Finding the Ratio of LCM and HCF

The problem asks us to find the ratio of two quantities: the Least Common Multiple (LCM) of 44, 10, and 5, and the Highest Common Factor (HCF) of 50 and 75. To solve this, we need to calculate both the LCM and the HCF separately and then find their ratio.

Calculating the LCM of 44, 10, and 5

To find the LCM of a set of numbers, we can use the prime factorization method. We find the prime factors of each number and then take the highest power of each unique prime factor.

• Prime factorization of 44: \(44 = 2 \times 2 \times 11 = 2^2 \times 11^1\)
• Prime factorization of 10: \(10 = 2 \times 5 = 2^1 \times 5^1\)
• Prime factorization of 5: \(5 = 5 = 5^1\)

Now, let's identify all the unique prime factors involved: 2, 5, and 11. We take the highest power of each factor present in any of the numbers:

• Highest power of 2: \(2^2\) (from 44)
• Highest power of 5: \(5^1\) (from 10 and 5)
• Highest power of 11: \(11^1\) (from 44)

The LCM is the product of these highest powers:

\(\text{LCM}(44, 10, 5) = 2^2 \times 5^1 \times 11^1 = 4 \times 5 \times 11 = 20 \times 11 = 220\)

So, the LCM of 44, 10, and 5 is 220.

Calculating the HCF of 50 and 75

To find the HCF of a set of numbers, we also use the prime factorization method. We find the prime factors of each number and then take the lowest power of each common prime factor.

• Prime factorization of 50: \(50 = 2 \times 5 \times 5 = 2^1 \times 5^2\)
• Prime factorization of 75: \(75 = 3 \times 5 \times 5 = 3^1 \times 5^2\)

Now, let's identify the common prime factors. The only common prime factor is 5. We take the lowest power of this common factor:

• Lowest power of the common factor 5: \(5^2\) (present in both 50 and 75)

The HCF is the product of these lowest powers of common factors:

\(\text{HCF}(50, 75) = 5^2 = 25\)

So, the HCF of 50 and 75 is 25.

Finding the Ratio

The question asks for the ratio of the LCM of 44, 10, and 5 to the HCF of 50 and 75. This ratio is LCM : HCF.

Ratio = \(\text{LCM}(44, 10, 5)\) : \(\text{HCF}(50, 75)\)

Ratio = 220 : 25

To express the ratio in its simplest form, we divide both numbers by their greatest common divisor (GCD). We can see that both 220 and 25 are divisible by 5.

• \(220 \div 5 = 44\)
• \(25 \div 5 = 5\)

The simplified ratio is 44 : 5.

Therefore, the ratio of the LCM of 44, 10, and 5 to the HCF of 50 and 75 is 44 : 5.

Revision Table: LCM and HCF Concepts

Understanding LCM and HCF is fundamental for solving problems involving multiples and factors.

Additional Information: Ratio and Proportion Basics

A ratio is a comparison of two quantities. It can be written as \(a:b\) or \(\frac{a}{b}\). Ratios are often simplified to their lowest terms by dividing both parts by their greatest common divisor.

• Ratios represent a relationship between quantities, not their actual values.
• Simplifying ratios makes them easier to compare and understand.
• Proportion is an equality between two ratios.

In this problem, we compared the calculated LCM value with the calculated HCF value by forming a ratio and simplifying it.