The problem involves simplifying a complex fraction. Let's break it down step by step.
Problem:
$$
\frac{\left(\frac{3}{5} + \frac{2}{15}\right)}{\left(\frac{5}{7} - \frac{3}{10}\right)}
$$
Step 1: Simplify the numerator
The numerator is:
$$
\frac{3}{5} + \frac{2}{15}
$$
To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 15 is 15.
- Convert $\frac{3}{5}$ to a fraction with denominator 15:
$$
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
$$
- Now add the fractions:
$$
\frac{9}{15} + \frac{2}{15} = \frac{9 + 2}{15} = \frac{11}{15}
$$
So, the simplified numerator is:
$$
\frac{11}{15}
$$
Step 2: Simplify the denominator
The denominator is:
$$
\frac{5}{7} - \frac{3}{10}
$$
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 10 is 70.
- Convert $\frac{5}{7}$ to a fraction with denominator 70:
$$
\frac{5}{7} = \frac{5 \times 10}{7 \times 10} = \frac{50}{70}
$$
- Convert $\frac{3}{10}$ to a fraction with denominator 70:
$$
\frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70}
$$
- Now subtract the fractions:
$$
\frac{50}{70} - \frac{21}{70} = \frac{50 - 21}{70} = \frac{29}{70}
$$
So, the simplified denominator is:
$$
\frac{29}{70}
$$
Step 3: Simplify the complex fraction
Now we have:
$$
\frac{\frac{11}{15}}{\frac{29}{70}}
$$
To simplify this, multiply the numerator by the reciprocal of the denominator:
$$
\frac{\frac{11}{15}}{\frac{29}{70}} = \frac{11}{15} \times \frac{70}{29}
$$
Step 4: Multiply the fractions
Multiply the numerators and the denominators:
$$
\frac{11}{15} \times \frac{70}{29} = \frac{11 \times 70}{15 \times 29}
$$
Simplify the multiplication:
$$
11 \times 70 = 770
$$
$$
15 \times 29 = 435
$$
So, the fraction becomes:
$$
\frac{770}{435}
$$
Step 5: Simplify the resulting fraction
To simplify $\frac{770}{435}$, find the greatest common divisor (GCD) of 770 and 435. Using the Euclidean algorithm or prime factorization:
- Prime factorization of 770:
$$
770 = 2 \times 5 \times 7 \times 11
$$
- Prime factorization of 435:
$$
435 = 3 \times 5 \times 29
$$
The only common factor is 5. Divide both the numerator and the denominator by 5:
$$
\frac{770 \div 5}{435 \div 5} = \frac{154}{87}
$$
Final Answer:
The simplified form of the complex fraction is:
$$
\boxed{\frac{154}{87}}
$$
Parent Tip: Review the logic above to help your child master the concept of complex fraction.
