Think of the decimal as the fraction .

Now, multiply the numerator and denominator by for every digit after the decimal. In this case, because there are digits after the decimal, we will be multiplying by twice, or by .

Now, simplify this fraction.

Think of the decimal as the fraction .

Now, multiply the numerator and denominator by for every digit after the decimal. In this case, because there are digits after the decimal, we will be multiplying by twice, or by .

Now, simplify this fraction.

Write the decimal divided by .

Multiply the numerator and denominator by to get rid of the decimal, then simplify.

Write the decimal divided by .

Multiply the numerator and denominator by to get rid of the decimal, then simplify.

There is no other way but to analyze each answer choice. We do have a decimal choice so lets compare the decimal to all of the fractions. Choice I. is . Even if you don't see that, first divide the numerator and denominator by , then , and you will see that it's Choice I is wrong**.** Choice II is definitely bigger than . Reason is because if you look at the numerator, if I double it, that number is . Because this value is bigger than the denominator, this means the overall fraction is bigger than Remember, the bigger the denominator, the smaller the fraction. ( is greater than even though is bigger than ) The converse is the same. If apply this reasoning to both Choice III and V, only choice Vcan be eliminated. Choice III is hard to figure out the exact decimal value but if we didn't have a calculator, we can surely compare their values. Let's force choice IV into a fraction. The only way to compare these fractions easily is by having the same denominator. So, lets multiply with which gives us . So we are comparing with . Since is greater than this makes choice III bigger than and therefore makes choice IV the smallest value.

There is no other way but to analyze each answer choice. We do have a decimal choice so lets compare the decimal to all of the fractions. Choice I. is . Even if you don't see that, first divide the numerator and denominator by , then , and you will see that it's Choice I is wrong**.** Choice II is definitely bigger than . Reason is because if you look at the numerator, if I double it, that number is . Because this value is bigger than the denominator, this means the overall fraction is bigger than Remember, the bigger the denominator, the smaller the fraction. ( is greater than even though is bigger than ) The converse is the same. If apply this reasoning to both Choice III and V, only choice Vcan be eliminated. Choice III is hard to figure out the exact decimal value but if we didn't have a calculator, we can surely compare their values. Let's force choice IV into a fraction. The only way to compare these fractions easily is by having the same denominator. So, lets multiply with which gives us . So we are comparing with . Since is greater than this makes choice III bigger than and therefore makes choice IV the smallest value.

,

so the correct response is .

,

so the correct response is .

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.

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.