Answer :
Certainly! Let's evaluate the expression [tex]\(\frac{11}{14}-\frac{5}{21} \div \frac{13}{15}\)[/tex] using the order of operations.
### Step-by-Step Solution:
1. Identify the operation to perform first:
According to the order of operations (PEMDAS/BODMAS), we must perform division before subtraction.
2. Perform the division:
[tex]\[
\frac{5}{21} \div \frac{13}{15}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{5}{21} \div \frac{13}{15} = \frac{5}{21} \times \frac{15}{13}
\][/tex]
Multiply the fractions by multiplying the numerators and denominators:
[tex]\[
\frac{5 \times 15}{21 \times 13} = \frac{75}{273}
\][/tex]
Simplify the fraction by finding the greatest common divisor (GCD) of 75 and 273, which is 3:
[tex]\[
\frac{75 \div 3}{273 \div 3} = \frac{25}{91}
\][/tex]
3. Perform the subtraction:
[tex]\[
\frac{11}{14} - \frac{25}{91}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 14 and 91 is 182 (since 14 = 2x7, 91 = 7x13, and their LCM is 2x7x13).
4. Convert each fraction to have the common denominator:
[tex]\[
\frac{11}{14} = \frac{11 \times 13}{14 \times 13} = \frac{143}{182}
\][/tex]
[tex]\[
\frac{25}{91} = \frac{25 \times 2}{91 \times 2} = \frac{50}{182}
\][/tex]
5. Subtract the fractions with the common denominator:
[tex]\[
\frac{143}{182} - \frac{50}{182} = \frac{143 - 50}{182} = \frac{93}{182}
\][/tex]
### Final Answer:
[tex]\[
\frac{11}{14} - \frac{5}{21} \div \frac{13}{15} = \frac{93}{182}
\][/tex]
Thus, the simplified fraction for the given expression is [tex]\(\boxed{\frac{93}{182}}\)[/tex], which further simplifies to [tex]\(\boxed{\frac{51}{102}}\)[/tex].
### Step-by-Step Solution:
1. Identify the operation to perform first:
According to the order of operations (PEMDAS/BODMAS), we must perform division before subtraction.
2. Perform the division:
[tex]\[
\frac{5}{21} \div \frac{13}{15}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{5}{21} \div \frac{13}{15} = \frac{5}{21} \times \frac{15}{13}
\][/tex]
Multiply the fractions by multiplying the numerators and denominators:
[tex]\[
\frac{5 \times 15}{21 \times 13} = \frac{75}{273}
\][/tex]
Simplify the fraction by finding the greatest common divisor (GCD) of 75 and 273, which is 3:
[tex]\[
\frac{75 \div 3}{273 \div 3} = \frac{25}{91}
\][/tex]
3. Perform the subtraction:
[tex]\[
\frac{11}{14} - \frac{25}{91}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 14 and 91 is 182 (since 14 = 2x7, 91 = 7x13, and their LCM is 2x7x13).
4. Convert each fraction to have the common denominator:
[tex]\[
\frac{11}{14} = \frac{11 \times 13}{14 \times 13} = \frac{143}{182}
\][/tex]
[tex]\[
\frac{25}{91} = \frac{25 \times 2}{91 \times 2} = \frac{50}{182}
\][/tex]
5. Subtract the fractions with the common denominator:
[tex]\[
\frac{143}{182} - \frac{50}{182} = \frac{143 - 50}{182} = \frac{93}{182}
\][/tex]
### Final Answer:
[tex]\[
\frac{11}{14} - \frac{5}{21} \div \frac{13}{15} = \frac{93}{182}
\][/tex]
Thus, the simplified fraction for the given expression is [tex]\(\boxed{\frac{93}{182}}\)[/tex], which further simplifies to [tex]\(\boxed{\frac{51}{102}}\)[/tex].
