To solve the problem [tex]\( 3 \frac{1}{2} + \frac{3}{4} \)[/tex] and give the answer in its simplest form, follow these detailed steps:
1. Convert the mixed number to an improper fraction:
- The mixed number [tex]\( 3 \frac{1}{2} \)[/tex] can be converted to an improper fraction.
- First, multiply the whole number part by the denominator of the fractional part and add the numerator of the fractional part:
[tex]\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \][/tex]
2. Identify the second fraction:
- The second fraction is already given as [tex]\(\frac{3}{4}\)[/tex].
3. Find a common denominator:
- To add these fractions, we need a common denominator. The denominators here are 2 and 4.
- The least common denominator (LCD) of 2 and 4 is 4.
4. Convert fractions to the common denominator:
- Convert [tex]\(\frac{7}{2}\)[/tex] to a fraction with the denominator 4:
[tex]\[ \frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4} \][/tex]
- [tex]\(\frac{3}{4}\)[/tex] already has the denominator 4, so it remains [tex]\(\frac{3}{4}\)[/tex].
5. Add the fractions:
- Now, add the fractions with the common denominator:
[tex]\[ \frac{14}{4} + \frac{3}{4} = \frac{14 + 3}{4} = \frac{17}{4} \][/tex]
6. Simplify if necessary:
- The fraction [tex]\(\frac{17}{4}\)[/tex] is in its simplest form, as 17 and 4 have no common factors other than 1.
7. Convert to a mixed number (if required):
- Although the question doesn't specify it, sometimes it's useful to convert the improper fraction back to a mixed number.
- Divide 17 by 4:
[tex]\[ 17 ÷ 4 = 4 \text{ with a remainder of } 1 \][/tex]
- This means [tex]\( \frac{17}{4} = 4 \frac{1}{4} \)[/tex].
So, the result of [tex]\( 3 \frac{1}{2} + \frac{3}{4} \)[/tex] is:
[tex]\[ \frac{17}{4} \][/tex]
Or, alternatively in mixed number form:
[tex]\[ 4 \frac{1}{4} \][/tex]
Finally, in decimal form, the answer is:
[tex]\[ 3.875 \][/tex]
