First, try these practice DS questions:

**Q1**. If x and y are positive integers, is

Statement #1:

Statement #2:

**Q2**. If x and y are positive integers, is

Statement #1:

Statement #2:

Throughout this post, assume that I am talking about positive fractions with a positive numerator and positive denominator. If the fraction is negative, use the information below to figure out what happens to the absolute value of the fraction, and judge from there.

Ironically, it’s a bit easier if we add to part of the fraction and subtract from the other. The BIG idea here: if you increase the numerator and/or decrease the denominator of any positive fraction, that fraction will get bigger; if you decrease the numerator and/or increase the denominator of any positive fraction, that fraction will get smaller. Add a positive number to the numerator and/or subtract a positive number from the denominator of any positive fraction, and the new fraction will be greater than the starting fraction. Subtract a positive number from the numerator and/or add a positive number to the denominator of any positive fraction, and the new fraction will be smaller than the starting fraction. Though not relevant in the two practice problems above, this is a golden rule that will help you in a panoply of fraction and ratio problems.

Suppose we start with the positive fraction

Well, the rule here is a bit subtle. When you add the same number to numerator and denominator, the resultant fraction **is closer to \bold{1}** than is the starting fraction. This means, if the starting fraction

Here are a couple of examples.

*Example #1*

Add five to the numerator and the denominator.

Since

*Example #2*

Add two to the numerator and the denominator.

Since 1 is less than

Actually, this case is simply a generalization of the previous case. Suppose we start with a fraction

Well, the general rule is: adding a to the numerator and b to the denominator moves the resultant fraction closer to the fraction

Here are some example:

*Example #3*

Add

Resultant fraction

On the number line —

Because

*Example #4*

Add

Resultant fraction

On the number line —

Because

Now that you know these rules, go back to the practice problems at the beginning and see whether they make more sense now.

**Q1**. Statement #1: We are adding

Statement #2: Now, all we know is that the numerator of the starting fraction is less than

Now, combine the statements. We know

Neither statement is sufficient individually, but together, they are sufficient.

**Q2**. We are adding the same number,

Statement #1:

Statement #2:

Statement #1 is insufficient and Statement #2 is sufficient.