How to order fractions from least to greatest or from greatest to least - GRE Quantitative Reasoning

estimate or calculate decimals after simplifying

One option is to do the actual math to find the answer. Quantity A = 3/7 + 3/8 = 45/56. Quantity B = 13/14 + 4/5 = 121/70. Here Quantity A < 1 and Quantity B > 1, so it is easy to tell which is bigger. This isn't always the case however, so often doing the math is NOT the quickest or easiest way to solve GRE questions. For example, if the two answers had been 45/56 and 72/83, it would have been very difficult to tell which was the larger fraction just by looking at them.

Instead of computationally figuring this question out, we can break it into parts.

Let's look at the first fraction in the two columns: 3/7 < 1/2 and 13/14 > 1/2, so the first fraction in Quantity B is bigger.

Next look at the second fraction: 3/8 < 1/2 and 4/5 > 1/2, so the second fraction in Quantity B is bigger.

Now we know that Quantity B is bigger without having to do any real math!

At first glance, it appears that both values are equal.7/8 and x will be between 6/7 and 8/9.

However do a quick calculation on the GRE calculator to figure out that 6/7 is approximately 0.857 and and 8/9 is approximately 0.889. 7/8 (Quantity B) is 0.875. x could equal 0.86 and be less than Quantity B or x could equal 0.88 and be greater than Quantity B. Because at least two possibilities exist (x could also equal 7/8), there is not enough information given to determine the relationship.

The trick here is to set 1/2 as a baseline; all other choices are greater than 1/2.

Next is to see how much the difference is between each numerator and denominator (if you multiply 8/11 by 2/2, it will become 16/22) - each fraction besides 1/2 has a numerator that is 6 units less than the denominator. The trick is that the largest numerator will be the largest fraction when the numerator and denominator are the same units apart on all fractions.

To add fractions with the same denominator, add their numerators and leave the denominator the same: $\frac{3}{4}$ + $\frac{3}{4}$ = $\frac{6}{4}$

To square a fraction, square its numerator and denominator: left ( $\frac{3}{4}$ right $)^{2}$ = $$\frac{3^{2}$$$}{4^{2}$} = $\frac{9}{16}$

For comparison, rewrite the first fraction with 16 as its denominator also: $\frac{6}{4}$= $\frac{6times 4}{4times 4}$ = $\frac{24}{16}$

$\frac{24}{16}$ > $\frac{9}{16}$ so $\frac{3}{4}$ + $\frac{3}{4}$ > left ( $\frac{3}{4}$ right ) ^{2}

Since the fraction is the only one with a numerator larger than its denominator, it is the only one greater than one. That means it is the largest fraction.

estimate or calculate decimals after simplifying

The trick here is to set 1/2 as a baseline; all other choices are greater than 1/2.

Next is to see how much the difference is between each numerator and denominator (if you multiply 8/11 by 2/2, it will become 16/22) - each fraction besides 1/2 has a numerator that is 6 units less than the denominator. The trick is that the largest numerator will be the largest fraction when the numerator and denominator are the same units apart on all fractions.

Now that the GRE has a built-in calculator, simply find the decimal equivalent of each of the answer choices to see which is the largest!

One option is to do the actual math to find the answer. Quantity A = 3/7 + 3/8 = 45/56. Quantity B = 13/14 + 4/5 = 121/70. Here Quantity A < 1 and Quantity B > 1, so it is easy to tell which is bigger. This isn't always the case however, so often doing the math is NOT the quickest or easiest way to solve GRE questions. For example, if the two answers had been 45/56 and 72/83, it would have been very difficult to tell which was the larger fraction just by looking at them.

Instead of computationally figuring this question out, we can break it into parts.

Let's look at the first fraction in the two columns: 3/7 < 1/2 and 13/14 > 1/2, so the first fraction in Quantity B is bigger.

Next look at the second fraction: 3/8 < 1/2 and 4/5 > 1/2, so the second fraction in Quantity B is bigger.

Now we know that Quantity B is bigger without having to do any real math!

At first glance, it appears that both values are equal.7/8 and x will be between 6/7 and 8/9.

However do a quick calculation on the GRE calculator to figure out that 6/7 is approximately 0.857 and and 8/9 is approximately 0.889. 7/8 (Quantity B) is 0.875. x could equal 0.86 and be less than Quantity B or x could equal 0.88 and be greater than Quantity B. Because at least two possibilities exist (x could also equal 7/8), there is not enough information given to determine the relationship.

To add fractions with the same denominator, add their numerators and leave the denominator the same: $\frac{3}{4}$ + $\frac{3}{4}$ = $\frac{6}{4}$

To square a fraction, square its numerator and denominator: left ( $\frac{3}{4}$ right $)^{2}$ = $$\frac{3^{2}$$$}{4^{2}$} = $\frac{9}{16}$

For comparison, rewrite the first fraction with 16 as its denominator also: $\frac{6}{4}$= $\frac{6times 4}{4times 4}$ = $\frac{24}{16}$

$\frac{24}{16}$ > $\frac{9}{16}$ so $\frac{3}{4}$ + $\frac{3}{4}$ > left ( $\frac{3}{4}$ right ) ^{2}

Since the fraction is the only one with a numerator larger than its denominator, it is the only one greater than one. That means it is the largest fraction.

Now that the GRE has a built-in calculator, simply find the decimal equivalent of each of the answer choices to see which is the largest!

estimate or calculate decimals after simplifying

One option is to do the actual math to find the answer. Quantity A = 3/7 + 3/8 = 45/56. Quantity B = 13/14 + 4/5 = 121/70. Here Quantity A < 1 and Quantity B > 1, so it is easy to tell which is bigger. This isn't always the case however, so often doing the math is NOT the quickest or easiest way to solve GRE questions. For example, if the two answers had been 45/56 and 72/83, it would have been very difficult to tell which was the larger fraction just by looking at them.

Instead of computationally figuring this question out, we can break it into parts.

Let's look at the first fraction in the two columns: 3/7 < 1/2 and 13/14 > 1/2, so the first fraction in Quantity B is bigger.

Next look at the second fraction: 3/8 < 1/2 and 4/5 > 1/2, so the second fraction in Quantity B is bigger.

Now we know that Quantity B is bigger without having to do any real math!

At first glance, it appears that both values are equal.7/8 and x will be between 6/7 and 8/9.

However do a quick calculation on the GRE calculator to figure out that 6/7 is approximately 0.857 and and 8/9 is approximately 0.889. 7/8 (Quantity B) is 0.875. x could equal 0.86 and be less than Quantity B or x could equal 0.88 and be greater than Quantity B. Because at least two possibilities exist (x could also equal 7/8), there is not enough information given to determine the relationship.

The trick here is to set 1/2 as a baseline; all other choices are greater than 1/2.

Next is to see how much the difference is between each numerator and denominator (if you multiply 8/11 by 2/2, it will become 16/22) - each fraction besides 1/2 has a numerator that is 6 units less than the denominator. The trick is that the largest numerator will be the largest fraction when the numerator and denominator are the same units apart on all fractions.

To add fractions with the same denominator, add their numerators and leave the denominator the same: $\frac{3}{4}$ + $\frac{3}{4}$ = $\frac{6}{4}$

To square a fraction, square its numerator and denominator: left ( $\frac{3}{4}$ right $)^{2}$ = $$\frac{3^{2}$$$}{4^{2}$} = $\frac{9}{16}$

For comparison, rewrite the first fraction with 16 as its denominator also: $\frac{6}{4}$= $\frac{6times 4}{4times 4}$ = $\frac{24}{16}$

$\frac{24}{16}$ > $\frac{9}{16}$ so $\frac{3}{4}$ + $\frac{3}{4}$ > left ( $\frac{3}{4}$ right ) ^{2}