To solve the problem of subtracting two mixed fractions, [tex]$29 \frac{1}{4} - 12 \frac{5}{12}$[/tex], we can follow a detailed step-by-step process:
1. Convert the mixed fractions to improper fractions.
- For the first mixed fraction [tex]\(29 \frac{1}{4}\)[/tex]:
[tex]\[
29 \frac{1}{4} = \frac{29 \cdot 4 + 1}{4} = \frac{116 + 1}{4} = \frac{117}{4}
\][/tex]
- For the second mixed fraction [tex]\(12 \frac{5}{12}\)[/tex]:
[tex]\[
12 \frac{5}{12} = \frac{12 \cdot 12 + 5}{12} = \frac{144 + 5}{12} = \frac{149}{12}
\][/tex]
2. Find a common denominator for the two fractions.
- The denominators are 4 and 12. The least common multiple (LCM) of 4 and 12 is 12.
- Convert [tex]\(\frac{117}{4}\)[/tex] to a fraction with a denominator of 12:
[tex]\[
\frac{117}{4} = \frac{117 \times 3}{4 \times 3} = \frac{351}{12}
\][/tex]
- The fraction [tex]\(\frac{149}{12}\)[/tex] already has a denominator of 12, so we leave it as it is.
3. Subtract the fractions.
- With both fractions now having a common denominator:
[tex]\[
\frac{351}{12} - \frac{149}{12} = \frac{351 - 149}{12} = \frac{202}{12}
\][/tex]
4. Simplify the resulting fraction if possible.
- Factorize the numerator and the denominator to find the greatest common divisor (GCD).
[tex]\[
\text{GCD of 202 and 12 is 2.}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{202 \div 2}{12 \div 2} = \frac{101}{6}
\][/tex]
5. Convert the improper fraction back to a mixed fraction, since the original problem involves mixed fractions.
- Divide the numerator by the denominator to find the whole number part:
[tex]\[
101 \div 6 = 16 \text{ remainder } 5
\][/tex]
Thus,
[tex]\[
\frac{101}{6} = 16 \frac{5}{6}
\][/tex]
Thus, the result of the subtraction [tex]\(29 \frac{1}{4} - 12 \frac{5}{12}\)[/tex] is:
[tex]\[
16 \frac{5}{6}
\][/tex]
