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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 #115 : How To Find The Solution To An Equation

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

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

is a positive integer. Which is the greater quantity?

(A) $2 t \oplus 4t$

(B) $3t \oplus 3t$

Possible Answers:

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

(A) and (B) are equal

(A) is greater

(B) is greater

Correct answer:

(B) is greater

Explanation:

Substitute for both in the expression:

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

$2 t\oplus 4 t = \frac{2 t \cdot 4t + 1}{2t + 4t + 1} = \frac{8 t ^{2}+ 1}{6t + 1}$

$3t \oplus 3t = \frac{3t \cdot 3t + 1}{3t + 3t + 1}= \frac{9t^{2}+ 1}{6t + 1}$

Since is positive, so is common denominator 6t + 1 , so we need only compare numerators. Since $t^{2}$ must be positive,

$8 t ^{2}< 9t ^{2}$

$8 t ^{2}+ 1 < 9t ^{2} + 1$

$\frac{8 t ^{2}+ 1}{6t + 1} < \frac{9t ^{2}+ 1}{6t + 1}$

and

$2 t\oplus 4 t < 3t\oplus 3 t$

making (B) greater.

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Example Question #116 : How To Find The Solution To An Equation

and are both positive.

20 subtracted from five-sevenths of is equal to . Which is the greater quantity?

(A)

(B)

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

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

Correct answer:

(A) is greater

Explanation:

The statement is equivalent to

$\frac{5}{7}x -20 = y$ or, alternatively,

$y = \frac{5}{7}x -20$

$y+ 20 = \frac{5}{7}x -20 + 20$

$y+ 20 = \frac{5}{7}x$

Since is positive and $\frac{5}{7}x > y$ , we can derive:

$y < y+ 20 = \frac{5}{7}x < x$

making greater than . This makes (A) greater.

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Example Question #117 : How To Find The Solution To An Equation

Define $f (x) = \sqrt{14 - 2x}$ and $g(x) = \sqrt{3x - 21}$

What is the domain of the function f + g ?

Possible Answers:

$\varnothing$

$(- \infty,\infty )$

None of the other responses gives a correct answer.

$[7,\infty )$

$(- \infty, 7]$

Correct answer:

None of the other responses gives a correct answer.

Explanation:

$f (x) = \sqrt{14 - 2x}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$14 - 2x \geq 0$

$14 - 2x- 14 \geq 0 - 14$

$- 2x \geq - 14$

$- 2x \div (-2) \leq - 14 \div (-2)$

$x \leq 7$ or $(- \infty, 7]$

Similarly, $g(x) = \sqrt{3x - 21}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$3x - 21 \geq 0$

$3x - 21+ 21 \geq 0 + 21$

$3x \geq 21$

$3x \div 3 \geq 21 \div 3$

$x \geq 7$ or $[7,\infty )$

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

$(- \infty, 7] \cap [7,\infty ) = \left \{ 7\right \}$

The domain of f + g is the single value $\left \{ 7\right \}$ , which is not among the choices.

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Example Question #118 : How To Find The Solution To An Equation

Three consecutive integers add up to 36. What is the greatest integer of the three?

Possible Answers:

Correct answer:

Explanation:

To solve this problem, you can translate the question into an equation. It should look like: x+x+1+x+2=36 . Since we don't know the first number, we name it as x. Then, we add one to each following integer, which gives us x+1 and x+2. Then, combine like terms to get 3x+3=36 . Solve for x and you get 11. However, the question is asking for the greatest integer of the set, so the answer is actually 13 (because it is the x+2 term).

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Example Question #119 : How To Find The Solution To An Equation

Column A Column B

The value of x in The value of x in

$\frac{x}{5}+2=18$ 4x-3=13

Possible Answers:

The quantity in Column A is greater.

The quantities in both columns are equal.

The quantity in Column B is greater.

The relationship between the columns cannot be determined.

Correct answer:

The quantity in Column A is greater.

Explanation:

First, solve for x in Column A. Subtract 2 from both sides so that $\frac{x}{5}=16$ . Multiply by 5 on both sides so that X equals 80. Then, solve for x in Column B. Add 3 to both sides so that 4x=16 . X equals 4 in this equation. Therefore, Column A is greater.

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Example Question #120 : How To Find The Solution To An Equation

On a 70-question exam, Lisa answered 60 percent correctly. How many answers did she get right?

Possible Answers:

Correct answer:

Explanation:

If Lisa answered 60 percent of the questions on a 70-question exam correctly, the following equation can be used to determine how many quesitons she got right. is equal to the number of questions she answered correctly.

$\frac{60}{100}=\frac{x}{70}$

Given that $\frac{60}{100}=\frac{6}{10}=\frac{3}{5}$ , it follows that

$\frac{3}{5}=\frac{x}{70}$

Next, we cross multiply, which gives us:

5x=210

Now, we divide each side by 5, resulting in:

x=42

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Example Question #121 : How To Find The Solution To An Equation

If is negative and |4y - 27| = 80 , then what is ?

Possible Answers:

y = -428

The equation has no negative solution.

$y = - 26 \frac{3}{4}$

$y = -13 \frac{1}{4}$

y = -212

Correct answer:

$y = -13 \frac{1}{4}$

Explanation:

|4y - 27| = 80 , so either

4y - 27 = - 80 or 4y - 27 = 80

We solve both equations separately:

4y - 27 = - 80

4y - 27 + 27 = - 80 + 27

4y = -53

$\frac{4y}{4} = \frac{-53}{4}$

$y = -13 \frac{1}{4}$

4y - 27 = 80

4y - 27 + 27 = 80 + 27

4y = 107

$\frac{4y}{4} = \frac{107}{4}$

$y = 26 \frac{3}{4}$

Since the negative solution is being requested, we choose $y = -13 \frac{1}{4}$ .

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Example Question #122 : How To Find The Solution To An Equation

Give the solution set of the equation:

|3t + 12| - 15 = 34

Possible Answers:

$\left \{-12\frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-10\frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-10\frac{1}{3},10\frac{1}{3} \right \}$

$\left \{-20 \frac{1}{3},20\frac{1}{3} \right \}$

Correct answer:

$\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

Explanation:

|3t + 12| - 15 = 34

|3t + 12| - 15 + 15 = 34 + 15

|3t + 12| =49

Either

3t + 12 = -49 or 3t + 12 = 49 ,

so we solve the equations separately:

3t + 12 = -49

3t + 12 - 12 = -49 - 12

3t = -61

$\frac{3t}{3} =\frac{ -61}{3}$

$t = -20 \frac{1}{3}$

3t + 12 = 49

3t + 12 - 12 = 49 - 12

3t = 37

$\frac{3t}{3} =\frac{37}{3}$

$t = 12\frac{1}{3}$

The solution set is $\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

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Example Question #123 : How To Find The Solution To An Equation

|x + 5| = |x | - 5

Which of the folllowing is a true statement?

Possible Answers:

The equation has three solutions.

The equation has infinitely many solutions.

The equation has two solutions.

The equation has one solution.

The equation has no solution.

Correct answer:

The equation has infinitely many solutions.

Explanation:

Since the absolute value of a nonnegative number is the number itself, and the absolute value of a negative number is its (positive) opposite, we have to examine up to three cases: $x \ge 0$ , $-5 \le x < 0$ , and x < -5 .

However, let us examine that third case.

This makes and x+5 negative, so the equation can be rewritten:

|x + 5| = |x | - 5

-(x+5) = -x - 5

-x-5= -x - 5

This statement is identically true. Therefore, all values of less than work, and we have already proved that there are infinitely many solutions. We do not need to go further.

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Example Question #124 : How To Find The Solution To An Equation

t+ 13 = u + 8

u + 4 = v - 7

v- 11 = w - 6

Which of the following is a true statement?

Possible Answers:

t = w -19

t = w -21

t = w -13

t = w -27

t = w -11

Correct answer:

t = w -11

Explanation:

t+ 13 = u + 8 ,

t+ 13- 13 = u + 8 - 13

t = u - 5

u + 4 = v - 7

u + 4 - 4 = v - 7 - 4

u = v-11

v- 11 = w - 6

v- 11 + 11= w - 6 + 11

v = w+5

Using two substitutions:

t = u - 5

t = v-11 - 5 = v-16

t = v-16 = w+5 -16 = w -11

The correct choice is t = w -11 .

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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 #115 : How To Find The Solution To An Equation

Example Question #116 : How To Find The Solution To An Equation

Example Question #117 : How To Find The Solution To An Equation

Example Question #118 : How To Find The Solution To An Equation

Example Question #119 : How To Find The Solution To An Equation

Example Question #120 : How To Find The Solution To An Equation

Example Question #121 : How To Find The Solution To An Equation

Example Question #122 : How To Find The Solution To An Equation

Example Question #123 : How To Find The Solution To An Equation

Example Question #124 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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