Convert the easy fractions to a decimal. Only choice IV is simple and that value is . Lets compare to choice III which also has a decimal. Choice III is greater than choice IV so thats elminated. Lets apply a techique to determine the strengths of fractions. Lets compare choice I and II. We will cross-multiply these values but when we cross-multiply, multiply the denominator of the left fraction with the numerator of the right fraction and the product will be written next to the numerator of the right fraction. Same is done with multiplying the denominator of the right fraction with the numerator of the left fraction and the product will be written next to the numerator of the left fraction. Whichever product is greater means that fraction is greater than the other. So with choice I and II, we have products of 10200 versus 10201. Clearly 10201 is greater and that corresponds to choice II so choice I is eliminated. Lets compare now choice II and choice V. Applying this method, gives choice II the edge here. So now lets compare choice III and II. Lets convert the decimal to a fraction with a denominator of . This gives us a comparison of  to  which means choice III is clearly the biggest.  

By inspection, each answer choice has a denominator of  with the exception of the fraction of . This can be fixed by dividing the fraction by  which will be . To find the correct numerator value, just multiply  by  which is . 

Lets try to get these values into whole numbers. To do that, we need to make sure both numerator and denominator are whole numbers and we will look for the number with the most decimal places. This is found in the numerator. To get it into a whole number, we will multiply by  or move the decimal places  to the right. When you do that for the numerator, the same applies for the denominator. The new fraction is . Just reduce the fraction by factoring out a  and the answer is shown. 

Convert each fraction to decimal. We have  and . Add these values. Remember when adding decimals, make sure the decimal places are all lined up. 

Lets place a  under the decimal. We are going to make a fraction. Essentially, it's still the same value. Now, we want a whole number in the numerator and what we do to the numerator is done to the denominator as well. Lets multiply the top by  or move the decimal place  spots to the right. Now we have . This needs to be reduced, so divide by  twice until it can't be reduced any further. The answer should be . 

First, lets convert the exponent into a fraction. Any negative exponent means it's the reciprocal of the positive exponent. So  means . Now lets multiply it with the . This means  or  or .

Let  be .  

Lets multiply  by . Now we have:

Lets subtract this equation with the first one and we get:

 We do this because we want to get rid of the repeating decimals and now we have a simple equation, isolate  and we arrive at the final answer. 

Let  be .  

Lets multiply  by . I chose , because there is a set of  numbers that make the repeating decimal. To determine the value to multiply the repeating decimal, we do  to the power of number in a set before it repeats.

Now we have:

Lets subtract this equation with the first one and we get:

 We do this because we want to get rid of the repeating decimals and now we have a simple equation, isolate  and we arrive at the final answer.