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Answer :
To solve the expression [tex]\frac{140}{187} \cdot \left(\frac{13}{15} + \frac{16}{21}\right)[/tex], follow these steps:
Simplify the expression inside the parentheses:
First, find the sum of [tex]\frac{13}{15}[/tex] and [tex]\frac{16}{21}[/tex]. To add these fractions, you need a common denominator. The least common multiple (LCM) of 15 and 21 is 105.
Convert [tex]\frac{13}{15}[/tex] to a fraction with denominator 105:
[tex]\frac{13}{15} = \frac{13 \times 7}{15 \times 7} = \frac{91}{105}[/tex]Convert [tex]\frac{16}{21}[/tex] to a fraction with denominator 105:
[tex]\frac{16}{21} = \frac{16 \times 5}{21 \times 5} = \frac{80}{105}[/tex]
Now add the two fractions:
[tex]\frac{91}{105} + \frac{80}{105} = \frac{91 + 80}{105} = \frac{171}{105}[/tex]Simplify [tex]\frac{171}{105}[/tex]:
Find the greatest common divisor (GCD) of 171 and 105, which is 3.
- Simplify the fraction:
[tex]\frac{171}{105} = \frac{171 \div 3}{105 \div 3} = \frac{57}{35}[/tex]
- Simplify the fraction:
Multiply [tex]\frac{140}{187}[/tex] by [tex]\frac{57}{35}[/tex]:
Multiply the numerators and denominators:
[tex]\frac{140}{187} \times \frac{57}{35} = \frac{140 \times 57}{187 \times 35}[/tex]Calculate [tex]140 \times 57 = 7980[/tex] and [tex]187 \times 35 = 6545[/tex].
The resulting fraction is [tex]\frac{7980}{6545}[/tex].
Simplify [tex]\frac{7980}{6545}[/tex] if possible:
Find the GCD of 7980 and 6545, which is 5.
- Simplify:
[tex]\frac{7980}{6545} = \frac{7980 \div 5}{6545 \div 5} = \frac{1596}{1309}[/tex]
- Simplify:
The final simplified result of the expression [tex]\frac{140}{187} \cdot \left(\frac{13}{15} + \frac{16}{21}\right)[/tex] is [tex]\frac{1596}{1309}[/tex].
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