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ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Factor A Number

What is the sum of all of the factors of 27?

Possible Answers:

$\displaystyle 12$

$\displaystyle 39$

$\displaystyle 40$

$\displaystyle 13$

$\displaystyle 36$

Correct answer:

$\displaystyle 40$

Explanation:

27 has four factors: $\displaystyle 1, 3, 9, 27.$

Their sum is $\displaystyle 1 + 3 + 9 + 27 = 40$.

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Example Question #1 : Other Factors / Multiples

Add all of the factors of 30.

Possible Answers:

$\displaystyle 41$

$\displaystyle 42$

$\displaystyle 55$

$\displaystyle 71$

$\displaystyle 72$

Correct answer:

$\displaystyle 72$

Explanation:

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is

$\displaystyle 1+2+3+5+ 6+10+15 + 30 = 72$.

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Example Question #7 : Other Factors / Multiples

Which is the greater quantity?

(a) The number of factors of 169

(b) The number of factors of 121

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

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Example Question #8 : Other Factors / Multiples

Which is the greater quantity?

(a) The number of factors of 15

(b) The number of factors of 17

Possible Answers:

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

(a) 15 has four factors, 1, 3, 5, and 15.

(b) 17, as a prime, has two factors, 1 and 17.

Therefore, (a) is greater.

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Example Question #9 : Other Factors / Multiples

Which is the greater quantity?

(a) The product of the integers between $\displaystyle -1,000,000$ and $\displaystyle 1,000,000$ inclusive

(b) The sum of the integers between $\displaystyle -1,000,000$ and $\displaystyle 1,000,000$ inclusive

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The quanitites are equal, as both can be demonstrated to be equal to .

(a) One of the integers in the given range is , so one of the factors will be , making the product .

(b) The sum of the numbers will be:

$\displaystyle 0 + (-1 + 1) + (-2 + 2) + (-3 + 3) + ... + (-1,000,000+ 1,000,000)$

$\displaystyle 0 + 0 + 0+...+0 = 0$

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Example Question #16 : Factors / Multiples

Which is the greater quantity?

(a) The sum of the factors of $\displaystyle 18$

(b) The sum of the factors of $\displaystyle 20$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

(a) The factors of $\displaystyle 18$ are $\displaystyle 1, 2, 3, 6, 9, 18.$ Their sum is

$\displaystyle 1+ 2+3+ 6+ 9+18 = 39$.

(b) The factors of $\displaystyle 20$ are $\displaystyle 1, 2, 4, 5, 10, 20.$ Their sum is

$\displaystyle 1+ 2+4+5+10+20 = 42$.

(b) is greater.

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Example Question #17 : Numbers And Operations

Which is the greater quantity?

(a) The sum of all of the two-digit even numbers

(b) 2,500

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

The sum of the integers from $\displaystyle m$ to $\displaystyle n$ is equal to $\frac{\left ( n+m \right )\left ( n-m+1 \right )}{2}$ $\displaystyle \frac{\left ( n+m \right )\left ( n-m+1 \right )}{2}$. We take advantage of the fact that the sum of the even numbers from 10 to 98 is equal to twice the sum of the integers from 5 to 49, as seen here:

$\displaystyle 10+12 + 14+...+98$

$=2 \left ( 5+ 6+7+...+49\right )$ $\displaystyle =2 \left ( 5+ 6+7+...+49\right )$

$=2\cdot \frac{\left ( 49+5 \right )\left ( 49-5+1 \right )}{2}$ $\displaystyle =2\cdot \frac{\left ( 49+5 \right )\left ( 49-5+1 \right )}{2}$

$=54 \times 45 = 2,430 < 2,500$ $\displaystyle =54 \times 45 = 2,430 < 2,500$

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Example Question #11 : Other Factors / Multiples

What is the prime factorization of 16ab^2 $\displaystyle 16ab^2$?

Possible Answers:

$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

$2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b$

$2\cdot 2\cdot 2\cdot 2$ $\displaystyle 2\cdot 2\cdot 2\cdot 2$

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

Correct answer:

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

Explanation:

First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example: $4\times 4$ $\displaystyle 4\times 4$, which then yields $2\times 2\times 2\times 2$ $\displaystyle 2\times 2\times 2\times 2$). Then, at the end, remember to factor the variables as well. Since the b term is squared, that means there are two of them. Therefore, the final answer is $2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$.

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Example Question #12 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ number of factors of 100 $\displaystyle 100$

$(b)\ 10$ $\displaystyle (b)\ 10$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

it is not possible to tell based on the information given.

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ is greater

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

Factors of 100 $\displaystyle 100$ are:

$1,2,4,5,10,20,25,50\ and\ 100$ $\displaystyle 1,2,4,5,10,20,25,50\ and\ 100$. So it has $\displaystyle 9$ factors which is less than $\displaystyle 10$.

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Example Question #13 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ The sum of all of the factors of $\displaystyle 72$

$(b)\ 200$ $\displaystyle (b)\ 200$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(a) $\displaystyle (a)$ is greater

(b) $\displaystyle (b)$ is greater

it is not possible to tell based on the information given.

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

Factors of $\displaystyle 72$ are:

$1,2,3,4,6,8,9,12,18,24,36\ and\ 72$ $\displaystyle 1,2,3,4,6,8,9,12,18,24,36\ and\ 72$. So we can write:

$\displaystyle 1+2+3+4+6+8+9+12+18+24+36+72=195$

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

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ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Factor A Number

Example Question #1 : Other Factors / Multiples

Example Question #7 : Other Factors / Multiples

Example Question #8 : Other Factors / Multiples

Example Question #9 : Other Factors / Multiples

Example Question #16 : Factors / Multiples

Example Question #17 : Numbers And Operations

Example Question #11 : Other Factors / Multiples

Example Question #12 : Other Factors / Multiples

Example Question #13 : Other Factors / Multiples

All ISEE Upper Level Quantitative Resources

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