Alright, let’s work through this problem step by step:
### Finding the Least Common Denominator (LCD):
Firstly, we need to find the least common denominator (LCD) for the fractions [tex]\(\frac{3}{10}\)[/tex] and [tex]\(\frac{2}{15}\)[/tex].
1. Identify the denominators: We have
- Denominator 1: [tex]\(10\)[/tex]
- Denominator 2: [tex]\(15\)[/tex]
2. Compute the LCD:
- The prime factorizations of [tex]\(10\)[/tex] and [tex]\(15\)[/tex] are:
[tex]\(10 = 2 \times 5\)[/tex]
[tex]\(15 = 3 \times 5\)[/tex]
- The LCD will be the smallest number that both [tex]\(10\)[/tex] and [tex]\(15\)[/tex] divide into, which would include the highest powers of all the prime factors appearing in either number:
- [tex]\(2\)[/tex] (from [tex]\(10\)[/tex])
- [tex]\(3\)[/tex] (from [tex]\(15\)[/tex])
- [tex]\(5\)[/tex] (common to both)
- Therefore, the LCD is [tex]\(2 \times 3 \times 5 = 30\)[/tex].
### Expressing Each Fraction Using the LCD:
Next, we express each fraction with the common denominator of [tex]\(30\)[/tex].
1. For [tex]\(\frac{3}{10}\)[/tex]:
- To convert [tex]\(\frac{3}{10}\)[/tex] to a fraction with denominator [tex]\(30\)[/tex], we need to multiply both the numerator and the denominator by [tex]\(3\)[/tex] (since [tex]\(30 \div 10 = 3\)[/tex]):
[tex]\[
\frac{3}{10} \times \frac{3}{3} = \frac{9}{30}
\][/tex]
2. For [tex]\(\frac{2}{15}\)[/tex]:
- To convert [tex]\(\frac{2}{15}\)[/tex] to a fraction with denominator [tex]\(30\)[/tex], we need to multiply both the numerator and the denominator by [tex]\(2\)[/tex] (since [tex]\(30 \div 15 = 2\)[/tex]):
[tex]\[
\frac{2}{15} \times \frac{2}{2} = \frac{4}{30}
\][/tex]
So, the fractions [tex]\(\frac{9}{30}\)[/tex] and [tex]\(\frac{4}{30}\)[/tex] are the original fractions [tex]\(\frac{3}{10}\)[/tex] and [tex]\(\frac{2}{15}\)[/tex] expressed with the common denominator [tex]\(30\)[/tex].
### Expressing Fractions with a New Common Denominator of 150:
Now, let's express both fractions using a new common denominator of [tex]\(150\)[/tex].
1. For [tex]\(\frac{3}{10}\)[/tex]:
- To convert [tex]\(\frac{3}{10}\)[/tex] to a fraction with a new denominator [tex]\(150\)[/tex], we need to multiply both the numerator and denominator by [tex]\(15\)[/tex] (since [tex]\(150 \div 10 = 15\)[/tex]):
[tex]\[
\frac{3}{10} \times \frac{15}{15} = \frac{45}{150}
\][/tex]
2. For [tex]\(\frac{2}{15}\)[/tex]:
- To convert [tex]\(\frac{2}{15}\)[/tex] to a fraction with a new denominator [tex]\(150\)[/tex], we need to multiply both the numerator and the denominator by [tex]\(10\)[/tex] (since [tex]\(150 \div 15 = 10\)[/tex]):
[tex]\[
\frac{2}{15} \times \frac{10}{10} = \frac{20}{150}
\][/tex]
Therefore, the fractions [tex]\(\frac{45}{150}\)[/tex] and [tex]\(\frac{20}{150}\)[/tex] are the original fractions [tex]\(\frac{3}{10}\)[/tex] and [tex]\(\frac{2}{15}\)[/tex] expressed with the new common denominator [tex]\(150\)[/tex].
### Summary of Results:
- Least Common Denominator (LCD): [tex]\(30\)[/tex]
- Fractions with LCD [tex]\(30\)[/tex]: [tex]\(\frac{9}{30}\)[/tex] and [tex]\(\frac{4}{30}\)[/tex]
- Fractions with new denominator [tex]\(150\)[/tex]: [tex]\(\frac{45}{150}\)[/tex] and [tex]\(\frac{20}{150}\)[/tex]
And there you have it! We have expressed the given fractions with the least common denominator and a new common denominator.
