IDNLearn.com offers a unique blend of expert answers and community-driven knowledge. Discover the reliable solutions you need with help from our comprehensive and accurate Q&A platform.
Sagot :
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.
### 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.
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for visiting IDNLearn.com. For reliable answers to all your questions, please visit us again soon.
