Data Analysis - ISEE Upper Level Quantitative Reasoning

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Question

An arithmetic sequence begins

$5, 35, \square, \bigtriangleup , \bigcirc...$

What number replaces the circle?

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Answer

Since this is an arithmetic sequence, each entry in the sequence is obtained by adding the same number to the previous entry - this number is

.

The next three entries in the sequence are computed as follows:

, which replaces the square

, which replaces the triangle

, which replaces the circle

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Question

A geometric sequence begins

$5, 35, \square, \bigtriangleup , \bigcirc...$ .

What number replaces the circle?

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Answer

Since this is a geometric sequence, each entry in the sequence is obtained by multiplying the previous entry by the same number - this number is

$35 \div 5 = 7$ .

Now we can find the next three entries in the sequence:

$35 \times 7 = 245$

This replaces the square.

$245 \times 7 = 1,715$

replaces the triangle.

$1,715\times 7 = 12,005$

replaces the circle and is therefore the correct answer.

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Question

The first two terms of an arithmetic sequence are

Which of the following expressions is equivalent to the fifth term?

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Answer

An arithmetic sequence is formed by adding the same expression to each term to get the next term; this common difference is

.

To obtain the fifth term, add to the second term three times - equivalently, add three times this to the second term;

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Question

The Fibonacci sequence is defined as follows:

$F_{1} = F_{2} = 1$

For integers , $F_{n} = F_{n-1}+ F_{n-2}$ .

Which is the greater quantity?

(a) $2 \cdot F_{50}$

(b) $F_{51}$

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Answer

The Fibonacci sequence begins as follows:

This sequence is seen to be an increasing sequence. Therefore, each term is greater than its preceding term. In particular,

$F_{50 }> F_{49}$

If we substitute 51 for in the rule of the sequence, we get

$F_{n} = F_{n-1}+ F_{n-2}$

$F_{51} = F_{51-1}+ F_{51-2}$

$F_{51} = F_{50}+ F_{49}$

$F_{50 }> F_{49}$ , so

$F_{50 }+ F_{50} > F_{49} + F_{50 }$

$2 \cdot F_{50} > F_{51 }$

This makes (a) greater.

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Question

The Fibonacci sequence is formed as follows:

$F_{1} = 1$

$F_{2}= 1$

For all integers , $F_{n}= F_{n-1}+ F_{n-2}$

Which of the following is true of $F_{1,000}$ , the one-thousandth number in this sequence?

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Answer

To express $F_{1,000}$ , the one-thousandth term of the sequence, in terms of $F_{997}$ and $F_{998}$ alone, we note that, by definition of the sequence, each term, except for the first two, is equal to the sum of the previous two. Therefore,

$F_{999} = F_{997}+ F_{998}$

Also

$F_{1,000} = F_{998}+ F_{999}$ , and, substituting:

$F_{1,000} = F_{998}+ \left ( F_{997}+ F_{998} \right )$

and

$F_{1,000} = F_{997}+ 2 F_{998}$ ,

the correct choice.

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Question

Define a sequence as follows:

$a_{1} = 3$

$a_{2} = 1$

For all integers , $a_{n} = 3a_{n-1} - a_{n-2}$ .

Which of the following expressions is equal to $a_{1,000}$ ?

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Answer

Setting :

$a_{n} = 3a_{n-1} - a_{n-2}$

$a_{1,000} = 3a_{1,000-1} - a_{1,000-2}$

$a_{1,000} = 3a_{999} - a_{998}$

Similarly,

$a_{999} = 3a_{998 } - a_{997}$

Substituting:

$a_{1,000} = 3 (3a_{998 } - a_{997}) - a_{998}$

$=9a_{998 } - 3 a_{997} - a_{998}$

$=8a_{998 } - 3 a_{997}$

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Question

The following Venn diagram depicts the number of students who play hockey, football, and baseball. How many students play football and baseball?

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Answer

The number of students who play football or baseball can by finding the summer of the number of students who play football alone, baseball alone, baseball and football, and all three sports.

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Question

A class of students was asked whether they have cats, dogs, or both.The results are depicted in the following Venn diagram. How many students do not have a dog?

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Answer

First, calculate the number of students with a dog:

Next, subtract the number of students with a dog from the total number of students.

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Question

If a standard die is rolled, what is the probability of getting a 1 or a 2?

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Answer

We need to know the total number of possibilities, and the total number of ways to achieve our goal.

A standard die has 6 faces, so there are a total of 6 numbers that we could roll.

We want to roll a 1 or a 2, which means there are 2 ways that we can succeed (rolling a 1 or a 2).

Thus, we have a probability of success as $\dpi{100} \frac{2}{6}$ which reduces to $\dpi{100} \frac{1}{3}$ .

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Question

The beverage menu from a restaurant reads as follows:

$\begin{matrix} \textrm{\underline{Beverage}} & \textrm{\underline{Price}} \ \textrm{Soda}& \$1.29 \ \textrm{Iced Tea} & \$1.39 \ \textrm{Coffee} & $1.69 \ \textrm{Hot Tea} & $1.59 \ \textrm{Milk} & \$1.49 \end{matrix}$

Ten friends go out for drinks. They order four sodas, two coffees, two iced teas, a milk, and a hot tea. One of the ten agrees to leave the entire amount of the tip, which is to be 20% of the check. What is the amount of the tip (round to the nearest dime)?

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Answer

The costs of the beverages will be as follows:

$\begin{matrix} \textrm{4 Sodas}& \4 \times $1.29 = $5.16 \ \textrm{2 Iced Teas} & 2 \times $1.39 = $2.78 \ \textrm{2 Coffees} &2 \times $1.69 = $3.38 \ \textrm{1 Hot Tea} & \$1.59 \ \textrm{1 Milk} & \$1.49 \end{matrix}$

Add these amounts:

$\begin{matrix} ; $5.16\ ; ; 2.78\ ; ; 3.38\ ; ; 1.59\; ; \underline{1.49}\ $14.40 \end{matrix}$

20% of $14.40 is $14.40 \times 0.20 = 2.88$ , or $2.90 when rounded to the nearest dime.

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Question

The following is a portion of the menu at a coffee shop:

$\begin{matrix} \textrm{\underline{Item}}& \textrm{\underline{Price}}\ \textrm{Latte} & \$2.89\ \textrm{Americano} &\$2.49 \ \textrm{Cappucino} & \$2.79\ \textrm{Espresso} & \$3.19\ \textrm{Turkish} & \$ 3.09 \end{matrix}$

A boss treats her employees to coffee at this coffee shop. She orders three lattes, three Americanos, two espressos, a cappucino, and a Turkish coffee. The tax is . The boss hands the cashier a $\$50$ bill. How much change will she get back?

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Answer

Add the prices of the ten items:

$\begin{matrix} \$2.89 \ \ 2.89\ \2.89\ \2.49\ \2.49\ \2.49\ \3.19\ \3.19\ \2.79\ \underline{\3.09}\28.40 \end{matrix}$

of this is $\$ 28.40 \times 0.07 = 1.988 \approx 1.99$ , so the tax is $\$1.99.$

The price of the beverages after tax is $\$28.40 + 1.99 = \$30.39$ .

The change out of a $\$50$ bill is .

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Question

Refer to the above circle graph. There are voters registered as independents in Smith County; to the nearest hundred, how many voters in all are registered?

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Answer

Set to the number of voters total. of them are independent - these number , so we can set up the equation:

$0.4 V \div 0.4= 3,471 \div 0.4$

$V\approx 8,678$ , so the correct choice is .

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Question

Refer to the above graph, which shows the high temperatures in the town of Smithville over seven days.

Which day of this week saw the highest peak temperature?

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Answer

The highest point on the line graph is located at Tuesday.

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Question

Refer to the above graph, which shows the high temperatures in the town of Smithville over seven days.

On how many days that week did Smithville see a peak temperature 75 degrees or greater?

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Answer

There were four days that saw a peak temperature of 75 degrees or greater:

Monday: 76 degrees

Tuesday: 80 degrees

Wednesday: 77 degrees

Friday: 78 degrees

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Question

Nine students are running for student council; each member of the student body will vote for four. Freida's boyfriend Greg is running and she wants to vote for him. How many ways can she cast a ballot so that she can include Greg among her choices?

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Answer

Since Freida has already chosen one of the nine candidates, she will choose three of the remaining eight, Since order is unimportant, the number of ways to do this is the number of combinations of three chosen from eight.

$C (8,3) = \frac{8!}{(8-3)!; 3!}= \frac{8!}{5!; 3!} = \frac{40,320}{120 \cdot 6 } = 56$

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Question

The above is an annual income tax table for a given state. Married couples will pay the tax rate for the tax bracket in which their entire income falls.

Mr. Jackson earned $34,287 last year; Mrs. Jackson earned $25,879. How much will the couple pay in income tax for that year (nearest hundred dollars)?

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Answer

The Jackson's income totaled

$$34,287+$25,879= \$60,166$ .

This puts them in the 1.7% tax bracket, so they will pay

$\$60,166 \times 0.017 = \$1,022.82$ in taxes.

The correct response is $1,000.

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Question

Washington Elementary School has seven eighth-grade teachers; each teacher has the number of boys and girls listed above.

An eighth-grade girl has transferred from another school. It is desired that no teacher have more than thirty students overall or eighteen girls in his or her class. Of the seven teachers, how many could accept the girl into his or her class?

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Answer

We can eliminate Cermak and Georghiou immediately since each has eighteen girls in his class. Of the other five:

Adams has students;

Boyer has students;

Donovan has students;

Esterhaus has students;

Finnegan has students.

Only Mrs. Finnegan's class could accept the new girl without going against the wishes of the school. The correct response is one.

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Question

Jefferson Elementary School has seven eighth-grade teachers; each teacher has the number of boys and girls listed above.

How many teachers have classes in which at least 40% of the students are boys?

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Answer

Five teachers each have 30 students, and 40% of 30 is

$30 \times 40 % = 30 \times 0.4 = 12$ .

Two teachers each have 29 students, and 40% of 29 is

$2 9 \times 40 % = 30 \times 0.4 = 11.6$ .

Therefore, to have a class that is at least 40% boys, a teacher must have 12 boys at minimum. Only Mr. Georghiou does not have at least 12 boys, so the correct response is six.

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Question

Adams Elementary School has seven fifth-grade teachers; each teacher has been assigned the number of boys and girls listed above.

However, just before the start of the school year, Mrs. Henry has been hired as an additional fifth-grade teacher. It is desired that some students be reassigned to her class so that each class have the same number of students, or as close to the same number of students as possible. If this is done, how many students will be in Mrs. Henry's class?

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Answer

This problem is essentially figuring out what will be the average number of students in each class if there are eight teachers instead of seven.

Add the fourteen numbers in the table:

$\begin{matrix} 13\ 17\ 15\ 15\ 12\ 18\ 17\ 13\ 14\ 16\ 18\ 11\ 11\ \underline{18}\end{matrix}$

students.

If there are eight teachers, then each teacher, including Mrs. Henry, will have

$208 \div 8 = 26$ students.

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Question

A popular word game uses one hundred tiles, each of which is marked with a letter or a blank. The distribution of the tiles is shown above, with each letter paired with the number of tiles marked with that letter. Notice that there are two blank tiles.

What percent of the tiles are marked with consonants?

Note: for purposes of this question, "Y" is a consonant.

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Answer

Counting the number of tiles with vowels is much easier.

Out of the 100 tiles, there are nine "A" tiles, twelve "E" tiles, nine "I" tiles, eight "O" tiles, and four "U" tiles. This is a total of

tiles out of 100.

There are also two blanks, so the number of tiles marked with consonants is

56 out of 100 is 56%.

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