Finding the Larger Number Using HCF, LCM, and Ratio

This problem involves finding two numbers when their Highest Common Factor (HCF), Least Common Multiple (LCM), and ratio are given. We will use the fundamental relationship between these concepts to solve it.

Understanding the Key Concepts

• HCF (Highest Common Factor): The largest number that divides two or more numbers exactly.
• LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more numbers.
• Relationship between HCF, LCM, and Numbers: For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM. That is, if the two numbers are \(a\) and \(b\), then \(a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)\).
• Ratio of Numbers: If the ratio of two numbers is \(p : q\), the numbers can be represented as \(px\) and \(qx\), where \(x\) is a common factor. Notably, if \(px\) and \(qx\) are the numbers and their ratio in simplest form is \(p:q\), then their HCF will be \(x\).

Applying the Given Information

We are given:

• HCF of the two numbers = 12
• LCM of the two numbers = 72
• Ratio of the two numbers = 2 : 3

Solving the Problem Step-by-Step

Let the two numbers be \(a\) and \(b\).

Since the ratio of the two numbers is 2 : 3, we can represent the numbers as \(2x\) and \(3x\), where \(x\) is a common factor. As discussed earlier, if the numbers are in the simplified ratio \(2:3\), their HCF will be \(x\). We are given that the HCF is 12. Therefore, \(x = 12\).

Now we can find the two numbers:

• First number = \(2x = 2 \times 12 = 24\)
• Second number = \(3x = 3 \times 12 = 36\)

Let's verify this using the relationship \(a \times b = \text{HCF} \times \text{LCM}\).

Product of the numbers = \(24 \times 36\)

HCF × LCM = \(12 \times 72\)

Calculating the products:

• \(24 \times 36 = 864\)
• \(12 \times 72 = 864\)

Since \(864 = 864\), the numbers 24 and 36 satisfy the given HCF and LCM.

The two numbers are 24 and 36.

The question asks for the larger of the two numbers.

Comparing 24 and 36, the larger number is 36.

Summary of the Solution Steps

1. Represent the two numbers using the given ratio \(2x\) and \(3x\).
2. Recognize that if the ratio is in simplest form, the HCF is the common factor \(x\).
3. Equate the given HCF (12) to the common factor \(x\), so \(x = 12\).
4. Calculate the two numbers using \(x = 12\).
5. Identify the larger number from the calculated values.

Revision Table: HCF, LCM, and Ratio Problem

Additional Information: Properties of HCF and LCM

Understanding the properties of HCF and LCM is crucial for solving number theory problems.

• The HCF of two numbers is always a factor of their LCM. In this case, 12 is a factor of 72 (\(72 \div 12 = 6\)). This is a useful check.
• If two numbers are co-prime (their HCF is 1), then their LCM is simply the product of the numbers.
• The product of two numbers is always equal to the product of their HCF and LCM. This property was key to solving the problem.
• When numbers are given in a ratio \(a:b\) (in simplest form), the numbers can be written as \(ax\) and \(bx\), where \(x\) is their HCF. Their LCM will be \((a \times b \times x)\). Let's verify with our numbers 24 and 36, with ratio 2:3 and HCF 12. Here \(a=2\), \(b=3\), \(x=12\). LCM = \(2 \times 3 \times 12 = 6 \times 12 = 72\), which matches the given LCM. This confirms our method of setting \(x\) as the HCF was correct when the ratio was in simplest form.