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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

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Example Questions

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

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

and are integers. Which is greater?

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

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

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 is an integer, then $\left \lfloor N\right \rfloor = N$ by definition.

Since x, y , and, by closure, x + y are all integers,

$\left \lfloor x+ y\right \rfloor = x + y$ and $\left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor = x + y$ , 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 5x + 4y = -200 .

Which is the greater quantity?

(a) The -coordinate of the -intercept.

(b) The -coordinate of the -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 -coordinate of the -intercept, substitute y = 0 :

5x + 4y = -200

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

5x + 0 = -200

5x = -200

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

x = -40

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

5x + 4y = -200

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

0 + 4y = -200

4y = -200

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

y = -50

Therefore (a) is the greater quantity.

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

| 45 - t | = 60

| 45 - u | = 50

Which is the greater quantity?

(a)

(b)

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:

| 45 - t | = 60

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

45 - t =- 60

45 - t -45 =- 60 -45

-t = -105

t = 105

45 - t = 60

45 - t -45 =60 -45

-t = 15

t = -15

Similarly:

| 45 - u | = 50

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

45 - u = -50

45 - u -45 = -50 -45

- u = -95

u =95

45 - u = 50

45 - u -45 = 50 -45

- u = 5

u = -5

Therefore, it cannot be determined with certainty which of and is the greater.

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

| t - 10 | = 40

Which is the greater quantity?

(a)

(b)

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 | t - 10 | = 40 , then either t - 10 = 40 or t - 10 = -40 . Solve for in both equations:

t - 10 = 40

t - 10 + 10 = 40 + 10

t = 50

t - 10 = -40

t - 10 + 10 = -40 + 10

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 #12 : Equations

t^3 = -125

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

t^3 = -125

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

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

t= -5

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

Consider the line of the equation 4x + 5y = 100 .

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -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 -coordinate of the -intercept, substitute y = 0 :

4x + 5y = 100

$4x + 5 \cdot 0 = 100$

4x + 0 = 100

4x = 100

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

x = 25

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

4x + 5y = 100

$4\cdot 0 + 5y = 100$

5y = 100

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

y = 20

(a) is the greater quantity.

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

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

and are integers. C = A - 0.01; D = B + 0.01

Which is the greater quantity?

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

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

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 is an integer, $\left \lceil C\right \rceil = \left \lceil A - 0.01 \right \rceil = A$ .

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

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

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

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

(a) is the greater quantity.

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Example Question #21 : Equations

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

and are integers.

a = x + 0.5, b= y + 0.5

Which is greater?

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

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

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

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

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

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

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

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

This makes (b) greater.

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Example Question #21 : Equations

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

Which is the greater quantity?

(a)

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

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

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

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

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

t-3= 0

t = 3

$\sqrt{x} = 3$

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

x=9

t +10 = 0

t = -10

$\sqrt{x} = -10$

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

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

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Example Question #21 : Equations

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

Which is the greater quantity?

(a) The -coordinate of the -intercept of this line

(b) The -coordinate of the -intercept of this line

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

The slope of this line is

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

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

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

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

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

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

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

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

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

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

x = 11

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

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

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

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

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

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

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 #12 : Equations

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 : Equations

Example Question #21 : Equations

Example Question #21 : Equations

All ISEE Upper Level Quantitative Resources

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