The ISEE: Quantitative Comparisons

Quantitative Comparisons, or QCs, appear on only the Upper and Middle Level ISEE. If you are not taking one of those tests, don’t worry about QCs.
 

Of the approximately 37 math questions in the Quantitative Reasoning section, about 15 are QCs. In a QC, instead of solving for a particular value, you need to compare two quantities. You’ll see two mathematical expressions: one in Column A, the other in Column B. Your job is to compare them.
Some questions include additional information about one or both quantities. This information is centered, unboxed, and essential to making the comparison.
The directions will look something like this:


Choice (D) is the only choice that represents a relationship that cannot be determined. (A), (B), and (C) all mean that a definite relationship can be found between the quantities in Columns A and B.
There are three things to remember about choice (D):
Rule 1: (D) is rarely correct for the first few QC questions.
Rule 2: (D) is never correct if the two columns contain only numbers.
Rule 3: (D) is correct if there’s more than one possible relationship between the two columns.



 
(A) If the quantity in Column A is greater
(B) If the quantity in Column B is greater
(C) If the two quantities are equal
(D) If the relationship cannot be determined from the information given
If x is a positive number, then Column A is larger. If x is equal to zero, then the quantities in Columns A and B are equal. If x is a negative number, then Column B is larger.
There is more than one possible relationship between Columns A and B here, so according to rule 3, (D) is the correct choice. As soon as you realize that there is more than one possible relationship, choose (D) and move on.

The following five strategies will help you to make quick comparisons. The key is to compare the values rather than calculate them.
Strategy 1: Compare piece by piece.
Strategy 2: Make one column look like the other.
Strategy 3: Do the same thing to both columns.
Strategy 4: Pick numbers.
Strategy 5: Avoid QC traps.
Let’s look at each strategy in detail.

This applies to QCs that compare two sums or two products.

a > b > c > d


 
We’re given four variables, or “pieces,” in the above example, as well as the relationship between these pieces. We’re told that a is greater than all of the other pieces, while c is greater than only d, etc. The next step is to compare the value of each piece in each column. If every piece in one column is greater than the corresponding piece in the other column, and if addition is the only mathematical operation involved, the column with the greater individual values (a > b and c > d ) will have the greater total value (a + c > b + d).
In other words, we know from the information given that a > b, and c > d. Therefore, the first term in Column A, a, is greater than its corresponding term in Column B, b. Likewise, the second term in Column A, c, is greater than d, its corresponding term in Column B. Since each individual “piece” in Column A is greater than its corresponding “piece” in Column B, the total value of Column A must be greater. The answer is (A).

Use this strategy when the quantities in the two columns look so different that a direct comparison would be impossible.
If the quantities in Column A and B are expressed differently, or if one looks more complicated than the other, try to make a direct comparison easier by changing one column to look more like the other.
Let’s try an example in which the quantities in Column A and Column B are expressed differently.



 
In the example above, it’s difficult to make a direct comparison as the quantities are written. However, if you get rid of the parentheses in Column A so that the quantity more closely resembles that in Column B, you should see the relationship right away. If you multiply to get rid of the parentheses in Column A, you’ll end up with 2x + 2 in both columns. Therefore, the columns are equal in value, and the answer is (C).
This strategy is also useful when one column looks more complicated than the other.

By adding or subtracting the same amount from both columns, you can often unclutter a comparison and make the relationship more apparent. You can also multiply or divide both columns by the same positive number. This keeps the relationship between the columns the same. If the quantities in both columns are positive, you can square both columns. This also keeps the relationship between the columns the same.
Changing the values, and not just the appearances of the quantities in both columns, is often helpful in tackling QC questions. Set up the problem as an inequality with the two columns as opposing sides of the inequality.
To change the values of the columns, add or subtract the same amount from both columns and multiply or divide by a positive number without changing the absolute relationship. But be careful. Remember that the direction of an inequality sign will be reversed if you multiply or divide by a negative number. Since this reversal will alter the relationship between the two columns, avoid multiplying or dividing by a negative number.
You can also square the quantities in both columns when both columns are positive. But be careful. Do not square both columns unless you know for certain that both columns are positive. Remember these two things when squaring the quantities in both columns: (1) the direction of an inequality sign can be reversed if one or both quantities are negative, and (2) the inequality sign can be changed to an equals sign if one quantity is positive and the other quantity is negative, with one quantity being the negative of the other.

Substitute numbers into those abstract algebra QCs. Try using a positive, a negative, and zero. A fraction can also be a handy choice for high difficulty problems.
If a QC involves variables, Pick Numbers to clarify the relationship. Here’s what to do:

1. Pick Numbers that are easy to work with: positive, negative, zero, and fraction.
2. Plug in the numbers and calculate the values. What’s the relationship between the columns?
3. Pick a different number for each variable and recalculate. See if you get a different relationship.

Never assume that all variables represent positive integers. Unless you’re told otherwise, variables can be positive or negative, and they can be zero or fractions. Because different kinds of numbers behave differently, you should always choose a different kind of number the second time around.

Keep your eyes open for those trick questions designed to fool you into the obvious but wrong answer. Questions are arranged in order of increasing difficulty, so chances are you’ll see traps toward the end of the set.
To avoid these nasty traps, always be on your toes. Never assume anything. Be particularly careful toward the end of a QC set.
Check out some qualitative comparison practice questions to test your knowledge!