\(\displaystyle \left ( \frac{5}{6}\times \frac{12}{13} \right )^{2}\div \frac{3}{4}\)

First we will evaluate the terms in the parentheses:

\(\displaystyle \left ( \frac{5}{6}\times \frac{2\times 6}{13} \right )^{2}\div \frac{3}{4}\)

\(\displaystyle \left ( \frac{5}{1}\times \frac{2}{13} \right )^{2}\div \frac{3}{4}\)

\(\displaystyle \left ( \frac{10}{13} \right )^{2}\div \frac{3}{4}\)

Next, we will square the first fraction:

\(\displaystyle \left ( \frac{10^{2}}{13^{2}} \right )\div \frac{3}{4}\)

\(\displaystyle \frac{100}{169}\div \frac{3}{4}\)

We can evaluate the division as such:

\(\displaystyle \frac{100}{169}\times\frac{4}{3}=\frac{400}{507}\)

When dividing fractions, you invert the second term and multiply the numbers. 

\(\displaystyle \frac{3}{9}\times\frac{15}{4}\) 

You can reduce the numbers that are diagonal from each other to make the numbers smaller and easier to multiply. 

\(\displaystyle \frac{3}{3}\times\frac{5}{4}=\frac{5}{4}=1\frac{1}{4}\)

When you dividing fractions, multiply by the reciprocal of the denominator.

\(\displaystyle \frac{\frac{13x}{5y}}{\frac{3x}{8y}}=\frac{13x}{5y}*\frac{8y}{3x}=\frac{13*x*8*y}{5*y*3*x}=\frac{13*8}{5*3}=\frac{104}{15}\)

First, start by putting changing the fractions in the equation so that they have the same denominator.

\(\displaystyle x=\frac{4}{3}-\frac{7}{12}\)

\(\displaystyle x=\frac{16}{12}-\frac{7}{12}\)

Then, you can easily subtract the numerators. The denominator stays the same at this point:

\(\displaystyle x=\frac{9}{12}\)

At this point, you can reduce the fraction by dividing the numerator and the denominator by \(\displaystyle 3\):

\(\displaystyle x=\frac{3}{4}\)

Then, to easily compare the fraction to the answer choices, you can change the fraction to a decimal:

\(\displaystyle \frac{3}{4}=0.75\)

By changing the answer choices to decimals as well, you can easily figure out which value is greater than \(\displaystyle 0.75\):

\(\displaystyle \frac{3}{5}=0.6< 0.75\)

\(\displaystyle \frac{5}{7}=0.71< 0.75\)

\(\displaystyle \frac{2}{3}=0.667< 0.75\)

\(\displaystyle \frac{3}{4}=0.75=0.75\)

\(\displaystyle \frac{7}{8}=0.875>0.75\), so \(\displaystyle \frac{7}{8}\) is the correct answer.

1/3 of the movies are action movies. 3/5 of 2/3 of the movies are comedies, or (3/5)*(2/3), or 6/15. Combining the comedies and the action movies (1/3 or 5/15), we get 11/15 of the movies being either action or comedy. Thus, 4/15 of the movies remain and all of them have to be historical.

Substitute the values of x and y into the given expression:

2(1/3) + 3(1/2)

= 2/3 + 3/2

= 4/6 + 9/6

= 13/6

\(\displaystyle x=\frac{7}{9}+\frac{3}{5}+\frac{2}{15}+\frac{7}{45}=\frac{35}{45}+\frac{27}{45}+\frac{6}{45}+\frac{7}{45}=\frac{75}{45}=\frac{5}{3}\)

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140)  = 43/140, which cannot be reduced.

Find the least common denominator to solve this problem

Multiply 27 with \(\displaystyle x\), and multiply \(\displaystyle 9x\) with 3 to obtain common denominators.

Convert the fractions.

\(\displaystyle \frac{2}{9x}+\frac{4}{27}=\frac{6}{27x}+\frac{4x}{27x}\)

Combine the terms as one fraction.

The answer is:  \(\displaystyle \frac{4x+6}{27x}\)