If $a:b::\frac{12}{5}:\frac{3}{10}$ and $b:c::\frac{7}{9}:\frac{12}{7}$ then $a:c$ is

$\frac{98}{27}$

The correct answer is Option (1) → $\frac{98}{27}$

To find the ratio $a:c$, we can use the mathematical property that the ratio of the first term to the third term is the product of the first two ratios:

$\frac{a}{c} = \frac{a}{b} \times \frac{b}{c}$

Step 1: Simplify the first ratio ($a:b$)

Given $a:b = \frac{12}{5} : \frac{3}{10}$

$\frac{a}{b} = \frac{12}{5} \div \frac{3}{10}$

$\frac{a}{b} = \frac{12}{5} \times \frac{10}{3}$

$\frac{a}{b} = \frac{12 \times 10}{5 \times 3} = \frac{120}{15} = 8$

Step 2: Simplify the second ratio ($b:c$)

Given $b:c = \frac{7}{9} : \frac{12}{7}$

$\frac{b}{c} = \frac{7}{9} \div \frac{12}{7}$

$\frac{b}{c} = \frac{7}{9} \times \frac{7}{12}$

$\frac{b}{c} = \frac{49}{108}$

Step 3: Calculate $a:c$

Multiply the two results:

$\frac{a}{c} = 8 \times \frac{49}{108}$

Divide $8$ and $108$ by their greatest common divisor, which is $4$:

$\frac{a}{c} = 2 \times \frac{49}{27}$

$\frac{a}{c} = \frac{98}{27}$