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

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Example Questions

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

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

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

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

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

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 $\left \lfloor N\right \rfloor = N$ $\displaystyle \left \lfloor N\right \rfloor = N$ by definition.

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

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

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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 y = 0 $\displaystyle y = 0$:

$\displaystyle 5x + 4y = -200$

$5x + 4 \cdot 0 = -200$ $\displaystyle 5x + 4 \cdot 0 = -200$

$\displaystyle 5x + 0 = -200$

$\displaystyle 5x = -200$

$5x\div 5= -200\div 5$ $\displaystyle 5x\div 5= -200\div 5$

x = -40 $\displaystyle x = -40$

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

$\displaystyle 5x + 4y = -200$

$5 \cdot 0 + 4y = -200$ $\displaystyle 5 \cdot 0 + 4y = -200$

$\displaystyle 0 + 4y = -200$

$\displaystyle 4y = -200$

$4y \div 4= -200 \div 4$ $\displaystyle 4y \div 4= -200 \div 4$

y = -50 $\displaystyle y = -50$

Therefore (a) is the greater quantity.

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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:

$\displaystyle | 45 - t | = 60$

$45 - t =- 60 \textrm{ or }45 - t = 60$ $\displaystyle 45 - t =- 60 \textrm{ or }45 - t = 60$

$\displaystyle 45 - t =- 60$

$\displaystyle 45 - t -45 =- 60 -45$

$\displaystyle -t = -105$

t = 105 $\displaystyle t = 105$

$\displaystyle 45 - t = 60$

$\displaystyle 45 - t -45 =60 -45$

-t = 15 $\displaystyle -t = 15$

t = -15 $\displaystyle t = -15$

Similarly:

$\displaystyle | 45 - u | = 50$

$45 - u = -50 \textrm{ or } 45 - u = 50$ $\displaystyle 45 - u = -50 \textrm{ or } 45 - u = 50$

$\displaystyle 45 - u = -50$

$\displaystyle 45 - u -45 = -50 -45$

$\displaystyle - u = -95$

u =95 $\displaystyle u =95$

$\displaystyle 45 - u = 50$

$\displaystyle 45 - u -45 = 50 -45$

- u = 5 $\displaystyle - u = 5$

u = -5 $\displaystyle u = -5$

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

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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:

$\displaystyle t - 10 = 40$

$\displaystyle t - 10 + 10 = 40 + 10$

t = 50 $\displaystyle t = 50$

$\displaystyle t - 10 = -40$

$\displaystyle t - 10 + 10 = -40 + 10$

t = -30 $\displaystyle t = -30$

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

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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:

$\displaystyle t^3 = -125$

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

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

t= -5 $\displaystyle t= -5$

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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 y = 0 $\displaystyle y = 0$:

$\displaystyle 4x + 5y = 100$

$4x + 5 \cdot 0 = 100$ $\displaystyle 4x + 5 \cdot 0 = 100$

$\displaystyle 4x + 0 = 100$

$\displaystyle 4x = 100$

$4x \div 4 = 100 \div 4$ $\displaystyle 4x \div 4 = 100 \div 4$

x = 25 $\displaystyle x = 25$

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

$\displaystyle 4x + 5y = 100$

$4\cdot 0 + 5y = 100$ $\displaystyle 4\cdot 0 + 5y = 100$

$\displaystyle 5y = 100$

$5y \div 5 = 100\div 5$ $\displaystyle 5y \div 5 = 100\div 5$

y = 20 $\displaystyle y = 20$

(a) is the greater quantity.

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

$\left \lceil N\right \rceil$ $\displaystyle \left \lceil N\right \rceil$ 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) $\left \lceil C \right \rceil + \left \lceil D\right \rceil$ $\displaystyle \left \lceil C \right \rceil + \left \lceil D\right \rceil$

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

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, $\left \lceil C\right \rceil = \left \lceil A - 0.01 \right \rceil = A$ $\displaystyle \left \lceil C\right \rceil = \left \lceil A - 0.01 \right \rceil = A$.

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

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

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

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

(a) is the greater quantity.

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

$\left \lfloor N\right \rfloor$ $\displaystyle \left \lfloor N\right \rfloor$ 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) $\left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor$ $\displaystyle \left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor$

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

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, $\left \lfloor a \right \rfloor = \left \lfloor x + 0.5 \right \rfloor = x$ $\displaystyle \left \lfloor a \right \rfloor = \left \lfloor x + 0.5 \right \rfloor = x$.

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

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

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

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

This makes (b) greater.

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Example Question #21 : Equations

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

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 $t = \sqrt {x}$ $\displaystyle t = \sqrt {x}$ and, subsequently, $t^{2} = x$ $\displaystyle t^{2} = x$:

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

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

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

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

t-3= 0 $\displaystyle t-3= 0$

t = 3 $\displaystyle t = 3$

$\sqrt{x} = 3$ $\displaystyle \sqrt{x} = 3$

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

x=9

$\displaystyle t +10 = 0$

t = -10 $\displaystyle t = -10$

$\sqrt{x} = -10$ $\displaystyle \sqrt{x} = -10$

This has no solution, since $\sqrt {x}$ $\displaystyle \sqrt {x}$ must be nonnegative. x=9

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

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Example Question #21 : Algebraic Concepts

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

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

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

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

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

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

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

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

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

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

$- \frac{11}{2} = -\frac{1}{2} x$ $\displaystyle - \frac{11}{2} = -\frac{1}{2} x$

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

x = 11 $\displaystyle x = 11$

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

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

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

$y - 4 = \frac{3}{2}$ $\displaystyle y - 4 = \frac{3}{2}$

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

$y = \frac{11}{2} = 5 \frac{1}{2}$ $\displaystyle y = \frac{11}{2} = 5 \frac{1}{2}$

This makes (a) the greater quantity.

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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 #15 : How To Find The Solution To An Equation

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

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

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

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

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

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

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

Example Question #21 : Equations

Example Question #21 : Algebraic Concepts

All ISEE Upper Level Quantitative Resources

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