Answer :
To find the sum of [tex]\(\frac{6}{8}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex], we need to first find a common denominator for the fractions.
1. Convert [tex]\(\frac{6}{8}\)[/tex] to a common denominator:
- Simplify [tex]\(\frac{6}{8}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 2. This simplifies to [tex]\(\frac{3}{4}\)[/tex].
2. Convert [tex]\(\frac{9}{12}\)[/tex] to a common denominator:
- Simplify [tex]\(\frac{9}{12}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 3. This simplifies to [tex]\(\frac{3}{4}\)[/tex].
3. Find the least common denominator (LCD):
- The least common multiple of 8 and 12 is 24.
4. Express [tex]\(\frac{3}{4}\)[/tex] using the LCD:
- Convert [tex]\(\frac{3}{4}\)[/tex] to have a denominator of 24.
- [tex]\(\frac{3}{4} = \frac{18}{24}\)[/tex].
5. Express [tex]\(\frac{3}{4}\)[/tex] using the LCD for the second fraction:
- We already showed that [tex]\(\frac{3}{4} = \frac{18}{24}\)[/tex].
The sum of the fractions is:
[tex]\[
\frac{18}{24} + \frac{18}{24} = \frac{36}{24} = \frac{3}{2}
\][/tex]
Now, we examine the given expressions to see which ones are equivalent to this sum:
- (A) [tex]\(\frac{36}{48} + \frac{36}{48}\)[/tex]:
[tex]\[
\frac{36}{48} = \frac{3}{4} \quad \text{(since \(\frac{36}{48}\) simplifies to \(\frac{3}{4}\))}
\][/tex]
Thus,
[tex]\[
\frac{36}{48} + \frac{36}{48} = \frac{3}{4} + \frac{3}{4} = \frac{6}{4} = \frac{3}{2}
\][/tex]
- (B) [tex]\(\frac{24}{36} + \frac{27}{36}\)[/tex]:
[tex]\[
\frac{24}{36} + \frac{27}{36} = \frac{51}{36} = \frac{17}{12} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (C) [tex]\(\frac{14}{16} + \frac{13}{16}\)[/tex]:
[tex]\[
\frac{14}{16} + \frac{13}{16} = \frac{27}{16} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (D) [tex]\(\frac{18}{20} + \frac{17}{20}\)[/tex]:
[tex]\[
\frac{18}{20} + \frac{17}{20} = \frac{35}{20} = \frac{7}{4} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (E) [tex]\(\frac{18}{24} + \frac{18}{24}\)[/tex]:
[tex]\[
\frac{18}{24} + \frac{18}{24} = \frac{36}{24} = \frac{3}{2}
\][/tex]
The expressions that correctly represent the sum are (A) [tex]\(\frac{36}{48} + \frac{36}{48}\)[/tex] and (E) [tex]\(\frac{18}{24} + \frac{18}{24}\)[/tex].
1. Convert [tex]\(\frac{6}{8}\)[/tex] to a common denominator:
- Simplify [tex]\(\frac{6}{8}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 2. This simplifies to [tex]\(\frac{3}{4}\)[/tex].
2. Convert [tex]\(\frac{9}{12}\)[/tex] to a common denominator:
- Simplify [tex]\(\frac{9}{12}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 3. This simplifies to [tex]\(\frac{3}{4}\)[/tex].
3. Find the least common denominator (LCD):
- The least common multiple of 8 and 12 is 24.
4. Express [tex]\(\frac{3}{4}\)[/tex] using the LCD:
- Convert [tex]\(\frac{3}{4}\)[/tex] to have a denominator of 24.
- [tex]\(\frac{3}{4} = \frac{18}{24}\)[/tex].
5. Express [tex]\(\frac{3}{4}\)[/tex] using the LCD for the second fraction:
- We already showed that [tex]\(\frac{3}{4} = \frac{18}{24}\)[/tex].
The sum of the fractions is:
[tex]\[
\frac{18}{24} + \frac{18}{24} = \frac{36}{24} = \frac{3}{2}
\][/tex]
Now, we examine the given expressions to see which ones are equivalent to this sum:
- (A) [tex]\(\frac{36}{48} + \frac{36}{48}\)[/tex]:
[tex]\[
\frac{36}{48} = \frac{3}{4} \quad \text{(since \(\frac{36}{48}\) simplifies to \(\frac{3}{4}\))}
\][/tex]
Thus,
[tex]\[
\frac{36}{48} + \frac{36}{48} = \frac{3}{4} + \frac{3}{4} = \frac{6}{4} = \frac{3}{2}
\][/tex]
- (B) [tex]\(\frac{24}{36} + \frac{27}{36}\)[/tex]:
[tex]\[
\frac{24}{36} + \frac{27}{36} = \frac{51}{36} = \frac{17}{12} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (C) [tex]\(\frac{14}{16} + \frac{13}{16}\)[/tex]:
[tex]\[
\frac{14}{16} + \frac{13}{16} = \frac{27}{16} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (D) [tex]\(\frac{18}{20} + \frac{17}{20}\)[/tex]:
[tex]\[
\frac{18}{20} + \frac{17}{20} = \frac{35}{20} = \frac{7}{4} \quad \text{(not equal to \(\frac{3}{2}\))}
\][/tex]
- (E) [tex]\(\frac{18}{24} + \frac{18}{24}\)[/tex]:
[tex]\[
\frac{18}{24} + \frac{18}{24} = \frac{36}{24} = \frac{3}{2}
\][/tex]
The expressions that correctly represent the sum are (A) [tex]\(\frac{36}{48} + \frac{36}{48}\)[/tex] and (E) [tex]\(\frac{18}{24} + \frac{18}{24}\)[/tex].
