Solving Arithmetic Expression with Fractions
To solve the expression shown in the image, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
The expression is as follows:
\( \frac{3}{5} - \left( \frac{8}{3} - \frac{9}{2} \right) + \frac{1}{4} \)
Step 1: Solve the parentheses first.
Inside the parentheses, we have two fractions that need to be subtracted:
\[ \frac{8}{3} - \frac{9}{2} \]
To subtract these fractions, we need a common denominator, which in this case would be 6 (the least common multiple of 3 and 2). First, convert both fractions to have the same denominator:
\[ \frac{8}{3} \times \frac{2}{2} = \frac{16}{6} \]
\[ \frac{9}{2} \times \frac{3}{3} = \frac{27}{6} \]
Now subtract the second fraction from the first:
\[ \frac{16}{6} - \frac{27}{6} = -\frac{11}{6} \]
Step 2: Apply the result back into the expression with the correct sign (negative, since we are subtracting the result of the parentheses):
\[ \frac{3}{5} - (-\frac{11}{6}) + \frac{1}{4} \]
Step 3: Add \(-(-\frac{11}{6})\) to \(\frac{3}{5}\). First, find a common denominator for \(\frac{3}{5}\) and \(\frac{11}{6}\), which is 30:
\[ \frac{3}{5} \times \frac{6}{6} = \frac{18}{30} \]
\[ \frac{11}{6} \times \frac{5}{5} = \frac{55}{30} \]
Now convert the subtraction of a negative to addition and add the two fractions:
\[ \frac{18}{30} + \frac{55}{30} = \frac{73}{30} \]
Step 4: Add the remaining fraction \(\frac{1}{4}\) to the result. Again, we need a common denominator, which is 120 for 30 and 4:
\[ \frac{73}{30} \times \frac{4}{4} = \frac{292}{120} \]
\[ \frac{1}{4} \times \frac{30}{30} = \frac{30}{120} \]
Now add them:
\[ \frac{292}{120} + \frac{30}{120} = \frac{322}{120} \]
Step 5: Reduce the fraction to its simplest form:
\[ \frac{322}{120} \text{ can be simplified by dividing by 2:} \frac{161}{60} \]
Therefore, the final simplified answer is:
\[ \frac{161}{60} \]
This fraction can be left as an improper fraction, or it can be converted to a mixed number by dividing the numerator by the denominator (161 ÷ 60), which would be 2 remainder 41, resulting in:
\[ 2 \frac{41}{60} \]
So the expression simplifies to \( 2 \frac{41}{60} \).
