Sure, let's work through the given expression step-by-step:
1. Identify the fractions in the expression:
[tex]\[
\frac{2}{9} - \frac{1}{12} + \frac{5}{3}
\][/tex]
2. Find a common denominator:
- The denominators here are 9, 12, and 3.
- The least common multiple (LCM) of 9, 12, and 3 is 36.
3. Convert all fractions to have this common denominator of 36:
- Convert [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[
\frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36}
\][/tex]
- Convert [tex]\(\frac{1}{12}\)[/tex]:
[tex]\[
\frac{1}{12} = \frac{1 \times 3}{12 \times 3} = \frac{3}{36}
\][/tex]
- Convert [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[
\frac{5}{3} = \frac{5 \times 12}{3 \times 12} = \frac{60}{36}
\][/tex]
4. Perform the arithmetic operations:
- First, handle the subtraction and addition:
[tex]\[
\frac{8}{36} - \frac{3}{36} + \frac{60}{36}
\][/tex]
- Subtract:
[tex]\[
\frac{8}{36} - \frac{3}{36} = \frac{5}{36}
\][/tex]
- Add:
[tex]\[
\frac{5}{36} + \frac{60}{36} = \frac{65}{36}
\][/tex]
5. Simplify the fraction if necessary:
- [tex]\(\frac{65}{36}\)[/tex] is already in its simplest form since 65 and 36 have no common factors other than 1.
6. Convert improper fraction to mixed number (if desired):
- Divide 65 by 36:
[tex]\[
65 \div 36 = 1 \text{ remainder } 29
\][/tex]
- So,
[tex]\[
\frac{65}{36} = 1 \frac{29}{36}
\][/tex]
Thus, the result is:
[tex]\[
1 \frac{29}{36}
\][/tex]
The numerical result for the given expression is:
[tex]\[
1.8055555555555556
\][/tex]
So, the work we did corresponds to the floating-point approximation, confirming our steps.
