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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 #105 : How To Find The Solution To An Equation

One-third of the sum of a number and sixty is ninety-three. What is the number?

Possible Answers:

339

219

459

Correct answer:

219

Explanation:

If we let be the number, "the sum of a number and sixty" can be written as

x+60

"One-third of the sum of a number and sixty" can be written as

$\frac{1}{3} (x+60)$

Set this equal to ninety-three and solve:

$\frac{1}{3} (x+60) = 93$

$3 \cdot \frac{1}{3} (x+60) =3 \cdot 93$

x+60 = 279

x+60 - 60= 279 - 60

x = 219

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Example Question #106 : How To Find The Solution To An Equation

Twelve added to two-fifths of a number is equal to sixty. What is that number?

Possible Answers:

120

162

180

102

Correct answer:

120

Explanation:

If we let be the number, "two-fifths of a number" can be written as

$\frac{2}{5} x$ .

"Twelve added to two-fifths of a number" can be written as

$\frac{2}{5} x + 12$ .

Then "Twelve added to two-fifths of a number is equal to sixty" can be written and solved for as follows:

$\frac{2}{5} x + 12 = 60$

$\frac{2}{5} x + 12 - 12 = 60 - 12$

$\frac{2}{5} x =48$

$\frac{5} {2}\cdot \frac{2}{5}x = \frac{5} {2}\cdot 48$

x = 120

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Example Question #101 : Equations

Ninety-seven is five less than two-fifths of a number. What is the number?

Possible Answers:

$237\frac{1}{2}$

255

$247\frac{1}{2}$

The correct answer is not among the other choices.

230

Correct answer:

255

Explanation:

If we let be the number, "two-fifths of a number" can be written as

$\frac{2}{5}x$ .

"Five less than two-fifths of a number can be written as

$\frac{2}{5}x -5$

"Ninety-seven is five less than two-fifths of a number" can be written and solved as follows:

$97 = \frac{2}{5}x -5$

$97 + 5 = \frac{2}{5}x -5 + 5$

$102 = \frac{2}{5}x$

$\frac{5}{2} \cdot 102 = \frac{5}{2} \cdot \frac{2}{5}x$

255 = x

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Example Question #108 : How To Find The Solution To An Equation

Define $f (x) = \sqrt{3x+8 }$ and $g(x) = \sqrt{2x+7}$ .

What is the domain of the function f + g ?

Possible Answers:

$\left [-2 \frac{2}{3}, \infty \right )$

$\left [-3 \frac{1} {2},-2 \frac{2}{3} \right ]$

$\left [-3 \frac{1} {2}, \infty \right )$

$\varnothing$

$(- \infty,\infty )$

Correct answer:

$\left [-2 \frac{2}{3}, \infty \right )$

Explanation:

$f (x) = \sqrt{3x+8 }$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$3x+8 \geq 0$

$3x+8-8 \geq 0 -8$

$3x \geq -8$

$3x\div 3 \geq -8 \div 3$

$x \geq -2 \frac{2}{3}$ or $\left [-2 \frac{2}{3}, \infty \right )$

Similarly, $g(x) = \sqrt{2x+7}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$2x+7 \geq 0$

$2x+7-7 \geq 0 -7$

$2x \geq -7$

$2x \div 2 \geq -7 \div 2$

$x \geq -3\frac{1}{2}$ or $\left [-3 \frac{1} {2}, \infty \right )$

The domain of the sum of two functions is the intersection of the domains of the two individual functions. This intersection is

$\left [-2 \frac{2}{3}, \infty \right ) \cap \left [-3 \frac{1} {2}, \infty \right ) = \left [-2 \frac{2}{3}, \infty \right )$

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Example Question #109 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\ast$ as follows:

$a \ast b = \frac{a b + 1}{a+ b + 1 }$

For which value of is the expression $N \ast \left (-7 \right )$ undefined?

Possible Answers:

$N = \frac{1}{7}$

N = 7

$N = - \frac{1}{7}$

N = 8

N = 6

Correct answer:

N = 6

Explanation:

$a \ast b = \frac{a b + 1}{a+ b + 1 }$ , so

$N \ast \left (-7 \right ) = \frac{N (-7) + 1}{N+ (-7) + 1 } = \frac{-7N + 1}{N-6 }$

This is is undefined if and only if the denominator is equal to zero, which happens when

N - 6 = 0 , or N = 6

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Example Question #110 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\Lambda$ as follows:

$a \Lambda b = \frac{a^{2} b - 16}{16a^{2} + b}$

For which value of is the expression $N \Lambda 4$ undefined?

Possible Answers:

N = -2

$N = \frac{1}{2 }$

None of the other responses gives a correct answer.

N = 2

$N = - \frac{1}{2 }$

Correct answer:

None of the other responses gives a correct answer.

Explanation:

$a \Lambda b = \frac{a^{2} b - 16}{16a^{2} + b}$ , so

$N \Lambda 4 = \frac{N^{2} \cdot 4 - 16}{16 \cdot N^{2} +4} = \frac{4N^{2} - 16}{16 N^{2} +4}$

This expression is undefined if and only if the denominator is equal to 0. However, for all values of ,

$16 N^{2} \geq 0$ , so

$16 N^{2} +4 > 0$

It is impossible for $N \Lambda 4$ to be undefined, so none of the four values of given gives a correct response.

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Example Question #111 : How To Find The Solution To An Equation

Define f (x)= 2x+ 5 and g (x) = 2x+3 .

Evaluate $\left ( g \circ f \right )(3)$ .

Possible Answers:

$\left ( g \circ f \right )(3) = 20$

$\left ( g \circ f \right )(3) = 33$

$\left ( g \circ f \right )(3) = 28$

$\left ( g \circ f \right )(3) = 23$

$\left ( g \circ f \right )(3) = 25$

Correct answer:

$\left ( g \circ f \right )(3) = 25$

Explanation:

$\left ( g \circ f \right )(3) = g (f(3))$

f (x)= 2x+ 5 , so

$f (3)= 2 \cdot 3 + 5 = 6 + 5 = 11$

g (x) = 2x+3 , so

$g (f(3)) = g (11) = 2 \cdot 11 + 3 = 22 + 3 = 25$ ,

which is the correct response.

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Example Question #112 : How To Find The Solution To An Equation

Define f (x)= 2x+ 5 and g (x) = 2x+3 .

If $\left ( f \circ g\right ) (N) = 100$ , evaluate .

Possible Answers:

None of the other responses gives a correct answer.

$N = 21 \frac{3}{4}$

$N = 22 \frac{3}{4}$

$N = 21 \frac{1}{4}$

$N = 22 \frac{1}{4}$

Correct answer:

$N = 22 \frac{1}{4}$

Explanation:

$\left ( f \circ g\right ) (x) = f(g(x)) = f(2x+3) = 2 (2x+3) +5 = 4x + 6 + 5 = 4x + 11$

Therefore, if

$\left ( f \circ g\right ) (N) = 100$ ,

we solve for in the equation

4N + 11 = 100

4N + 11- 11 = 100 - 11

4N = 89

$4N \div 4 = 89 \div 4$

$N = 22 \frac{1}{4}$

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Example Question #113 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\Leftrightarrow$ as follows:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

is a positive number. Which is the greater quantity?

(A) $t \Leftrightarrow 4t$

(B) $2t \Leftrightarrow 2t$

Possible Answers:

(B) is greater

(A) is greater

(A) and (B) are equal

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

Correct answer:

(A) is greater

Explanation:

Substitute for both in the expression:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

$t\Leftrightarrow 4t = \frac{t + 4t}{ t \cdot 4t +1} = \frac{5t}{4t^{2}+1}$

$2t\Leftrightarrow 2t = \frac{2t + 2t}{ 2t \cdot 2t +1} = \frac{4t}{4t^{2}+1}$

The quantities share a denominator that is positve, being one greater than for times a square of a positive. Therefore, we need only compare numerators. For any positive , 5t > 4t . Therefore,

$\frac{5t}{4t^{2}+1} > \frac{4t}{4t^{2}+1}$

and

$t\Leftrightarrow 4t > 2t\Leftrightarrow 2t$

making (A) greater.

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Example Question #114 : How To Find The Solution To An Equation

and are both positive.

20 added to four-thirds of is equal to . Which is the greater quantity?

(A)

(B)

Possible Answers:

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

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

The statement is equivalent to

$y = \frac{4}{3}x + 20$

If is positive, then

$\frac{4}{3} x > x$

and

$y = \frac{4}{3} x + 20 > \frac{4}{3} x > x$

and y > x ,

making (B) greater.

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All ISEE Upper Level Quantitative Resources

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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 #105 : How To Find The Solution To An Equation

Example Question #106 : How To Find The Solution To An Equation

Example Question #101 : Equations

Example Question #108 : How To Find The Solution To An Equation

Example Question #109 : How To Find The Solution To An Equation

Example Question #110 : How To Find The Solution To An Equation

Example Question #111 : How To Find The Solution To An Equation

Example Question #112 : How To Find The Solution To An Equation

Example Question #113 : How To Find The Solution To An Equation

Example Question #114 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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