With the numerator having more decimal spots than the denominator, we need to move the decimal point in the numerator two places to the right.

Then in the denominator, we move the decimal point also two to the right. Since there's only one decimal place we just add one more zero.

Then we can reduce by dividing top and bottom by $\displaystyle 2$.

$\displaystyle \frac{0.78}{0.8}=\frac{7.8}{8}=\frac{78}{80}=\frac{39}{40}$

Since there are four decimal places, we shift the decimal point in the numerator four places to the right.

For the denominator, since there is no decimal point, we just add four more zeroes.

Then reduce by dividing top and bottom by $\displaystyle 25$.

$\displaystyle \frac{0.6025}{8}=\frac{6025}{80000}=\frac{241}{3200}$

We need to convert the sentence into a math expression. Anytime there is "of" means we need to multiply. Let's first convert the decimal to a fraction. We need to move the decimal point two places to the right.

Since $\displaystyle 0.26$ is the same as $\displaystyle \frac{0.26}{1}$ we can add two more zeroes to the denominator.

$\displaystyle \frac{26}{100}\cdot\frac{1}{13}$ 

We can reduce the $\displaystyle 26$ to a $\displaystyle 2$ and the $\displaystyle 13$ to a $\displaystyle 1$.

$\displaystyle \frac{26}{100}\cdot \frac{1}{13}=\frac{2}{100}\cdot \frac{1}{1}$

Then reduce the $\displaystyle 2$ to $\displaystyle 1$ and the $\displaystyle 100$ to $\displaystyle 50$.

$\displaystyle \frac{2}{100}=\frac{2 \cdot 1}{2 \cdot 50}=\frac{1}{50}$.

Then dividing $\displaystyle 50$ into $\displaystyle 1$ and we get $\displaystyle 0.02$. 

We need to convert this sentence into a math expression. Anytime there is "of" in a sentence it means we need to multiply. Let's convert $\displaystyle \frac{1}{10}$ into a decimal which is $\displaystyle 0.1$.

Thus our mathematical expression becomes:

$\displaystyle 0.35x=0.1$.

 Divide both sides by $\displaystyle 0.35$.

$\displaystyle x=\frac{0.1}{0.35}$ 

Move decimal point two places to the right. The numerator will become $\displaystyle 10$. Then simplify by dividing top and bottom by $\displaystyle 5$.

$\displaystyle x=\frac{10}{35}=\frac{2}{7}$ 

Let's convert the decimal into a fraction.

$\displaystyle \frac{1}{2}x^2+x+\frac{1}{2}=0$ 

If we multiply everything by $\displaystyle 2$, we should have an easier quadratic.

$\displaystyle x^2+2x+1=0$ 

Remember, we need to find two terms that are factors of the c term that add up to the b term. 

$\displaystyle (x+1)(x+1)=0$

$\displaystyle x=-1$ This is the only value.

Let's actually simplify the top of the fraction. $\displaystyle 3$ divides into $\displaystyle 0.48$.

We should have: 

$\displaystyle \frac{\frac{0.48}{3}}{5}=\frac{0.16}{5}$.

Then move the decimal two spots to the right and add two zeroes to the denominator.

$\displaystyle \frac{16}{500}$ 

Let's actually multiply top and bottom by $\displaystyle 2$ to get:  $\displaystyle \frac{32}{1000}$.

Now we want to eliminate those zeroes. By dividing, the decimal point in the numerator moves to the left three places to get an answer of $\displaystyle \frac{0.032}{1}$ or $\displaystyle 0.032$.

Since each decimal has two digits, we can convert easily to integers.

$\displaystyle \frac{\frac{0.39}{0.81}}{\frac{0.52}{0.27}}=\frac{\frac{39}{81}}{\frac{52}{27}}$ 

Then multiply top and bottom by $\displaystyle \frac{27}{52}$ to get: $\displaystyle \frac{39}{81}\cdot \frac{27}{52}$

$\displaystyle 81$ is reduced to $\displaystyle 3$ and $\displaystyle 27$ is reduced to $\displaystyle 1.$ 

Then $\displaystyle 39$ and $\displaystyle 52$ can be divided by $\displaystyle 13$ to get $\displaystyle 3$ and $\displaystyle 4$ respectively.

$\displaystyle \frac{39}{81}\cdot\frac{27}{52}=\frac{3}{3}\cdot\frac{1}{4}=\frac{1}{4}$

Let $\displaystyle 3.567567...$ be $\displaystyle x$. Let's multiply that value by $\displaystyle 1000$. The reason is when we subtract it, we will get us an integer instead and the repeating decimals will disappear. 

$\displaystyle 3.567567..=x$

$\displaystyle 3567.567567...=1000x$

If we subtract, we get $\displaystyle 3564=999x$.

Divide both sides by $\displaystyle 999$ and we get $\displaystyle \frac{3564}{999}$.

If you divide by $\displaystyle 27$ on top and bottom, you should get the answer. Otherwise, just divide top and bottom by $\displaystyle 3$ three times based on the divisibility rules for $\displaystyle 3$. If the sum is divisible by $\displaystyle 3$, then the number is divisible by $\displaystyle 3$.

$\displaystyle \frac{3564}{999}=\frac{27 \cdot 132}{27\cdot 37}=\frac{132}{37}$

To compare these two quantities, we'll want to simplify Quantity A.

The fraction

$\displaystyle \frac{0.0002(0.09)(0.4)}{0.000006(0.2)}$

may be a bit daunting; let's convert it to scientific notation:

$\displaystyle \frac{2\cdot10^{-4}(9\cdot10^{-2})(4\cdot 10^{-1})}{6\cdot10^{-6}(2\cdot10^{-1})}$

Now multiply the non-ten terms, and the ten terms (add the exponents together):

$\displaystyle \frac{72\cdot(10^{-4-2-1})}{12\cdot10^{-6-1}}$

$\displaystyle \frac{72\cdot10^{-7}}{12\cdot10^{-7}}$

Now cancel like factors in the numerator and denominator:

$\displaystyle \frac{6(12)\cdot10^{-7}}{12\cdot10^{-7}}$

$\displaystyle 6$

The two quantities are equal.

To begin, it can be useful to convert the values in the fraction

$\displaystyle \frac{300(0.00002)(70)}{0.0021(40)}$

into a modified scientific notationnotation:

$\displaystyle \frac{3\cdot10^2(2\cdot10^{-5})(7\cdot10^1)}{21\cdot10^{-4}(4\cdot10^1)}$

Now multiply the ten terms (adding exponents together) and the non-ten terms:

$\displaystyle \frac{42\cdot 10^{-2}}{84\cdot10^{-3}}$

From here, reduce the terms, subtracting the bottom tens exponent from the top tens exponent:

$\displaystyle \frac{1\cdot 10^{-2-(-3)}}{2}$

$\displaystyle \frac{1\cdot 10^{1}}{2}$

$\displaystyle 5$