Answer :
We start with the expression
[tex]$$
\frac{5}{14} \times \frac{4}{15} + \frac{5}{14} \times \frac{12}{25}.
$$[/tex]
Step 1. Factor the Common Term
Both terms have a common factor of [tex]$\frac{5}{14}$[/tex], so we factor it out:
[tex]$$
\frac{5}{14} \left( \frac{4}{15} + \frac{12}{25} \right).
$$[/tex]
Step 2. Add the Terms Inside the Parentheses
To add [tex]$\frac{4}{15}$[/tex] and [tex]$\frac{12}{25}$[/tex], we need a common denominator. The least common denominator (LCD) of 15 and 25 is 75. We convert each fraction:
[tex]$$
\frac{4}{15} = \frac{4 \times 5}{15 \times 5} = \frac{20}{75}, \quad \frac{12}{25} = \frac{12 \times 3}{25 \times 3} = \frac{36}{75}.
$$[/tex]
Now, add the fractions:
[tex]$$
\frac{20}{75} + \frac{36}{75} = \frac{20 + 36}{75} = \frac{56}{75}.
$$[/tex]
Step 3. Multiply by the Common Factor
Multiply the common factor by the sum obtained:
[tex]$$
\frac{5}{14} \times \frac{56}{75}.
$$[/tex]
Multiply the numerators and denominators:
[tex]$$
\frac{5 \times 56}{14 \times 75} = \frac{280}{1050}.
$$[/tex]
Step 4. Simplify the Fraction
Divide the numerator and the denominator by their greatest common divisor. Dividing both by 70 (since [tex]$280 \div 70 = 4$[/tex] and [tex]$1050 \div 70 = 15$[/tex]) gives:
[tex]$$
\frac{280}{1050} = \frac{4}{15}.
$$[/tex]
Final Answer
The simplified expression is
[tex]$$
\boxed{\frac{4}{15}}.
$$[/tex]
[tex]$$
\frac{5}{14} \times \frac{4}{15} + \frac{5}{14} \times \frac{12}{25}.
$$[/tex]
Step 1. Factor the Common Term
Both terms have a common factor of [tex]$\frac{5}{14}$[/tex], so we factor it out:
[tex]$$
\frac{5}{14} \left( \frac{4}{15} + \frac{12}{25} \right).
$$[/tex]
Step 2. Add the Terms Inside the Parentheses
To add [tex]$\frac{4}{15}$[/tex] and [tex]$\frac{12}{25}$[/tex], we need a common denominator. The least common denominator (LCD) of 15 and 25 is 75. We convert each fraction:
[tex]$$
\frac{4}{15} = \frac{4 \times 5}{15 \times 5} = \frac{20}{75}, \quad \frac{12}{25} = \frac{12 \times 3}{25 \times 3} = \frac{36}{75}.
$$[/tex]
Now, add the fractions:
[tex]$$
\frac{20}{75} + \frac{36}{75} = \frac{20 + 36}{75} = \frac{56}{75}.
$$[/tex]
Step 3. Multiply by the Common Factor
Multiply the common factor by the sum obtained:
[tex]$$
\frac{5}{14} \times \frac{56}{75}.
$$[/tex]
Multiply the numerators and denominators:
[tex]$$
\frac{5 \times 56}{14 \times 75} = \frac{280}{1050}.
$$[/tex]
Step 4. Simplify the Fraction
Divide the numerator and the denominator by their greatest common divisor. Dividing both by 70 (since [tex]$280 \div 70 = 4$[/tex] and [tex]$1050 \div 70 = 15$[/tex]) gives:
[tex]$$
\frac{280}{1050} = \frac{4}{15}.
$$[/tex]
Final Answer
The simplified expression is
[tex]$$
\boxed{\frac{4}{15}}.
$$[/tex]