Understanding the Least Common Multiple (LCM) for Decimals

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. Finding the LCM is straightforward for integers, but how do we find the LCM for decimal numbers like 1.2 and 2.7?

To find the LCM of decimals, we can convert them into integers by multiplying them by a suitable power of 10. After finding the LCM of the resulting integers, we divide the result by the same power of 10 used initially.

Step-by-Step LCM Calculation for 1.2 and 2.7

Let's find the LCM of 1.2 and 2.7 using the conversion method.

Step 1: Convert Decimals to Integers

To convert 1.2 and 2.7 into integers, we need to multiply both numbers by a power of 10 such that all decimal places are removed. In this case, multiplying by 10 is sufficient.

• \(1.2 \times 10 = 12\)
• \(2.7 \times 10 = 27\)

Now, the problem reduces to finding the LCM of the integers 12 and 27.

Step 2: Find the LCM of the Integers (12 and 27)

We can find the LCM of 12 and 27 using prime factorization or listing multiples.

Method A: Prime Factorization

• Prime factorization of 12: \(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)
• Prime factorization of 27: \(27 = 3 \times 3 \times 3 = 3^3\)

To find the LCM, take the highest power of all prime factors that appear in either factorization:

• Highest power of 2 is \(2^2\).
• Highest power of 3 is \(3^3\).

\(\text{LCM}(12, 27) = 2^2 \times 3^3 = 4 \times 27 = 108\).

Method B: Listing Multiples (for smaller numbers)

List multiples of 12:

12, 24, 36, 48, 60, 72, 84, 96, 108, ...

List multiples of 27:

27, 54, 81, 108, ...

The smallest common multiple is 108.

So, \(\text{LCM}(12, 27) = 108\).

Step 3: Convert the LCM back to the original scale

Since we multiplied the original numbers (1.2 and 2.7) by 10 in Step 1 to get integers, we now need to divide the LCM of the integers (108) by the same factor (10) to get the LCM of the original decimal numbers.

\(\text{LCM}(1.2, 2.7) = \frac{\text{LCM}(12, 27)}{10} = \frac{108}{10} = 10.8\).

Final Answer for LCM of 1.2 and 2.7

The Least Common Multiple of 1.2 and 2.7 is 10.8.

Revision Table: Key Concepts for LCM of Decimals

Additional Information: LCM and HCF of Fractions

Another way to approach finding the LCM of decimals is to first convert them into fractions. For example, \(1.2 = \frac{12}{10} = \frac{6}{5}\) and \(2.7 = \frac{27}{10}\).

The formula for the LCM of two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) is:

\(\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}\)

Using this formula for \(\frac{6}{5}\) and \(\frac{27}{10}\):

• Numerator LCM: \(\text{LCM}(6, 27) = 54\) (as calculated before).
• Denominator HCF: HCF (Highest Common Factor) of 5 and 10 is 5.

\(\text{LCM}\left(\frac{6}{5}, \frac{27}{10}\right) = \frac{54}{5} = 10.8\).

Both methods yield the same result, 10.8.

Similarly, you can find the HCF of decimals by converting them to fractions and using the formula:

\(\text{HCF}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}\)

Or, by converting to integers, finding the HCF, and then dividing by the power of 10. For 1.2 (12) and 2.7 (27), HCF(12, 27) = 3. Dividing by 10 gives 0.3. So, HCF(1.2, 2.7) = 0.3.