ISEE Upper Level Quantitative Reasoning - Data Analysis

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the sum will be at least eight

(b) The probability that the sum will be at most six

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

The following table illustrates the ways that the dice can come up. The orange cells indicate a roll of 6 or less; the pink cells indicate a roll of eight or more.

Dice_table

Since each of these outcomes are equally likely, and there are fifteen outcomes that are favorable to each of the two events, the probabilities are the same.

An arithmetic sequence begins as follows:

Which is the greater quantity?

(a) The fourth term of the sequence

(b) 200

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

The common difference of the sequence is , so the next two terms of the sequence are:

215 is the fourth term. This makes (a) greater.

Define set $A = \left { 1, 3, 5, 6, 7, 8, 10 \right }$ . How could we define set so that $A \cap B = \left { 1, 5, 6, 8\right }$ ?

$B = \left { 1, 2, 5, 6, 8, 9, 11, 12\right }$

$B = \left { 1, 2, 5, 6, 9, 12, 13, 15\right }$

$B = \left { 1, 2, 4, 5, 6, 10, 11, 12\right }$

$B = \left { 1,5, 10\right }$

$B = \left {5, 6,8, 9, 16\right }$

Explanation

$A \cap B$ is the set of all elements in both and .

We can test each set and determine which elements are shared by that set and :

If $B = \left { 1, 2, 5, 6, 9, 12, 13, 15\right }$ :

then

$A \cap B$

$= \left { \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, 10 \right } \cap \left { \underline{1}, 2,\underline{ 5}, \underline{6}, 9, 12, 13, 15\right }$

$= \left { 1,5,6\right }$

If $B = \left { 1, 2, 4, 5, 6, 10, 11, 12\right }$ :

then

$A \cap B$

$= \left { \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, \underline{10} \right } \cap \left { \underline{1}, 2, 4, \underline{5}, \underline{6}, \underline{10}, 11, 12\right }$

$= \left { 1,5,6, 10\right }$

If $B = \left { 1,5, 10\right }$ :

then

$A \cap B$

$= \left { \underline{1}, 3, \underline{5},6, 7, 8, \underline{10} \right } \cap \left { \underline{1}, \underline{5}, \underline{10}\right }$

$= \left { 1,5, 10\right }$

If $B = \left {5, 6,8, 9, 16\right }$ :

then

$A \cap B$

$= \left { 1, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right } \cap\left {\underline{5}, \underline{6},\underline{8},9, 16\right }$

$= \left { 5,6,8\right }$

If $B = \left { 1, 2, 5, 6, 8, 9, 11, 12\right }$ :

then

$A \cap B$

$= \left { \underline{1}, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right } \cap \left { \underline{1}, 2, \underline{5}, \underline{6}, \underline{8}, 9, 11, 12\right }$

$= \left { 1,5,6, 8\right }$

This is the correct choice.

An arithmetic sequence begins as follows:

$4 \frac{1}{2} , 6 \frac{3}{4},...$ '

Which is the greater quantity?

(a) The fifth number in the sequence

(b)

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

The common difference of the sequence is

$6 \frac{3}{4} - 4 \frac{1}{2} = 6 \frac{3}{4} - 4 \frac{2}{4} = 2\frac{1}{4}$ .

The fifth number in the sequence is

$4 \frac{1}{2} + (5-1) \times 2\frac{1}{4} =4 \frac{1}{2} + 4 \times 2\frac{1}{4} = 4 \frac{1}{2}+ 9 = 13\frac{1}{2}$ .

This makes (b) greater.

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region.

What is $A \cup B$ ?

$A \cup B = \left { a,b,d,e,f,g,h\right }$

$A \cup B = \left { a,f \right }$

$A \cup B = \left { b,d,e,g,h\right }$

$A \cup B = \left { a,c,f \right }$

$A \cup B = \left { a,b,c, d,e,f,g,h\right }$

Explanation

$A \cup B$ is the union of sets and - that is, the set of all elements of that are elements of either or . We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except . Therefore,

$A \cup B = \left { a,b,d,e,f,g,h\right }$

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the sum will be at least eight

(b) The probability that the sum will be at most six

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

The following table illustrates the ways that the dice can come up. The orange cells indicate a roll of 6 or less; the pink cells indicate a roll of eight or more.

Dice_table

Since each of these outcomes are equally likely, and there are fifteen outcomes that are favorable to each of the two events, the probabilities are the same.

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the sum will be at least eight

(b) The probability that the sum will be at most six

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

The following table illustrates the ways that the dice can come up. The orange cells indicate a roll of 6 or less; the pink cells indicate a roll of eight or more.

Dice_table

Since each of these outcomes are equally likely, and there are fifteen outcomes that are favorable to each of the two events, the probabilities are the same.

A geometric sequence begins as follows:

Which is the greater quantity?

(a) The fourth element of the sequence

(b) 30

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

The common ratio of the sequence is

$\frac{125}{250} = \frac{1}{2}$

The next two terms of the sequence can be found as follows:

$125 \times \frac{1}{2} = 62 \frac{1}{2}$

$62 \frac{1}{2}\times \frac{1}{2} = 31 \frac{1}{4}$

This is the fourth term, which is greater than 30.

Below is a table that gives the population of Washington City for five census years.

$\begin{matrix} \textrm{\underline{Year }}& \textrm{\underline{Population}} \ 1970& 3,872\ 1980& 4,239\ 1990& 4,512\ 2000& 4,412 \ 2010& 4,832 \end{matrix}$

Which is the greater quantity?

(a) The population increase in Washington City from 1970 to 1975

(b) The population increase in Washington City from 1975 to 1980

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation

No information is given in the table about the population of Washington City in 1975. Therefore, it cannot be determined with any certainty which population change was the greater.

Below is a table that gives the population of Washington City for five census years.

$\begin{matrix} \textrm{\underline{Year }}& \textrm{\underline{Population}} \ 1970& 3,872\ 1980& 4,239\ 1990& 4,512\ 2000& 4,412 \ 2010& 4,832 \end{matrix}$

Which is the greater quantity?

(a) The population increase in Washington City from 1970 to 1975

(b) The population increase in Washington City from 1975 to 1980

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation

No information is given in the table about the population of Washington City in 1975. Therefore, it cannot be determined with any certainty which population change was the greater.

Data Analysis

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