To add the mixed fractions [tex]\(2 \frac{7}{12}\)[/tex] and [tex]\(3 \frac{5}{9}\)[/tex], follow these steps:
1. Convert the Mixed Fractions to Improper Fractions:
- For [tex]\(2 \frac{7}{12}\)[/tex]:
- Multiply the whole number part by the denominator: [tex]\(2 \times 12 = 24\)[/tex].
- Add the numerator: [tex]\(24 + 7 = 31\)[/tex].
- Therefore, [tex]\(2 \frac{7}{12}\)[/tex] converts to [tex]\(\frac{31}{12}\)[/tex].
- For [tex]\(3 \frac{5}{9}\)[/tex]:
- Multiply the whole number part by the denominator: [tex]\(3 \times 9 = 27\)[/tex].
- Add the numerator: [tex]\(27 + 5 = 32\)[/tex].
- Therefore, [tex]\(3 \frac{5}{9}\)[/tex] converts to [tex]\(\frac{32}{9}\)[/tex].
2. Find the Least Common Multiple (LCM) of the Denominators:
- The denominators are 12 and 9.
- The Least Common Multiple (LCM) of 12 and 9 is 36.
3. Convert Both Fractions to Have the Same Denominator:
- For [tex]\(\frac{31}{12}\)[/tex]:
- Multiply the numerator and the denominator by the necessary factor to make the denominator 36:
- [tex]\(31 \times \frac{36 \div 12}{36 \div 12} = 31 \times 3 = 93\)[/tex].
- Thus, [tex]\(\frac{31}{12} = \frac{93}{36}\)[/tex].
- For [tex]\(\frac{32}{9}\)[/tex]:
- Multiply the numerator and the denominator by the necessary factor to make the denominator 36:
- [tex]\(32 \times \frac{36 \div 9}{36 \div 9} = 32 \times 4 = 128\)[/tex].
- Thus, [tex]\(\frac{32}{9} = \frac{128}{36}\)[/tex].
4. Add the Numerators:
- With a common denominator, we add the numerators: [tex]\(93 + 128 = 221\)[/tex].
5. Combine the Numerator and the Denominator:
- The numerator is 221, and the common denominator is 36. The resulting fraction is [tex]\(\frac{221}{36}\)[/tex].
6. Convert Back to a Mixed Fraction:
- Divide the numerator by the denominator to get the whole part: [tex]\(221 \div 36 = 6\)[/tex] with a remainder of 5.
- Therefore, the improper fraction [tex]\(\frac{221}{36}\)[/tex] converts to the mixed fraction [tex]\(6 \frac{5}{36}\)[/tex].
Thus, the sum of [tex]\(2 \frac{7}{12}\)[/tex] and [tex]\(3 \frac{5}{9}\)[/tex] is [tex]\(6 \frac{5}{36}\)[/tex].
