Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Middle Level Quantitative : Numbers and Operations

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

Create An Account

All ISEE Middle Level Quantitative Resources

6 Diagnostic Tests 110 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Fractions

Which is the greater quantity?

(a) The reciprocal of $- \frac{5}{6}$

(b) The reciprocal of $-\frac{6}{7}$

Possible Answers:

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Correct answer:

(b) is greater

Explanation:

The reciprocal of any fraction can be found by switching numerator and denominator. Since both numbers are negative, both reciprocals will be negative.

(a) $-\frac{5}{6}$ will have reciprocal $-\frac{6}{5}$

(b) $-\frac{6}{7}$ will have reciprocal $-\frac{7}{6}$

We can compare these by writing them both with common denominator LCM (5,6) = 30 .

$-\frac{6}{5}= -\frac{6 \times 6}{5\times 6} = -\frac{36}{30}$

$-\frac{7}{6}= -\frac{5 \times 7}{5\times 6} = -\frac{35}{30}$

$-\frac{6}{5} = -\frac{36}{30} < -\frac{35}{30}-\frac{7}{6}$

making (b) greater

Report an Error

Example Question #1 : How To Divide Fractions

Which is the greater quantity?

(a) The reciprocal of $\frac{3}{4}$

(b) The reciprocal of 0.7

Possible Answers:

(a) and (b) are equal

It cannot be determined from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

The reciprocal of any fraction can be found by switching numerator and denominator.

(a) $\frac{3}{4}$ will have reciprocal $\frac{4}{3}$

(b) $0.7 = \frac{7}{10}$ and will have reciprocal $\frac{10}{7}$

We can compare these by writing them both with common denominator LCM (3,7) = 21 :

$\frac{4}{3}= \frac{4\times 7}{3\times 7} = \frac{28}{21}$

$\frac{10}{7}= \frac{3\times10}{3\times 7} = \frac{30}{21}$

$\frac{4}{3} = \frac{28}{21} <\frac{30}{21} = \frac{10}{7}$

making (b) greater

Report an Error

Example Question #2 : Numbers And Operations

$3\div \frac{1}{2} =$

Possible Answers:

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{3}{2}$

Correct answer:

Explanation:

When dealing with math problems that involve fractions and whole numbers or mixed numbers, you should first convert the non-fraction into a fraction:

$3\rightarrow \frac{3}{1}$

The problem should look like this:

$\frac{3}{1}\div \frac{1}{2}$

When dividing with fractions, you need to find the reciprocal of the second fraction before you can do anything else. In other words, flip the second number.

$\frac{1}{2} \rightarrow \frac{2}{1}$

When you find the reciprocal of the second number, change the problem from division to multiplication. The new problem should look like this:

$\frac{3}{1}\div\frac{1}{2}=\frac{3}{1} \times \frac{2}{1} =$ $\frac{6}{1}$

The answer is 6.

Report an Error

Example Question #3 : Numbers And Operations

$\frac{1}{2} \div \frac{1}{8} =$

Possible Answers:

$\frac{1}{16}$

$\frac{1}{6}$

$\frac{1}{4}$

Correct answer:

Explanation:

When dividing fractions, you must first find the reciprocal of the second number in the operation. In other words, flip the second fraction.

$\frac{1}{8}\rightarrow\frac{8}{1}$

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

$\frac{1}{2} \times \frac{8}{1} =$

Then multiply both the numerators and denominators.

$\frac{1}{2} \times \frac{8}{1} =\frac{1\times8}{2\times1}=\frac{8}{2}$

When possible, always reduce the fraction. In this case, both 2 and 8 are divisible by 2.

$\tfrac{8}{2}\rightarrow\tfrac{4}{1}\rightarrow4$

The result is your answer.

Report an Error

Example Question #3 : Numbers And Operations

$\frac{1}{3}\div\frac{3}{5} =$

Possible Answers:

$\frac{3}{5}$

$\frac{1}{5}$

$\frac{5}{9}$

$\frac{3}{2}$

$\frac{1}{15}$

Correct answer:

$\frac{5}{9}$

Explanation:

When dividing fractions, you must first find the reciprocal of the second fraction. In other words, flip the second fraction.

$\frac{3}{5} \rightarrow\frac{5}{3}$

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

$\frac{1}{3}\times\frac{5}{3} =$

Multiply the numerators and denominators. The result is your answer.

$\frac{1}{3}\times\frac{5}{3} =\frac{1\times5}{3\times3}=\frac{5}{9}$

Report an Error

Example Question #4 : Numbers And Operations

$4\div\tfrac{1}{4} =$

Possible Answers:

$\frac{1}{4}$

$\frac{1}{16}$

Correct answer:

Explanation:

When dealing with fractions and whole numbers, first convert the whole number to a fraction. This is easily done by putting the whole number over 1.

$4\rightarrow\tfrac{4}{1}$

When dividing fractions, you must first find the reciprocal of the second fraction in the operation. In other words, flip the second fraction.

$\tfrac{1}{4}\rightarrow \tfrac{4}{1}$

When you do this, the operation changes from division to multiplication. The problem should now look like this:

$\tfrac{4}{1}\times\tfrac{1}{4}=$

Solve the multiplication by multiplying the numerators and the denominators.

$\tfrac{4}{1}\times\tfrac{4}{1} =\tfrac{4\times4}{1\times1}= \tfrac{16}{1}\rightarrow16$

Since the result is $\tfrac{16}{1}$ , it reduces to 16 as any fraction with a denominator of 1 is equal to the value of its numerator.

Report an Error

Example Question #5 : Numbers And Operations

If Candy has $\small \frac{3}{4}$ a cup of trail mix and she wants to divide it evenly among herself and seven of her friends, how much does each person get?

Possible Answers:

$\small \frac{3}{32}$

$\small \frac{24}{4}$

$\small \frac{3}{34}$

$\small \frac{1}{32}$

$\small \frac{1}{16}$

Correct answer:

$\small \frac{3}{32}$

Explanation:

Candy has $\small \frac{3}{4}$ cups of trail mix. If she divides it amoung herself and seven of her friends, she is dividing it by $\small \frac{8}{1}$ . In order to solve this problem, we simply find the reciprocal of the second fraction and multiply.

To find the reciprical of a fraction, we switch the numerator and denominator.

$\small \frac{8}{1}$ becomes $\small \frac{1}{8}$ .

Now we multiply $\small \frac{3}{4}$ by $\small \frac{1}{8}$ .

$\small \frac{3}{4}\cdot \frac{1}{8}=\frac{3}{32}$

Each person gets $\small \small \frac{3}{32}$ of the trail mix.

Report an Error

Example Question #1 : Numbers And Operations

is a positive integer.

Which is the greater quantity?

(a) $A \div \frac{1}{3}$

(b) $A \cdot \frac{1}{3}$

Possible Answers:

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

Division by a fraction is equivalent to multiplication by its reciprocal, so

$A \div \frac{1}{3}$ can be rewritten as $A \cdot \frac{3}{1} = A \cdot 3$

$3 > \frac{1}{3}$ , and is positive, so, by the multiplication property of inequality,

$A \cdot 3 >A \cdot \frac{1}{3}$

$A \div \frac{1}{3}>A \cdot \frac{1}{3}$ .

Report an Error

Example Question #8 : Fractions

is the reciprocal of . Which of the following is the reciprocal of X + 1 in terms of ?

Possible Answers:

$\frac{Y} {Y+1}$

$\frac{Y} {Y-1}$

$\frac{Y+1}{Y}$

$\frac{Y-1}{Y}$

Correct answer:

$\frac{Y} {Y+1}$

Explanation:

If is the reciprocal of , then

$X = \frac{1}{Y}$

$X + 1 = \frac{1}{Y} + 1 = \frac{1}{Y} + \frac{Y}{Y} = \frac{Y+1}{Y}$

The reciprocal of X + 1 is $\frac{Y} {Y+1}$ .

Report an Error

Example Question #8 : Numbers And Operations

The reciprocal of is $1 \frac{2}{3}$ . Which is the greater quantity?

(a) The reciprocal of Y - 1

(b)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

The reciprocal of is $1 \frac{2}{3} = \frac{1 \times 3+ 2}{3} = \frac{5}{3}$ , so is the reciprocal of this, or

$Y = \frac{3}{5}$

$Y - 1 = \frac{3}{5} - 1 = \frac{3}{5} - \frac{5}{5} = - \frac{2}{5}$

The reciprocal of Y - 1 is $- \frac{5}{2}$ , which is less than 0.

Report an Error

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

View Tutors

Eva
Certified Tutor

Technicka Univerzita Liberec, Bachelor of Education, Education.

All ISEE Middle Level Quantitative Resources

6 Diagnostic Tests 110 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Atlanta, English Tutors in San Diego, GRE Tutors in Washington DC, GMAT Tutors in Atlanta, Math Tutors in Seattle, Math Tutors in New York City, Algebra Tutors in Boston, MCAT Tutors in Phoenix, SAT Tutors in Chicago, SAT Tutors in Philadelphia

Popular Courses & Classes

SSAT Courses & Classes in Los Angeles, GMAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Phoenix, GRE Courses & Classes in Chicago, ACT Courses & Classes in Houston, SAT Courses & Classes in Seattle, SSAT Courses & Classes in Boston, GRE Courses & Classes in Miami, MCAT Courses & Classes in Chicago, SAT Courses & Classes in Los Angeles

Popular Test Prep

SSAT Test Prep in Chicago, MCAT Test Prep in Chicago, LSAT Test Prep in Houston, GMAT Test Prep in New York City, ACT Test Prep in Denver, ACT Test Prep in Houston, LSAT Test Prep in Denver, MCAT Test Prep in Atlanta, GRE Test Prep in New York City, ISEE Test Prep in Boston

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Middle Level Quantitative : Numbers and Operations

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : Fractions

Example Question #1 : How To Divide Fractions

Example Question #2 : Numbers And Operations

Example Question #3 : Numbers And Operations

Example Question #3 : Numbers And Operations

Example Question #4 : Numbers And Operations

Example Question #5 : Numbers And Operations

Example Question #1 : Numbers And Operations

Example Question #8 : Fractions

Example Question #8 : Numbers And Operations

All ISEE Middle Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: