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

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

varsity tutors app store varsity tutors android store

Example Questions

Example Question #15 : How To Find The Solution To An Equation

\(\displaystyle \left \lfloor N\right \rfloor\) refers to the greatest integer less than or equal to \(\displaystyle N\).

\(\displaystyle x\) and \(\displaystyle y\) are integers. Which is greater?

(a) \(\displaystyle \left \lfloor x+ y\right \rfloor\)

(b) \(\displaystyle \left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor\)

Possible Answers:

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

If \(\displaystyle N\) is an integer, then \(\displaystyle \left \lfloor N\right \rfloor = N\) by definition.

Since \(\displaystyle x, y\), and, by closure, \(\displaystyle x + y\) are all integers, 

\(\displaystyle \left \lfloor x+ y\right \rfloor = x + y\) and \(\displaystyle \left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor = x + y\), making (a) and (b) equal.

Example Question #16 : How To Find The Solution To An Equation

Consider the line of the equation \(\displaystyle 5x + 4y = -200\).

Which is the greater quantity?

(a) The \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept.

(b) The \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept.

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

(a) To find the \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept, substitute \(\displaystyle y = 0\):

\(\displaystyle 5x + 4y = -200\)

\(\displaystyle 5x + 4 \cdot 0 = -200\)

\(\displaystyle 5x + 0 = -200\)

\(\displaystyle 5x = -200\)

\(\displaystyle 5x\div 5= -200\div 5\)

\(\displaystyle x = -40\)

(b) To find the \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept, substitute \(\displaystyle x = 0\):

\(\displaystyle 5x + 4y = -200\)

\(\displaystyle 5 \cdot 0 + 4y = -200\)

\(\displaystyle 0 + 4y = -200\)

\(\displaystyle 4y = -200\)

\(\displaystyle 4y \div 4= -200 \div 4\)

\(\displaystyle y = -50\)

Therefore (a) is the greater quantity.

Example Question #17 : How To Find The Solution To An Equation

\(\displaystyle | 45 - t | = 60\)

\(\displaystyle | 45 - u | = 50\)

Which is the greater quantity?

(a) \(\displaystyle t\)

(b) \(\displaystyle u\)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Each can be rewritten as a compound statement. Solve separately:

\(\displaystyle | 45 - t | = 60\)

\(\displaystyle 45 - t =- 60 \textrm{ or }45 - t = 60\)

\(\displaystyle 45 - t =- 60\)

\(\displaystyle 45 - t -45 =- 60 -45\)

\(\displaystyle -t = -105\)

\(\displaystyle t = 105\)

or 

\(\displaystyle 45 - t = 60\)

\(\displaystyle 45 - t -45 =60 -45\)

\(\displaystyle -t = 15\)

\(\displaystyle t = -15\)

 

Similarly:

\(\displaystyle | 45 - u | = 50\)

\(\displaystyle 45 - u = -50 \textrm{ or } 45 - u = 50\)

\(\displaystyle 45 - u = -50\)

\(\displaystyle 45 - u -45 = -50 -45\)

\(\displaystyle - u = -95\)

\(\displaystyle u =95\)

 

\(\displaystyle 45 - u = 50\)

\(\displaystyle 45 - u -45 = 50 -45\)

\(\displaystyle - u = 5\)

\(\displaystyle u = -5\)

 

Therefore, it cannot be determined with certainty which of \(\displaystyle t\) and \(\displaystyle u\) is the greater.

Example Question #18 : How To Find The Solution To An Equation

\(\displaystyle | t - 10 | = 40\)

Which is the greater quantity?

(a) \(\displaystyle t\)

(b) \(\displaystyle 50\)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

If \(\displaystyle | t - 10 | = 40\), then either \(\displaystyle t - 10 = 40\) or  \(\displaystyle t - 10 = -40\). Solve for \(\displaystyle t\) in both equations:

\(\displaystyle t - 10 = 40\)

\(\displaystyle t - 10 + 10 = 40 + 10\)

\(\displaystyle t = 50\)

or 

\(\displaystyle t - 10 = -40\)

\(\displaystyle t - 10 + 10 = -40 + 10\)

\(\displaystyle t = -30\)

Therefore, either (a) and (b) are equal or (b) is the greater quantity, but it cannot be determined with certainty.

Example Question #691 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

\(\displaystyle t^3 = -125\)

Which is the greater quantity?

(a) \(\displaystyle t\)

(b) \(\displaystyle -5\)

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

\(\displaystyle t^3 = -125\)

\(\displaystyle \sqrt[3]{t^3 }=\sqrt[3]{ -125}\)

\(\displaystyle \sqrt[3]{t^3 }= -\sqrt[3]{ 125}\)

\(\displaystyle t= -5\)

Example Question #20 : How To Find The Solution To An Equation

Consider the line of the equation \(\displaystyle 4x + 5y = 100\).

Which is the greater quantity?

(a) The \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept

(b) The \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) To find the \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept, substitute \(\displaystyle y = 0\):

\(\displaystyle 4x + 5y = 100\)

\(\displaystyle 4x + 5 \cdot 0 = 100\)

\(\displaystyle 4x + 0 = 100\)

\(\displaystyle 4x = 100\)

\(\displaystyle 4x \div 4 = 100 \div 4\)

\(\displaystyle x = 25\)

(b) To find the \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept, substitute \(\displaystyle x = 0\):

\(\displaystyle 4x + 5y = 100\)

\(\displaystyle 4\cdot 0 + 5y = 100\)

\(\displaystyle 5y = 100\)

\(\displaystyle 5y \div 5 = 100\div 5\)

\(\displaystyle y = 20\)

(a) is the greater quantity.

Example Question #21 : How To Find The Solution To An Equation

\(\displaystyle \left \lceil N\right \rceil\) refers to the least integer greater than or equal to \(\displaystyle N\).

\(\displaystyle A\) and \(\displaystyle B\) are integers. \(\displaystyle C = A - 0.01; D = B + 0.01\)

Which is the greater quantity?

(a) \(\displaystyle \left \lceil C \right \rceil + \left \lceil D\right \rceil\)

(b) \(\displaystyle \left \lceil C+ D \right \rceil\)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

(a) Since \(\displaystyle A\) is an integer, \(\displaystyle \left \lceil C\right \rceil = \left \lceil A - 0.01 \right \rceil = A\).

Since \(\displaystyle B\) is an integer, \(\displaystyle \left \lceil D\right \rceil = \left \lceil B+ 0.01 \right \rceil = B + 1\).

\(\displaystyle \left \lceil C \right \rceil + \left \lceil D\right \rceil = A + (B + 1) = (A+B)+ 1\)

(b) By closure, \(\displaystyle A + B\) is an integer, so 

\(\displaystyle \left \lceil C+ D \right \rceil = \left \lceil (A - 0.01) + (B + 0.01)\right \rceil = \left \lceil A + B\right \rceil = A + B\).

(a) is the greater quantity.

Example Question #21 : How To Find The Solution To An Equation

\(\displaystyle \left \lfloor N\right \rfloor\) refers to the greatest integer less than or equal to \(\displaystyle N\).

\(\displaystyle x\) and \(\displaystyle y\) are integers.

\(\displaystyle a = x + 0.5, b= y + 0.5\)

Which is greater?

(a) \(\displaystyle \left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor\)

(b) \(\displaystyle \left \lfloor a+ b \right \rfloor\)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

(a) Since \(\displaystyle x\) is an integer, \(\displaystyle \left \lfloor a \right \rfloor = \left \lfloor x + 0.5 \right \rfloor = x\).

Since \(\displaystyle y\) is an integer, \(\displaystyle \left \lfloor b \right \rfloor = \left \lfloor y + 0.5 \right \rfloor = y\).

\(\displaystyle \left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor = x + y\)

(b) By closure, \(\displaystyle x+y\) is an integer, so 

\(\displaystyle \left \lfloor a+b \right \rfloor = \left \lfloor x + 0.5 +y+0.5 \right \rfloor = \left \lfloor x +y+1 \right \rfloor = x +y+1\).

This makes (b) greater.

Example Question #21 : Equations

\(\displaystyle x + 7 \sqrt{x} - 30 = 0\) 

Which is the greater quantity?

(a) \(\displaystyle x\)

(b)

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

Substitute \(\displaystyle t = \sqrt {x}\) and, subsequently, \(\displaystyle t^{2} = x\):

\(\displaystyle t^{2} + 7 t- 30 = 0\)

Factor as \(\displaystyle (t + ?) (t + ?)\), replacing the two question marks with integers whose product is \(\displaystyle -30\) and whose sum is . These integers are \(\displaystyle -3,10\).

\(\displaystyle (t-3)(t +10) = 0\)

Break this up into two equations, replacing \(\displaystyle \sqrt{x}\) for \(\displaystyle t\):

\(\displaystyle t-3= 0\)

\(\displaystyle t = 3\)

\(\displaystyle \sqrt{x} = 3\)

\(\displaystyle \left ( \sqrt{x} \right )^{2} = 3^{2}\)

or

\(\displaystyle t +10 = 0\)

\(\displaystyle t = -10\)

\(\displaystyle \sqrt{x} = -10\)

This has no solution, since \(\displaystyle \sqrt {x}\) must be nonnegative.

is the only solution, so (a) and (b) must be equal.

 

Example Question #21 : Algebraic Concepts

Consider the line through points \(\displaystyle (3,4)\) and \(\displaystyle (7,2)\).

Which is the greater quantity?

(a) The \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept of this line

(b) The \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept of this line

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

The slope of this line is 

\(\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{4-2}{3-7}= \frac{2}{-4} = -\frac{1}{2}\).

We will use the point-slope form of the line, with this slope and point \(\displaystyle (3,4)\):

\(\displaystyle y - 4 = -\frac{1}{2} (x - 3)\)

The \(\displaystyle x\)-coordinate of the \(\displaystyle x\)-intercept of this line can be found by substituting \(\displaystyle y = 0\) and solving for \(\displaystyle x\):

\(\displaystyle y - 4 = -\frac{1}{2} (x - 3)\)

\(\displaystyle 0- 4 = -\frac{1}{2} \cdot x + \frac{1}{2} \cdot 3\)

\(\displaystyle - 4 = -\frac{1}{2} x + \frac{3}{2}\)

\(\displaystyle - 4- \frac{3}{2} = -\frac{1}{2} x + \frac{3}{2} - \frac{3}{2}\)

\(\displaystyle - \frac{11}{2} = -\frac{1}{2} x\)

\(\displaystyle - \frac{11}{2}\cdot (-2) = -\frac{1}{2} x \cdot (-2)\)

\(\displaystyle x = 11\)

 

The \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept of this line can be found by substituting \(\displaystyle x= 0\) and solving for \(\displaystyle y\):

\(\displaystyle y - 4 = -\frac{1}{2} (0 - 3)\)

\(\displaystyle y - 4 = -\frac{1}{2} (- 3)\)

\(\displaystyle y - 4 = \frac{3}{2}\)

\(\displaystyle y - 4 + 4 = \frac{3}{2} + 4\)

\(\displaystyle y = \frac{11}{2} = 5 \frac{1}{2}\)

This makes (a) the greater quantity.

Learning Tools by Varsity Tutors