How to Answer Quantitative Comparison Questions with Arithmetic

What this lesson covers

In a quantitative comparison, you compare the quantity in Column A with the quantity in Column B. These questions appear in the Quantitative Reasoning section of the ISEE Middle Level, where no calculator is allowed. This lesson uses numbers only, no letters. It teaches what the four answer choices mean, when the answer is D, and how to compare two columns without working out every product.

The four answer choices

• A: The quantity in Column A is greater.
• B: The quantity in Column B is greater.
• C: The two quantities are equal.
• D: The relationship cannot be determined from the information given.

The method

1. Decide first. Before you compute, ask whether the information fixes the relationship. If two cases that both fit the information give different results, the answer is D. Say a class has between 20 and 30 students. Column A is the number of students, and Column B is 25. A class of 22 makes Column B greater, and a class of 28 makes Column A greater. So the answer is D.
2. Fixed numbers rule out D. When each column is one fixed number, the comparison can only come out one way. Compute both columns and compare.
3. Unknown does not mean D. If every allowed case must come out the same way, give the reason and choose A, B or C. Say Lena is older than her brother. Column A is her age in 5 years, and Column B is his age in 3 years. She starts older and gains more years, so Column A is greater, whatever their ages are.
4. Remove equal parts. When both columns share a part, take the same amount away from both, or divide both by the same positive number. Then compare what's left. It works like a balance scale: take the same weight off both pans, and the heavier side stays heavier.

Worked example

Column A is $25 \times 31 + 40$. Column B is $25 \times 30 + 60$. Which column is greater?

First, the decision. Both columns are fixed numbers, so D is out.

You don't have to multiply everything out. Column B has a $25 \times 30$, so look for one in Column A. The shared-factor rule, also called the distributive property, gives $25 \times 31 = 25 \times (30 + 1) = 25 \times 30 + 25$. So Column A is $25 \times 30 + 25 + 40$, which is $25 \times 30 + 65$.

Now both columns contain $25 \times 30$. Take it away from both. Column A leaves 65, and Column B leaves 60. Since 65 is greater than 60, the answer is A. Remove only a part that really is in both columns.

To check, compute both columns: $775 + 40 = 815$ and $750 + 60 = 810$. Column A is greater.

The common mistake

The lesson's common mistake is the minus sign left behind. Say Column A is $70 - 26$ and Column B is $70 - 29$. A student takes the 70s away, compares 26 with 29, and picks B because 29 is bigger.

But subtracting 26 is the same as adding $-26$. When the 70s go, what's left is $-26$ and $-29$, not 26 and 29. On the number line, $-26$ is to the right of $-29$, so $-26$ is greater. The answer is A.

Check: $70 - 26 = 44$ and $70 - 29 = 41$, so Column A is greater. Another way to see it: Column A takes away less, so it keeps more.

Removing an equal part changes nothing else, so each sign stays with its number.