Gruber's Essential Guide to Test Taking: Grades 6-9 - Gruber Gary R. 2019

Quantitative comparison
Math strategies

The quantitative comparison question has been used more and more on standardized tests. In this question, the student is asked to compare quantities in two columns. He or she is asked to find whether the quantity in the first column is greater, less than, or equal to the quantity in the second column. On exams of this level (grades 6 to 9 and up), the student is also asked to state whether in fact a determination can even be made.

Here’s a very simple illustration of this:

EXAMPLE

SOLUTION

Since 2 is greater than 1, you would say Column A is greater than Column B and so Choice A is correct.

There are many questions of the quantitative comparison type much more difficult than the example above that will take a student a long time to answer and subject the student to mistakes if he or she doesn’t know certain strategies.

Here are some rules that you and your child will discover for yourselves as you become more familiar with the strategies, but it’s a good idea to see them now and be aware of them:

1Never choose D if just actual numbers (no algebraic letters) are present in the columns.

2You can always substitute numbers for the letters to see how the columns compare.

3You can always add or subtract the same quantity to both columns and still get the same comparison between the columns.

4You can always multiply or divide both columns by the same positive number and still get the same comparison between the columns.

QUANITTAITVE COMPARISON STRATEGY 1: Cancel Quantities Common to Both Colums

EXAMPLE 1

Remember:

Choose A if Column A is greater than Column B.

Choose B if Column A is less than Column B.

Choose C if Column A equals Column B.

Choose D if a definite comparison cannot be made.

How do you think this problem should be approached? Should we add the 124 4 376 under Column A and then compare the sum with that of 376 + 125 under Column B? That’s probably the way your child was taught to do such problems, but it’s not the easiest or most direct way. Since in quantitative comparison questions you are not asked to find actual results or answers, but only to compare the columns, it isn’t necessary to calculate what’s under each column. After all, if the test maker had wanted your child just to do a straight calculation, he or she would have used the question in the regular multiple-choice math section, not in the quantitative comparison section.

Here’s the strategy: Get rid of quantities common to both columns.

In the example above, get rid of the 376 that appears in both columns:

Since 124 is less than 125, Column A is less than Column B. Choice B is therefore correct.

This strategy also works for numbers that are multiplied. For example:

EXAMPLE 2

Whatever you do, don’t multiply the numbers in each column! Cancel the common 34 and 31 from both columns:

Column A is greater than Column B, so Choice A is correct. There’s just one note of caution: Remember, in multiplication or division problems, never cancel if you may be canceling a negative number or 0 :

EXAMPLE 3

If you canceled the a, you would find that 3 is greater than 2 and choose A as the answer. But that’s not correct. Because if a — 0, the columns are equal and if a is negative, Column A is less than Column B! So when using this strategy for multiplication or division problems (don’t worry about addition or subtraction problems) do not cancel a negative number or cancel 0 from both columns.

The way you would do Example 3 is to try different values for a : Let a = 0: Then the columns are equal. Column A = Column B = 0. Let a = 1: Column A = 3 and Column B = 2; therefore Column A > Column B.

Since you get two different comparisons, a definite comparison of the columns cannot be made and Choice D is correct according to the directions.

More on these Choice D questions will be given later, in connection with another strategy, so if this is not completely clear now, it will become clearer later when we do more of these problems.

Have your child try the following examples. Check to see whether his or her solutions match those given in the book, making sure that your child has used the strategies described here.

Remember:

Choose A if Column A is greater than Column B.

Choose B if Column A is less than Column B.

Choose C if Column A equals Column

Choose D if a definite comparison cannot be made.

PROBLEMS

SOLUTIONS

1 (C) Cancel common quantities from both sides:

Column A = Column B

2 (A) Cancel common quantities:

Column A is greater.

3 (A) Write columns like this:

Now cancel common quantities from both columns:

Column A is greater than Column B.

4 (A) Cancel common quantities:

Column A is greater than column B

5 (A) Rewrite as:

Cancel common quantities:

Column A is greater than Column B

6 (A) Cancel common quantities:

Column A is greater than Column B.

7 (B) Cancel a from both columns:

Column A is less than Column B.

8 (D)This is tricky: In multiplication you can’t just cancel the b because b may be 0 or be negative.

If b is not 0, then 5b is less than 6b.

If b is 0, then 5b = 6b.

Thus a definite comparison cannot be made and Choice D is correct.

QUANTITATIVE COMPARISON STRATEGY 2: To Simplify You Can Multiply, Divide, Ass, or Subtract the Same Number in Both Columns

Remember:

Choose A if Column A is greater than Column B.

Choose B if Column A is less than Column B.

Choose C if Column A equals Column B.

Choose D if a definite comparison cannot be made.

EXAMPLE 1

Column A is less than Column B, so Choice B is correct. We added 297 to both columns to get rid of the minus sign in Column A, since it’s easier to add than subtract in this case.

EXAMPLE 2

Here’s a problem where you’d subtract instead of add.

Don’t add yet! You can see that the 478 in Column A is just 1 unit less than the 479 in Column B. You can also see that the 424 in Column A is just 1 unit more than the 423 in Column B. Let’s subtract 423 from both columns.

Column A equals Column B, so Choice C is correct.

Here are a few examples that require multiplying each column by the same number. (This type of problem is explained in the “Math Shortcuts” section on page 175.)

EXAMPLE 3

Use the following strategy: Try to get rid of denominators—they make things difficult. Do this by multiplying Column A and Column B by 7. This cancels the 7 in the Column A denominator:

Now multiply both columns by 9 to get rid of the denominator in Column B:

Column A is Greater than Column B, so Choice A is correct.

EXAMPLE 4

Column A is greater than Column B and therefore Choice A is correct.

EXAMPLE 5

Since x is positive (given), Column A is less than Column B and Choice B is correct.

In the following example, both columns should be divided by the same number.

EXAMPLE 6

Column A is less than Column B, so Choice B is correct.

Have your child try the following questions. Make sure that he or she uses the method described in the book.

Remember:

Choose A if Column A is greater than Column B.

Choose B if Column A is less than Column B.

Choose C if Column A equals Column B.

Choose D if a definite comparison cannot be made.

PROBLEMS

SOLUTIONS

Column A equals Column B.

4 (D)Multiply by p and divide by 5:

If p = 1 (given), then Column A = Column B.

If ft > 1 (given), then p X p > 1, so Column A > Column B. Thus a definite relation cannot be determined.

5 (B)Don’t subtract! Get rid of the minus sign by adding to both columns.

Column A is less than Column B.

6 (B)You don’t have to add the numbers in the columns! Subtract 90 from both columns and then subtract 87 from both columns:

Column A is less than Column B.

QUANTITATIVE COMPARISON STRATEGY 3: Use Common Sense to Answer Questions

Often common sense can be used to answer quantitative comparison questions. Your child should use common sense whenever possible instead of racking his or her brains to solve certain problems. The less “brain-racking” there is on the test, the less exhausted your child will become and the more confident he or she will be when attacking the remaining questions.

Remember:

Choose A if Column A is greater than Column B

Choose B if Column A is less than Column B

Choose C if Column A equals Column B

Choose D if a definite comparison cannot be made.

EXAMPLE!

It is obvious that it would take longer to travel 60 kilometers than it would to travel 40 kilometers, if your rate of travel (30 kph) is the same. There is no need to memorize any formulas about rate and time. Choice A is correct.

EXAMPLE 2

Since the shortest distance between any two points is a straight line, each of the sides of the triangle has a shorter distance than the arc of the circle it cuts. So the sum of all the arcs (the whole circle) must be larger than the sum of the sides of the triangle (the perimeter of the triangle). Choice A is correct.

EXAMPLE 3

It should be clear that since a month is almost four times longer than a week, 7 months and 6 weeks is longer than 6 months and 7 weeks. There is no need to calculate the exact amount of days. Choice A is correct.

EXAMPLE 4

Common sense: The area of a circle with the smaller radius is smaller. Column A is less than Column B. Choice B is correct.

EXAMPLE 5

Common sense: If Harry’s sister is 12, that doesn’t tell Harry’s age. Similarly, if Mary’s brother is 13, that doesn’t tell Mary’s age. A definite comparison between Harry’s age and Mary’s age cannot be made. Choice D is correct.

EXAMPLE 6

If Q is greater than 6, it can be any number greater than 6.

If P is greater than 8, it can be any number greater than 8.

Any number greater than 6 can be greater than, less than, or equal to a number greater than 8. Thus a definite comparison cannot be made. Choice D is correct.

EXAMPLE 7

Use common sense: The average of 5, 5, 5, 5, and 2 is

The average of 2, 2, 2, 2, 5 is

You can see that four 5’s and a 2 is greater than four 2’s and a 5 without going through the addition or division. Column A is greater than Column B, and Choice A is correct.

Have your child try the following examples. Check to see whether his or her solutions match those given in the book. Make sure your child used the strategies presented here.

Remember:

Choose A if Column A is greater than Column B

Choose B if Column A is less than Column B

Choose C if Column A equals Column B

Choose D if a definite comparison cannot be made.

PROBLEMS

SOLUTIONS

1 (D)You don’t know what Mary’s allowance is, so of her allowance is not determined. What’s left is also riot determined. Tlie same situation applies to what’s left for Paul’s allowance. Thus a comparison of the columns cannot be made.

2 (B)Think, a whole number that is less than 6 is 5, 4, 3 etc. A whole number greater than 5 is 6, 7, 8, etc.

Any number in Column A is less than any number in Column B.

3 (A)5 2 is greater than 2.5> so Column A is greater than Column B.

4 (D)Let Phil’s wages = P. So in column A, Sam’s wages are 3P. In Column B, Phil’s wages are P. 3P>P, so Column A>Column B.

5 (B)You should realize that if AB is a diameter of the circle (it passes through the center O), any chord that is not a diameter, like CD is going to be shorter in length. So Column A is less than Column B.

6 (A)You should realize that the average of 17 and 31 is some where between 17 and 31. In Column B, however, the average of 17, 20, and 31 is going to be closer to the 17 than to the 31 because the 20 is going to drag the average down toward the 17. Thus the average of 17 and 31 is greater (Column A is greater than Column B.)

7 (A)This problem is tricky. Look at all of the odd numbers greater than 4 but less than 100:

5, 7, 9 ... all the way up to 99.

Then look at all of the even numbers:

6, 8, 10 ... all the way up to 98.

Now write the even numbers below the odd numbers:

You can see that there is one less even number than odd number.

So Column A is greater than Column B.

QUANTITATIVE COMPARISON STRATEGY 4: Try Numbers for Variables/Try to Get Different Comparisons

If you are given a comparison with variables like x or y in the columns, you may want to substitute numbers for them. Often a definite comparison is not possible. If you get different comparisons (like Column A > Column B and Column A = Column B), then a definite relationship cannot be made and Choice D is correct. The way to show that Choice D is correct is to choose numbers that give you comparisons like Column A > Column B. Then try to get another set of numbers for the variables, making something like Column A < Column B, or Column A = Column B. Then you will have proven that Choice D is correct.

Remember:

Choose A if Column A is greater than Column B

Choose B if Column A is less than Column B

Choose C if Column A equals Column B

Choose D if a definite comparison cannot be made.

EXAMPLE 1

Choose numbers consistent with what is given.

Let Column A = 4 and Column B = 1. Then Column A > Column B. Now let Column A = 1 and Column B = 1. Then Column A = Column B.

You got two different comparisons. Thus Choice D is correct.

EXAMPLE 2

Choose numbers for x and y.

Let x = 1. Then since x + y = 2, y = 1. Column A = Column B. Now let x = 2. Then since x + y = 2, y = 0, so Column A > Column B.

Two different comparisons were made, so Choice D is correct.

EXAMPLE 3

a, b, and c are consecutive whole numbers where a is smallest and c is greatest.

c + 1a + 3

Let the consecutive numbers be 1, 2, and 3. c is greatest, so c = 3. a is smallest, so a — 1.

Column A = c + 1 = 3 + 1 = 4

Column B = = a + 3 = l + 3: =4

Column A = Column B.

Now choose another set of numbers. Let the consecutive whole numbers be 2, 3, 4. a = 2 and c = 4.

So Column A = c + 1 = 4 + 1 = 5 and

Column B = a + 3 = 2 + 3 = 5.

Column A = Column B again. You’d be on safe ground selecting Choice C.

EXAMPLE 4

Choose a number for the lengths.

Let AC = 5 and CE = 5. Since the figure is not drawn to scale, BC could equal something like 2 and CD could equal something like 3, making Column A < Column B.

Or BC could equal 3 and CD could equal 2, making Column A > Column B.

Thus a definite comparison cannot be obtained and Choice D is correct.

EXAMPLE 5

Choose numbers for y. Let y = 1. Then Column A = 10 = Column B.

Now let y = 10. Then Column A = 100 and Column B = 1, so Column A > Column B. Choice D is correct.

Have your child try the following examples. Check to see whether his or her solutions match those given in the book. Make sure your child uses the strategies presented here.

Remember:

Choose A if Column A is greater than Column B

Choose B if Column A is less than Column B

Choose C if Column A equals Column B

Choose D if a definite comparison cannot be made.

SOLUTIONS

1 (B)Try different numbers.

Column A < Column B

It would be safe to say that Column A < Column B. 2 (D) Try x — 1

2 (D)Try x = 1

Column A > Column B

A definite comparison cannot be made.

3 (A)Let s = 1: Column A = 4s = 4 x 1 =4

Column B = 3s = 3 x 1 = 3

Column A > Column B

Let s = 2: Column A = 4s = 4 x 2 = 8

Column B = 3s = 3 x 2 = 6

Column A > Column B

It’s safe to say that Column A > Column B.

4 (D)Since x + y = 20, let x = 1. Then y = 19. Column A = 1, Column B = 19; Column A < Column B. Let x = 19- Then y = 1 (this is because x + y = 20). So Column A = 19, Column B = 1; Column A > Column B. You got two different comparisons, so Choice D is correct.

5 (D)Translate: The ratio of a to b is 2: = 2.

Let a = 2. Then b — 1, satisfying = 2.

Column A = Column B.

Let a = 4. Then b = 2, satisfying = 2.

Column A > Column B.

Thus a definite comparison cannot be made.