x = 48 \]

To find the value of \(x\), we divide both sides of the equation by 6:

\[ x = \frac{48}{6} \]

\[ x = 8 \]

So, the common factor \(x\) is 8. This is also the HCF of the two numbers.

Finding the Two Numbers

Now that we know \(x = 8\), we can find the two numbers:

• First Number = \(2x = 2 \times 8 = 16\)
• Second Number = \(3x = 3 \times 8 = 24\)

Let's quickly check if the LCM of 16 and 24 is indeed 48:

• Prime factorization of 16 = \(2^4\)
• Prime factorization of 24 = \(2^3 \times 3\)
• LCM(16, 24) = \(2^4 \times 3 = 16 \times 3 = 48\). This matches the given information.

Calculating the Sum of the Numbers

The question asks for the sum of the two numbers. The numbers are 16 and 24.

Sum = First Number + Second Number

Sum = \(16 + 24\)

Sum = \(40\)

Final Answer

The sum of the two numbers is 40.

Revision Table: LCM, Ratio, and Number Properties

Additional Information: Properties of LCM and HCF

Understanding LCM and HCF is fundamental in number theory. Here are some additional points:

• The HCF of two prime numbers is always 1. Their LCM is their product.
• If one number is a multiple of the other, the smaller number is the HCF and the larger number is the LCM. For example, for numbers 6 and 18, HCF is 6 and LCM is 18.
• The product of two numbers is equal to the product of their LCM and HCF. This property was used implicitly in solving the problem \(48 \times x = (2x) \times (3x)\).
• LCM is always greater than or equal to the largest number, and HCF is always less than or equal to the smallest number.

These concepts are crucial for solving various problems involving number properties and ratios.