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ISEE Upper Level Quantitative : Equations

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

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Example Question #41 : Equations

$\left \lfloor N \right \rfloor$ is defined as the greatest integer less than or equal to

$\left \lceil N \right \rceil$ is defined as the least integer greater than or equal to

N > 1 ; itself is not an integer.

Which is the greater quantity?

(a) $\left \lceil N \right \rceil \cdot \left \left \lfloor N+1 \right \rfloor$

(b) $\left \lceil N + 1 \right \rceil \cdot \left \left \lfloor N \right \rfloor$

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:

Let $x = \left \lfloor N \right \rfloor$ .

Then $\left \lfloor N + 1 \right \rfloor = x + 1$ .

Since is a noninteger, $\left \lceil N\right \rceil = x+ 1$ and $\left \lceil N + 1 \right \rceil = x+ 2$ .

(a) $\left \lceil N \right \rceil \cdot \left \left \lfloor N+1 \right \rfloor = \left ( x + 1 \right ) \left ( x + 1 \right ) = x^{2}+2x+1$

(b) $\left \lceil N + 1 \right \rceil \cdot \left \left \lfloor N \right \rfloor = (x+2) x = x ^{2}+ 2x$

(a) is greater than (b).

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Example Question #41 : Equations

Define f (x) = 4 - 5x and g (x) = 5 - 4x .

can be any real number.

Which is the greater quantity?

(a) $(f \circ g) (x)$

(b) $(g \circ f) (x)$

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) $(f \circ g) (x) = (f ( g (x) ) = f (5-4x )$

Substitute 5-4x for :

f (x) = 4 - 5x

f (5-4x ) = 4 - 5 (5-4x )

$= 4 - 5 \cdot 5-\left ( -5 \right ) \cdot 4x$

= 4 -25+20x

= 20x-21

(b) $(g \circ f) (x) = g(f(x)) = g (4-5x)$

Substitute 4-5x for :

g (x) = 5 - 4x

g (x) = 5 - 4(4-5x)

$= 5 - 4 \cdot 4- \left ( - 4 \right ) \cdot 5x$

= 5 - 16 +20x

= 20x - 11

-11 > -21 , so regardless of the value of , 20x - 11 > 20x - 21 , and (b) is greater.

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Example Question #43 : Equations

Veronica and Stephanie bought sweaters of the same type, but at different stores. Veronica bought hers for $\$60.00$ at Markham's; Stephanie bought hers for $\$75.00$ at Schuster's. Veronica got a better discount rate than Stephanie, and she has been bragging about it.

Which is the greater quantity?

(a) The price of the sweater at Markham's before discount

(b) The price of the sweater at Schuster's before discount

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:

We show that the question cannot be answered with certainty by taking two sample cases.

Case 1: The sweaters cost $\$100.00$ at both stores.

Veronica saved $\$100.00 - 60.00 = \$40.00$ and got a $40\%$ discount rate. Stephanie saved $\$100.00 - 75.00 = \$25.00$ and got a $25\%$ discount rate.

Case 2: The sweater cost $\$100.00$ at Markham's and $\$90.00$ at Schuster's.

In this case Veronica saved $\$100.00 - 60.00 = \$40.00$ and got a $40\%$ discount rate. Stephanie saved $\$90.00 - 75.00 = \$15.00$ and got a discount rate of $\frac{15}{90} \times 100 = 16 \frac{2}{3} \%$ .

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Example Question #44 : Equations

Define $g (x) = x^{4} + 2x^{2} - 1$ .

Which is the greater quantity?

(a) g(5)

(b) g(-5)

Possible Answers:

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

(a) $g (x) = x^{4} + 2x^{2} - 1$

$g (5) = 5^{4} + 2 \cdot 5^{2} - 1$

$= 625 + 2 \cdot 25 - 1$

= 625 + 50 - 1

= 674

(b) $g (x) = x^{4} + 2x^{2} - 1$

$g (-5) = (-5)^{4} + 2 \cdot (-5)^{2} - 1$

$= 625 + 2 \cdot 25 - 1$

= 674

g (5) = g (-5)

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Example Question #41 : Algebraic Concepts

The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is 125 miles.

Which is the greater quantity?

(a) The distance between Clark and Ferrell on a map

(b) Two inches

Possible Answers:

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

The actual distance between Clark and Ferrell is unknown, but it is at least 250 - 125 = 125 miles. If is the map distance between Clark and Ferrell, then its minimum can be calculated using a proportion:

$\frac{N}{125} = \frac{6}{250}$

$\frac{N}{125} \cdot 125 = \frac{6}{250}\cdot 125$

$N = \frac{750}{250} = 3$

At the very least, the two are three inches apart on the map, so (a) is greater regardless.

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Example Question #46 : Equations

Carl starts working for a bicycle store today. He wants to buy a certain bicycle, but his $20\%$ employee discount doesn't take effect until ninety days from now. He finds out that the price of the bicycle is due to increase by $30\%$ in sixty days.

Which is the greater quantity?

(a) The price Carl would pay for the bicycle today

(b) The price Carl would pay for the bicycle in ninety days, assuming that the bicycle does not change in price again

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

(a) Let be the current price of the bicycle. Without the discount and the markup, Carl would pay today.

(b) Now we examine what Carl would pay in ninety days. The increase would be $30\%$ of , or, equivalently, 0.3x , and the price of the bicycle is

x + 0.3x = 1.3x .

The employee discount will be $20\%$ of this, or $0.2 \cdot 1.3x = 0.26x$ , and the price Carl would pay is

1.3x - 0.26x = 1.04x .

Carl would pay more by waiting ninety days.

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Example Question #42 : Algebraic Concepts

The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is 125 miles.

Which is the greater quantity?

(a) The distance between Clark and Ferrell on a map

(b) Ten inches

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

The actual distance between Clark and Ferrell is unknown, but at most it is 250 + 125 = 375 miles. If is the map distance between Clark and Ferrell, then its maximum can be calculated using a proportion:

$\frac{N}{375} = \frac{6}{250}$

$\frac{N}{375} \cdot 375 = \frac{6}{250}\cdot 375$

$N = \frac{2,250}{250} = 9$

At the very most, the two are nine inches apart on the map, so (b) is greater regardless.

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Example Question #43 : Algebraic Concepts

$f(x) = x^{2} - x + 12$

Which is the greater quantity?

(a) f(4)

(b) f(-3)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

Substitute to evaluate each expression:

(a) $f(x) = x^{2} - x + 12$

$f(4) = 4^{2} - 4 + 12 = 16 - 4 + 12 = 24$

(b) $f(x) = x^{2} - x + 12$

$f(-3) = (-3)^{2} - (-3) + 12 = 9 + 3 + 12 = 24$

f(4)= f(-3)

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Example Question #49 : Equations

A set of tires costs $\$240.00$ after a $20\%$ discount.

Which is the greater quantity?

(a) The price of the tires before the discount

(b) $\$288.00$

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

A $20\%$ discount means that the tires sell for $80\%$ of their original price. If that original price is $\$288.00$ , then after the discount, they will sell for $80\%$ of $\$288.00$ . This is

$\$288.00 \times 0.8 = \$230.40$ ,

which is less than $\$240.00$ . Therefore, the original purchase price must be greater than $\$288.00$ .

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Example Question #50 : Equations

Define f (x) = 4x - 3 , g (x) = 3x + 5 .

Which is the greater quantity?

(a) $\left ( f \circ g \right )(0)$

(b) $\left ( g \circ f \right )(0)$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

(a) $\left ( f \circ g \right )(0) = f(g(0))$

First, evaluate g(0) :

g (x) = 3x + 5 ,

so $g (0) = 3 \cdot 0+ 5 = 5$ .

Now, evaluate f(g(0)) = f (5) :

f (x) = 4x - 3

so $f (5) = 4 \cdot 5 - 3 = 20 -3 = 17$ .

$\left ( f \circ g \right )(0) = 17$

(b) $\left ( g \circ f \right )(0) =g(f(0))$

First, evaluate f (0) :

f (x) = 4x - 3 ,

so $f (0) = 4 \cdot 0- 3 = -3$ .

Now evaluate g(f(0)) = g(-3)

g (x) = 3x + 5 ,

so $g (-3) = 3 \left ( -3 \right ) + 5 = -9 + 5 = -4$ .

$\left ( g \circ f \right )(0) = -4$

$\left ( f \circ g \right )(0) > \left ( g\circ f \right )(0)$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Equations

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #41 : Equations

Example Question #41 : Equations

Example Question #43 : Equations

Example Question #44 : Equations

Example Question #41 : Algebraic Concepts

Example Question #46 : Equations

Example Question #42 : Algebraic Concepts

Example Question #43 : Algebraic Concepts

Example Question #49 : Equations

Example Question #50 : Equations

All ISEE Upper Level Quantitative Resources

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