Sure! Let's break down each part of the problem step-by-step to provide a detailed solution:
### 1. [tex]\(\frac{1}{4} + \frac{2}{3}\)[/tex]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Thus,
[tex]\[
\frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}
\][/tex]
Adding the fractions:
[tex]\[
\frac{3}{12} + \frac{8}{12} = \frac{11}{12}
\][/tex]
The sum is [tex]\(\frac{11}{12}\)[/tex], and in decimal form, it is approximately 0.9167.
### 2. [tex]\(6 \frac{4}{3} + 7 \frac{1}{4}\)[/tex]
Convert mixed numbers to improper fractions first:
- [tex]\(6 \frac{4}{3}\)[/tex] is equivalent to [tex]\(\left(6 + \frac{4}{3}\right) = 6 + 1 \frac{1}{3} = 7 \frac{1}{3}\)[/tex]
- Convert [tex]\(7 \frac{1}{4}\)[/tex] to an improper fraction:
[tex]\[
7 \frac{1}{4} = 7 + \frac{1}{4} = 7.25
\][/tex]
Now add the fractions:
[tex]\[
7.3333 + 7.25 = 14.5833
\][/tex]
The sum is [tex]\(14 \frac{7}{12}\)[/tex] or 14.5833.
### 3. [tex]\(7^2 \cdot \frac{3}{5} + \frac{3}{10}\)[/tex]
First, calculate [tex]\(7^2 = 49\)[/tex]. Then,
[tex]\[
49 \cdot \frac{3}{5} = \frac{147}{5} = 29.4
\][/tex]
[tex]\[
29.4 + \frac{3}{10} = 29.4 + 0.3 = 29.7
\][/tex]
The sum is 29.7.
### 4. [tex]\(5 \frac{2}{5} + 8 \frac{3}{4}\)[/tex]
Convert the mixed numbers to improper fractions:
- [tex]\(5 \frac{2}{5} = 5 + \frac{2}{5} = 5.4\)[/tex]
- [tex]\(8 \frac{3}{4} = 8 + \frac{3}{4} = 8.75\)[/tex]
Adding the fractions:
[tex]\[
5.4 + 8.75 = 14.15
\][/tex]
The sum is [tex]\(14 \frac{3}{20}\)[/tex] or 14.15.
### 5. [tex]\(\frac{3}{4} + \frac{1}{2} + \frac{5}{8}\)[/tex]
Find a common denominator, which is 8:
[tex]\[
\frac{3}{4} = \frac{6}{8}
\][/tex]
[tex]\[
\frac{1}{2} = \frac{4}{8}
\][/tex]
[tex]\[
\text{So,} \quad \frac{3}{4} + \frac{1}{2} + \frac{5}{8} = \frac{6}{8} + \frac{4}{8} + \frac{5}{8} = \frac{15}{8} = 1 \frac{7}{8}
\][/tex]
The sum is [tex]\(1.875\)[/tex].
### 6. [tex]\(6 \frac{3}{8} + 9 \frac{1}{2}\)[/tex]
Convert mixed numbers to improper fractions:
- [tex]\(6 \frac{3}{8} = 6 + \frac{3}{8} = 6.375\)[/tex]
- [tex]\(9 \frac{1}{2} = 9 + \frac{1}{2} = 9.5\)[/tex]
Adding the values:
[tex]\[
6.375 + 9.5 = 15.875
\][/tex]
The sum is [tex]\(15 \frac{7}{8}\)[/tex] or 15.875.
### 7. [tex]\(23 \frac{7}{12} + 18 \frac{2}{3}\)[/tex]
Convert mixed numbers to improper fractions:
- [tex]\(23 \frac{7}{12} = 23 + \frac{7}{12} = 23.5833\)[/tex]
- [tex]\(18 \frac{2}{3} = 18 + \frac{2}{3} = 18.6667\)[/tex]
Adding the values:
[tex]\[
23.5833 + 18.6667 = 42.25
\][/tex]
The sum is [tex]\(42 \frac{1}{4}\)[/tex] or 42.25.
In summary:
[tex]\[
(\frac{11}{12}, 14.5833, 29.7, 14.15, 1.875, 15.875, 42.25)
\][/tex]
