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Answer :
To add the fractions [tex]\(-\frac{9}{25}\)[/tex] and [tex]\(\frac{14}{15}\)[/tex], we need to follow these steps:
1. Find a Common Denominator: The first step in adding or subtracting fractions is to find a common denominator. The denominators we have are 25 and 15. The least common denominator (LCD) can be found by determining the least common multiple (LCM) of these two numbers.
- The prime factorization of 25 is [tex]\(5^2\)[/tex].
- The prime factorization of 15 is [tex]\(3 \cdot 5\)[/tex].
To find the LCM, we take the highest power of each prime number that appears in these factorizations:
- The LCM of [tex]\(5^2\)[/tex] and [tex]\(3 \cdot 5\)[/tex] is [tex]\(3 \cdot 5^2 = 75\)[/tex].
2. Convert Each Fraction: Now we convert each fraction to have this common denominator (75).
- For [tex]\(-\frac{9}{25}\)[/tex], convert to a denominator of 75:
[tex]\[
-\frac{9}{25} = -\frac{9 \cdot 3}{25 \cdot 3} = -\frac{27}{75}
\][/tex]
- For [tex]\(\frac{14}{15}\)[/tex], convert to a denominator of 75:
[tex]\[
\frac{14}{15} = \frac{14 \cdot 5}{15 \cdot 5} = \frac{70}{75}
\][/tex]
3. Add the Fractions: Now that both fractions have the same denominator, you can simply add the numerators:
- Adding the fractions:
[tex]\[
-\frac{27}{75} + \frac{70}{75} = \frac{-27 + 70}{75} = \frac{43}{75}
\][/tex]
4. Final Result: The final result of adding [tex]\(-\frac{9}{25}\)[/tex] and [tex]\(\frac{14}{15}\)[/tex] is [tex]\(\frac{43}{75}\)[/tex].
This shows the step-by-step process of how the fraction addition results in [tex]\(\frac{43}{75}\)[/tex].
1. Find a Common Denominator: The first step in adding or subtracting fractions is to find a common denominator. The denominators we have are 25 and 15. The least common denominator (LCD) can be found by determining the least common multiple (LCM) of these two numbers.
- The prime factorization of 25 is [tex]\(5^2\)[/tex].
- The prime factorization of 15 is [tex]\(3 \cdot 5\)[/tex].
To find the LCM, we take the highest power of each prime number that appears in these factorizations:
- The LCM of [tex]\(5^2\)[/tex] and [tex]\(3 \cdot 5\)[/tex] is [tex]\(3 \cdot 5^2 = 75\)[/tex].
2. Convert Each Fraction: Now we convert each fraction to have this common denominator (75).
- For [tex]\(-\frac{9}{25}\)[/tex], convert to a denominator of 75:
[tex]\[
-\frac{9}{25} = -\frac{9 \cdot 3}{25 \cdot 3} = -\frac{27}{75}
\][/tex]
- For [tex]\(\frac{14}{15}\)[/tex], convert to a denominator of 75:
[tex]\[
\frac{14}{15} = \frac{14 \cdot 5}{15 \cdot 5} = \frac{70}{75}
\][/tex]
3. Add the Fractions: Now that both fractions have the same denominator, you can simply add the numerators:
- Adding the fractions:
[tex]\[
-\frac{27}{75} + \frac{70}{75} = \frac{-27 + 70}{75} = \frac{43}{75}
\][/tex]
4. Final Result: The final result of adding [tex]\(-\frac{9}{25}\)[/tex] and [tex]\(\frac{14}{15}\)[/tex] is [tex]\(\frac{43}{75}\)[/tex].
This shows the step-by-step process of how the fraction addition results in [tex]\(\frac{43}{75}\)[/tex].
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