One of many questions GMAT can ask you is how to compare two or more fractions (i.e., which fractions are bigger/smaller). We will discuss a very simple method for comparing fractions.
How to compare two fractions. To compare two fractions, we need two steps.
Step 1. Make the same positive bottoms.
For example, let's compare $1/13$ and $2/25$. To make the same bottom, you multiply the opposite bottom up and down. For $1/13$, the opposite bottom is $25$, we have
$$\frac{1}{13} = \frac{1 \cdot 25}{13 \cdot 25} = \frac{25}{13 \cdot 25}.$$
Why didn't we compute $13 \cdot 25$? Just wait for it now. We will explain why after comparing $1/13$ and $2/25$. For $2/25$, the opposite bottom is $13$, so
$$\frac{2}{25} = \frac{13 \cdot 2}{13 \cdot 25} = \frac{26}{13 \cdot 25}.$$
Step 2. Compare the tops.
We continue the example from Step 1. We have
$$\frac{1}{13} = \frac{25}{13 \cdot 25} < \frac{26}{13 \cdot 25} = \frac{2}{25}.$$
This implies that $1/13$ is smaller than $2/25$. In math notations, we have $1/13 < 2/25$.
Remark. Note that in the example we used, we did not need to calculate $13 \cdot 25 = 325$ because it served as the same positive bottom. As long as we have the same positive bottom, the only things we need to compute are the tops, not the bottom.
Exercise. Which one of the following choices are bigger?
Choice 1. $2/10$,
Choice 2. $4/91$.
Solution. We have
$$\frac{2}{10} = \frac{2 \cdot 91}{10 \cdot 91} = \frac{182}{10 \cdot 91}$$
and
$$\frac{4}{91} = \frac{10 \cdot 4}{10 \cdot 91} = \frac{40}{10 \cdot 91},$$
so comparing the tops with the same bottom $10 \cdot 91$, we wee that $2/10 > 4/91$, and this is the answer.
Exercise. Which one of the following choices are bigger?
Choice 1. $\dfrac{1}{23} + \dfrac{1}{4}$,
Choice 2. $\dfrac{2}{23} \cdot \dfrac{3}{2}$.
Solution. Recalling what we learned in this previous lesson about adding fractions, Choice 1 can be manipulated as
$$\frac{1}{23} + \frac{1}{4} = \frac{1 \cdot 4 + 23 \cdot 1}{23 \cdot 4} = \frac{4 + 23}{23 \cdot 4} = \frac{27}{23 \cdot 4}.$$
Why don't we multiply out $23 \cdot 4$? This is due to the same reason as in the example we worked out in the beginning of this post. In general, until you have to compute the answer, it is the best to leave the multiplication. Choice 2 can be manipulated as
$$\frac{2}{23} \cdot \frac{3}{2} = \frac{2 \cdot 3}{23 \cdot 2}.$$
Now notice that the first choice has the bottom $23 \cdot 4$, so to get the same bottom you only need to multiply $2$ up and down to the second choice. This gives you
$$\frac{2}{23} \cdot \frac{3}{2} = \frac{2 \cdot 3}{23 \cdot 2} = \frac{2 \cdot 3 \cdot 2}{23 \cdot 2 \cdot 2} = \frac{6 \cdot 2}{23 \cdot 4} = \frac{12}{23 \cdot 4}.$$
Because we have the same positive bottom $23 \cdot 4$, we can compare that
$$\frac{27}{23 \cdot 4} > \frac{12}{23 \cdot 4}$$
by comparing the tops. The left one was Choice 1, so it is bigger than Choice 2. The answer is: Choice 1 is bigger than Choice 2.
Remark. We again note that we did not have to compute $23 \cdot 4 = 92$. Of course, you can compute this, but it is NOT necessary for this problem.
Exercise. Order the following choices from the biggest to the smallest.
Choice 1. $\dfrac{1}{2}$,
Choice 2. $\dfrac{1}{3}$,
How to compare two fractions. To compare two fractions, we need two steps.
Step 1. Make the same positive bottoms.
For example, let's compare $1/13$ and $2/25$. To make the same bottom, you multiply the opposite bottom up and down. For $1/13$, the opposite bottom is $25$, we have
$$\frac{1}{13} = \frac{1 \cdot 25}{13 \cdot 25} = \frac{25}{13 \cdot 25}.$$
Why didn't we compute $13 \cdot 25$? Just wait for it now. We will explain why after comparing $1/13$ and $2/25$. For $2/25$, the opposite bottom is $13$, so
$$\frac{2}{25} = \frac{13 \cdot 2}{13 \cdot 25} = \frac{26}{13 \cdot 25}.$$
Step 2. Compare the tops.
We continue the example from Step 1. We have
$$\frac{1}{13} = \frac{25}{13 \cdot 25} < \frac{26}{13 \cdot 25} = \frac{2}{25}.$$
This implies that $1/13$ is smaller than $2/25$. In math notations, we have $1/13 < 2/25$.
Remark. Note that in the example we used, we did not need to calculate $13 \cdot 25 = 325$ because it served as the same positive bottom. As long as we have the same positive bottom, the only things we need to compute are the tops, not the bottom.
Exercise. Which one of the following choices are bigger?
Choice 1. $2/10$,
Choice 2. $4/91$.
Solution. We have
$$\frac{2}{10} = \frac{2 \cdot 91}{10 \cdot 91} = \frac{182}{10 \cdot 91}$$
and
$$\frac{4}{91} = \frac{10 \cdot 4}{10 \cdot 91} = \frac{40}{10 \cdot 91},$$
so comparing the tops with the same bottom $10 \cdot 91$, we wee that $2/10 > 4/91$, and this is the answer.
Exercise. Which one of the following choices are bigger?
Choice 1. $\dfrac{1}{23} + \dfrac{1}{4}$,
Choice 2. $\dfrac{2}{23} \cdot \dfrac{3}{2}$.
Solution. Recalling what we learned in this previous lesson about adding fractions, Choice 1 can be manipulated as
$$\frac{1}{23} + \frac{1}{4} = \frac{1 \cdot 4 + 23 \cdot 1}{23 \cdot 4} = \frac{4 + 23}{23 \cdot 4} = \frac{27}{23 \cdot 4}.$$
Why don't we multiply out $23 \cdot 4$? This is due to the same reason as in the example we worked out in the beginning of this post. In general, until you have to compute the answer, it is the best to leave the multiplication. Choice 2 can be manipulated as
$$\frac{2}{23} \cdot \frac{3}{2} = \frac{2 \cdot 3}{23 \cdot 2}.$$
Now notice that the first choice has the bottom $23 \cdot 4$, so to get the same bottom you only need to multiply $2$ up and down to the second choice. This gives you
$$\frac{2}{23} \cdot \frac{3}{2} = \frac{2 \cdot 3}{23 \cdot 2} = \frac{2 \cdot 3 \cdot 2}{23 \cdot 2 \cdot 2} = \frac{6 \cdot 2}{23 \cdot 4} = \frac{12}{23 \cdot 4}.$$
Because we have the same positive bottom $23 \cdot 4$, we can compare that
$$\frac{27}{23 \cdot 4} > \frac{12}{23 \cdot 4}$$
by comparing the tops. The left one was Choice 1, so it is bigger than Choice 2. The answer is: Choice 1 is bigger than Choice 2.
Remark. We again note that we did not have to compute $23 \cdot 4 = 92$. Of course, you can compute this, but it is NOT necessary for this problem.
Exercise. Order the following choices from the biggest to the smallest.
Choice 1. $\dfrac{1}{2}$,
Choice 2. $\dfrac{1}{3}$,
Choice 3. $\dfrac{1}{8}$.
Solution. You can compare Choices 1 and 2 and then Choices 2 and 3, using the method we have introduced. On the other hand, we can also look at the choices as the amount we get by dividing one pie into $2$ equal pieces, $3$ equal pieces, and $8$ equal pieces. Choice 1 gives you the largest share, and then Choices 2 and 3. Thus, we have Choice 1 > Choice 2 > Choice 3 as the answer.
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