Upper Level: Quantitative Reasoning Study Guide for the ISEE

General Information

Before you begin to review terms and concepts, you need to know that there are two types of questions included in this section: Word Problems and Quantitative Comparisons. The skills and concepts covered in this study guide may be used in either type of question. See pages 27-38 of this booklet for information on these.

Tips for Word Problems

You will not be allowed to bring a calculator to the test, so it is best to study and practice without one. Besides, the word problems will be a test of your reasoning ability, requiring minimal or no computation.

Familiarity with the properties of numbers, the rules of operation, and the relationship between operations cannot be over-emphasized as good preparation for in this section.

Brush up on your skills in estimating, logical reasoning, and interpreting graphs and data. Know when to eliminate irrelevant information and absurd answer choices. Answers to some questions may be exact quantities, but there will be many items that will ask you about which steps to take, operations to apply, or which statement is true (or not true).

Answer all items from the four choices given. A wrong answer will not be counted as a penalty against your score. If you have no idea about a question, give it the best guess that you can.

Tackling Quantitative Comparisons

Quantitative comparison items compare two values written under columns A and B. Take note of statements given before or after the values under the columns. Compare the values and decide which of the following statements is true:

(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 choices for all quantitative comparison items will always be stated and written in this form and order; so it would be to your advantage to memorize these answer choices.

Now, take some time to make sure you remember the ideas listed under each of the following mathematical areas.

Numbers and Operations

Study number properties, the relative value of numbers along the number line, the four basic arithmetic operations, PEMDAS, integers, decimals, and fractions. Study the properties of odd and even numbers, prime and composite numbers, multiples and factors, exponents, and radicals. Develop the skill of translating numbers from fractions to decimals, and vice versa, with ease.

Even as you master number properties, practice on questions that use variables instead of actual numbers, such as in the quantitative comparison example below:

\[\begin{array}{c|c} \text{Column A:} & \text{Column B} \\ \hline c \cdot d & d^{c} \\ \end{array}\]

where \(c\) is a positive even integer and \(d\) is an odd integer less than \(c\)

Mathematical reasoning can be used for this type of question. You may also try plugging in numbers for \(c\) and \(d\) if you’re more comfortable with that method, or you may combine both methods to see it through comparison.

By first taking \(d\) as a positive odd integer, say \(d = 1\), the value in Column A will be greater than the value in Column B. If \(d = 3\), the value in Column A will be less than that of B. If \(d\) were a negative integer, the value in Column A will be a negative integer, while that in Column B will be positive. With these inconsistent results, we say that the relationship cannot be determined from the information given; therefore, the correct answer is (D).

Estimation and Logic

Estimating is not just a skill for the ISEE; it is a skill for life. Estimating and applying logic allow us to get to the closest answer quickly. In ISEE, this is very useful when we’re not asked for the exact answer, or when the answer choices are well spaced apart. In quantitative comparison, for instance, exact values are very rarely required; so estimation and logic are highly applicable. The example below illustrates this.

\[\begin{array}{c|c} \text{Column A} & \text{Column B} \\ \hline 8^{\frac{1}{2}} & 8 \cdot (\frac{1}{2})^2 \\ \end{array}\]

With simple calculations, the value in Column B can be easily found to be equal to \(2\). The value in Column A, however, is the square root of \(8\). It is not necessary to painstakingly solve for this, though. You know that the square root of \(4\) is \(2\) and that of \(9\) is \(3\). This tells you that the square root of \(8\) is between \(2\) and \(3\), and definitely greater than \(2\). The value in Column A is greater, so the correct answer choice is (A).

Relationships Among Operations

The operation of numbers follows a rule, called PEMDAS (or BODMAS), in determining the order that operations are performed.

Going by PEMDAS:

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.
A good way to remember: Please Excuse My Dear Aunt Sally

Familiarity with the rules governing exponents and the four arithmetic operations is a must. In the example above, it’s noteworthy to observe how exponents affect the magnitudes of quantities.

Normally, we expect that raising \(8\) to an exponent will increase the number “exponentially”, as is the case in \(8^3 = 512\). With a fractional exponent, however, we decrease it by finding its root.

A similar thing happens with the fraction \(\frac{1}{2}\)—when squared, the fraction decreases instead of increases.

\[\frac{1}{2} \times \frac{1}{2}\] \[= \frac{1 \times 1}{2 \times 2}\] \[= \frac{1}{4}\]

These are governed by rules on roots and exponents, which will be covered under the appropriate heading below.

Factors and Multiples

A factor divides a number and results in a whole number; for example, \(2\) is a factor of \(24\). The multiples of a number are the results of the number multiplied by other whole numbers; for example, \(6\), \(9\), \(12\), and \(18\) are multiples of \(3\). These concepts will help you simplify complicated expressions. The Upper-Level ISEE Quantitative Reasoning questions will not be as simply phrased as this question:

If \(x\) is the greatest common factor of \(6\) and \(9\) and \(y\) is the least common multiple of \(18\) and \(24\), which of the following is divisible by \(\frac{y}{x}\)?

A. \(36\)
B. \(48\)
C. \(30\)
D. \(44\)

Answer:

Let’s list the factors of \(6\) and \(9\):

\[6: \ \ 1,\, 2,\, 3,\, 6\] \[9: \ \ 1,\, 3, \,9\]

The common factors are \(1\) and \(3\). The greatest is \(3\). Thus, the greatest common factor of \(6\) and \(9\) is \(3\). So, \(x = 3\).

Now, let’s list some multiples of \(18\) and \(24\):

\[18: \ \ 18, \,36, \,54,\, 72,\, 90, …\] \[24: \ \ 24, \,48,\, 72, \,96,\, 120, …\]

We can see that the common multiple is \(72\) and it is the least of all possible multiples. Thus, the least common multiple of \(18\) and \(24\) is \(72\). So, \(y = 72\).

So, the value of \(\frac{y}{x}\) is \(\frac{72}{3} = 24\).

Out of the four answer choices given, we see that only answer choice \(B\) is divisible by \(24\).

\[\frac{48}{24} = 2\]

All the other options are not divisible by \(24\).

Thus, the correct answer is \(B\).

Roots and Exponents

An exponent tells how many times a number (or base) is used in multiplication, such that

\[4^5 = 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 1024\]

Here are the three main rules of exponents you need to master:

\(x^n = x \cdot x \cdot … x\), where \(n\) is the number of times \(x\) is multiplied

\(x^{-n} = \frac {1}{x^n}\), where \(x\) is not equal to zero

\(x^{\frac{1}{n}} = \sqrt [n] {x}\) (Take note that a fractional exponent implies a root.)

The other rules can be memorized, too, or derived from the three main rules or laws:

\[x^a \cdot x^b = x^{a+b}\] \[(xy)^a = x^a \cdot y^a\] \[\frac {x^a}{x^b} = x^{a – b}\] \[(\frac{x}{y})^a = \frac{x^a}{y^a}\] \[(x^a)^b = x^{ab}\] \[x^0 = 1, \text{if} \,x \,\text{is not equal to zero}\] \[x^{\frac{m}{n}} = \sqrt[n]{x^m}\]

A very important reminder about an exponent: It is applied only to the base to which it directly attaches. So \(m^2\) means “\(m \cdot m\)”, and \(-m^2\) means “\(-(m \cdot m)\)” not “\(-m \cdot -m\)”.

Scientific Notation

It’s not enough to know that scientific notation is a manner of writing a very large or very small number. Consider this form:

\(N \cdot 10^a\), where \(N\) is a number between \(1\) and \(10\) (excluding \(10\) and \(1\)) and \(a\) can be positive or negative.

A multiple-choice question in this section may go like this:

Find the reciprocal of \(8.0 \cdot 10^9\) and express it in scientific notation.

Answer:

The reciprocal of \(a \cdot 10^{b}\) is \(\frac{1}{a} \cdot 10^{-b}\). So, the reciprocal of \(8.0 \cdot 10^9\) can be written as \(\frac{1}{8} \cdot 10^{-9}\). Further simplifying, we get:

\[0.125 \cdot 10^{-9}\] \[= 1.25 \cdot 10^{-10}\]