To find the Least Common Denominator (LCD) for the fractions [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] and express each fraction using this common denominator, we'll proceed step-by-step.
### Step 1: Identify the Denominators
First, identify the denominators of the given fractions:
- The denominators are [tex]\(10\)[/tex] and [tex]\(6\)[/tex].
### Step 2: Calculate the Least Common Denominator (LCD)
To find the LCD of the two denominators, we need to determine the smallest number that both [tex]\(10\)[/tex] and [tex]\(6\)[/tex] can divide without a remainder. The Least Common Denominator is:
[tex]\[ \text{LCD}(10, 6) = 30 \][/tex]
### Step 3: Convert Each Fraction to Use the LCD
Next, we convert each fraction to an equivalent fraction with the common denominator [tex]\(30\)[/tex].
#### Converting [tex]\(\frac{7}{10}\)[/tex]:
To convert [tex]\(\frac{7}{10}\)[/tex], multiply both the numerator and the denominator by the same number to get [tex]\(30\)[/tex] in the denominator:
[tex]\[ \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} \][/tex]
#### Converting [tex]\(\frac{5}{6}\)[/tex]:
Similarly, convert [tex]\(\frac{5}{6}\)[/tex] by multiplying the numerator and the denominator to get [tex]\(30\)[/tex] in the denominator:
[tex]\[ \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \][/tex]
### Final Equivalents:
Thus, the fractions [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] expressed with the Least Common Denominator [tex]\(30\)[/tex] are:
[tex]\[ \frac{7}{10} = \frac{21}{30} \][/tex]
[tex]\[ \frac{5}{6} = \frac{25}{30} \][/tex]
### Related Conversions for Given Fractions:
Given the additional fractions to be converted to a common denominator, follow similar steps.
#### Converting [tex]\(\frac{7}{30}, \frac{5}{30}\)[/tex]:
These fractions already have a common denominator of [tex]\(30\)[/tex] and are:
[tex]\[ \frac{7}{30} \quad \text{and} \quad \frac{5}{30} \][/tex]
#### Converting [tex]\(\frac{42}{60}, \frac{30}{60}\)[/tex]:
These fractions already have a common denominator of [tex]\(60\)[/tex] and are:
[tex]\[ \frac{42}{60} \quad \text{and} \quad \frac{30}{60} \][/tex]
#### Converting [tex]\(\frac{42}{60}, \frac{50}{60}\)[/tex]:
These fractions already have a common denominator of [tex]\(60\)[/tex] and are:
[tex]\[ \frac{42}{60} \quad \text{and} \quad \frac{50}{60} \][/tex]
Therefore, the results of these conversions are:
1. The Least Common Denominator is [tex]\(30\)[/tex].
2. The fractions [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] expressed with the Least Common Denominator are:
[tex]\[ \frac{7}{10} = \frac{21}{30} \][/tex]
[tex]\[ \frac{5}{6} = \frac{25}{30} \][/tex]
The provided fractions have been confirmed as:
[tex]\[ \frac{7}{30}, \frac{5}{30}, \frac{21}{30}, \frac{25}{30}, \frac{42}{60}, \frac{30}{60}, \frac{42}{60}, \frac{50}{60} \][/tex]
These results are consistent with the original question and the given solution.
