To solve the expression [tex]\(\frac{6}{7} \cdot \frac{5}{9} \div \frac{10}{21}\)[/tex], let's break it down into detailed steps:
1. Understand the order of operations:
- First, handle the division inside the expression [tex]\(\frac{5}{9} \div \frac{10}{21}\)[/tex].
- Then, multiply the result by [tex]\(\frac{6}{7}\)[/tex].
2. Divide the fractions:
- The division of two fractions [tex]\(\frac{a}{b} \div \frac{c}{d}\)[/tex] can be done by multiplying the first fraction by the reciprocal of the second fraction: [tex]\(\frac{a}{b} \cdot \frac{d}{c}\)[/tex].
- So, [tex]\(\frac{5}{9} \div \frac{10}{21}\)[/tex] becomes [tex]\(\frac{5}{9} \cdot \frac{21}{10}\)[/tex].
3. Multiply the fractions:
- Multiply the numerators and the denominators: [tex]\(\frac{5 \cdot 21}{9 \cdot 10}\)[/tex].
- Perform the multiplications: [tex]\(5 \cdot 21 = 105\)[/tex] and [tex]\(9 \cdot 10 = 90\)[/tex].
- This gives [tex]\(\frac{105}{90}\)[/tex].
4. Simplify the fraction:
- The greatest common divisor (GCD) of 105 and 90 is 15.
- Simplify [tex]\(\frac{105}{90}\)[/tex] by dividing both the numerator and the denominator by 15: [tex]\(\frac{105 \div 15}{90 \div 15} = \frac{7}{6}\)[/tex].
Thus, [tex]\(\frac{5}{9} \div \frac{10}{21} = \frac{7}{6}\)[/tex].
5. Multiply with [tex]\(\frac{6}{7}\)[/tex]:
- Now multiply [tex]\(\frac{6}{7} \cdot \frac{7}{6}\)[/tex].
6. Multiply the fractions:
- Multiply the numerators and the denominators: [tex]\(\frac{6 \cdot 7}{7 \cdot 6}\)[/tex].
- Perform the multiplications: [tex]\(6 \cdot 7 = 42\)[/tex] and [tex]\(7 \cdot 6 = 42\)[/tex].
- This gives [tex]\(\frac{42}{42}\)[/tex].
7. Simplify the fraction:
- [tex]\(\frac{42}{42} = 1\)[/tex].
Thus, [tex]\(\frac{6}{7} \cdot \frac{5}{9} \div \frac{10}{21} = 1\)[/tex].
