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

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Example Questions

Example Question #145 : How To Find The Solution To An Equation

$x^{3} + 7x^{2} = 49x + 343$

$y^{3} + 343 = 7y^{2}+ 49 y$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

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

Explanation:

$x^{3} + 7x^{2} = 49x + 343$

$x^{3} + 7x^{2} - 49x - 343 = 49x + 343 - 49x - 343$

$x^{3} + 7x^{2} - 49x - 343 = 0$

$(x^{3} + 7x^{2}) -( 49x + 343 )= 0$

$x ^{2}(x + 7 ) -49 (x + 7 )= 0$

$(x ^{2} -49 )(x + 7 ) = 0$

$(x ^{2} -7 ^{2})(x + 7 ) = 0$

(x + 7 ) (x - 7 ) (x + 7 ) = 0

By the Zero Product Principle, either x - 7 = 0 , in which case x = 7 , or x+ 7 = 0 , in which case x =- 7 .

$y^{3} + 343 = 7y^{2}+ 49 y$

$y^{3} + 343- 7y^{2}- 49 y = 7y^{2}+ 49 y - 7y^{2}- 49 y$

$y^{3} - 7y^{2}- 49 y + 343 =0$

$(y^{3} - 7y^{2} )- (49 y - 343 )=0$

$y^{2} (y - 7 )- 49 (y - 7)=0$

$(y ^{2} -49 )(y-7 ) = 0$

(y+7 ) (y-7 ) (y-7 ) = 0

By the Zero Product Principle, either y- 7 = 0 , in which case y= 7 , or y+ 7 = 0 , in which case y =- 7 .

Both and are equal to either or 7, but it is unclear which is which, or whether both are even the same. Therefore, it cannot be determined which, if either, is the greater.

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Example Question #141 : Equations

| 5x + 12 | < 30

| 5y+ 12 | >30

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

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

Explanation:

| 5x + 12 | < 30

-30 < 5x+12 < 30

-30- 12 < 5x+12 - 12 < 30 - 12

-42 < 5x < 18

$-42\div 5 < 5x \div 5 < 18 \div 5$

-8.4 < x < 3.6

| 5y+ 12 | >30

Either

5y + 12 > 30

5y + 12- 12 > 30 - 12

5y > 18

$5y \div 5 > 18 \div 5$

y > 3.6 ,

in which case y > x ,

5y + 12< - 30

5y + 12 - 12 < - 30 - 12

5y < - 42

$5y \div 5 < - 42 \div 5$

y < -8.4 ,

in which case x > y .

It is therefore unclear which is the greater, or

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Example Question #146 : How To Find The Solution To An Equation

|x+ 13| = | x + 7|

|y + 21 | = | y + 1|

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

Correct answer:

(a) is the greater quantity

Explanation:

|x+ 13| = | x + 7| , so either:

x+13 = x + 7

x+13 - x= x + 7 - x

13 = 7 , which is false, or

x+13 = - (x + 7)

x+13 = -x - 7

x+13+ x - 13 = -x - 7 + x - 13

2x = -20

$2x \div 2 = -20 \div 2$

x = -10

|y + 21 | = | y + 1| , so either

y + 21 = y + 1

y + 21 - y = y + 1 - y

21 = 1 , which is false, or

y + 21 = - ( y + 1)

y + 21 = - y - 1

y + 21 + y -21 = - y - 1 + y -21

2 y = -22

$2 y\div 2 = -22 \div 2$

y = -11

x = -10 and y = -11 , so x > y

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Example Question #147 : How To Find The Solution To An Equation

$\sqrt{x} + \frac{1}{2} = \frac{4}{7}$

Which is the greater quantity?

(a)

(b) $\frac{9}{25}$

Possible Answers:

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

$\sqrt{x} + \frac{1}{2} = \frac{4}{7}$

$\sqrt{x} + \frac{1}{2} - \frac{1}{2} = \frac{4}{7} - \frac{1}{2}$

$\sqrt{x} = \frac{4 \cdot 2}{7 \cdot 2} - \frac{1 \cdot 7 }{2 \cdot 7 }$

$\sqrt{x} = \frac{8}{14} - \frac{7 }{14 } = \frac{1}{14 }$

$(\sqrt{x} )^{2}= \left ( \frac{1}{14 } \right )^{2}$

$x= \frac{1^{2} }{14^{2} } = \frac{1}{196} < \frac{9}{25}$

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Example Question #148 : How To Find The Solution To An Equation

$x^{2} + 100 = 20 x$

$y^{2} +20y+ 100 = 0$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

$x^{2} + 100 = 20 x$

$x^{2} - 20 x + 100 = 20 x - 20 x$

$x^{2} - 20 x + 100 = 0$

The quadratic trinomial fits the perfect square pattern:

$x^{2} - 2 \cdot 10 \cdot x + 10 ^{2} = 0$

$(x-10) ^{2} = 0$

x- 10 = 0

x = 10

$y^{2} +20y+ 100 = 0$

The quadratic trinomial also fits the perfect square pattern:

$y^{2} + 2 \cdot 10 \cdot y + 10 ^{2} = 0$

$(y+10) ^{2} = 0$

y+ 10 = 0

y = -10

Therefore, x >y .

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Example Question #149 : How To Find The Solution To An Equation

| 4x + 7 | = 0

|4y-7|=0

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

| 4x + 7 | = 0 , so

4x+7 = 0

4x = -7

$x = - \frac{7}{4}$

|4y-7|=0 , so

4y - 7 = 0

4y = 7

$y= \frac{7}{4}$

$x = - \frac{7}{4}$ and $y= \frac{7}{4}$ , so y > x .

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Example Question #151 : How To Find The Solution To An Equation

$\sqrt{x} + \frac{1}{2} = \sqrt{y}$

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

It can be deduced that both and are nonnegative, since both are radicands of square roots.

$\sqrt{y} = \sqrt{x} + \frac{1}{2}$ , so

$( \sqrt{y})^{2} = \left (\sqrt{x} + \frac{1}{2} \right )^{2}$

$y = \left (\sqrt{x}\right ) ^{2} + 2 \cdot \sqrt{x} \cdot \frac{1}{2} + \left ( \frac{1}{2} \right ) ^{2}$

$y = x + \sqrt{x} + \frac{1}{4}$

$\sqrt{x} > 0$ , so

$\sqrt{x} + \frac{1}{4} > 0$ , and

y > x .

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Example Question #151 : How To Find The Solution To An Equation

$343 x^{3} + 7 x = 49x^{2} + 1$

$y^{3} + 49 y = 7y^{2} + 343$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

$343 x^{3} + 7 x = 49x^{2} + 1$

$343 x^{3} + 7 x - 49x^{2} - 1 = 49x^{2} + 1 - 49x^{2} - 1$

$343 x^{3}- 49x^{2} + 7 x - 1 = 0$

$(343 x^{3}- 49x^{2} )+( 7 x - 1 )= 0$

$49x^{2} ( 7 x - 1 ) + 1 ( 7 x - 1 )= 0$

$(49x^{2} + 1) ( 7 x - 1 )= 0$

By the Zero Product Principle, one of the factors is equal to 0:

$49x^{2}+ 1 = 0$

$49x^{2} = - 1$

$x^{2} = -\frac{1}{49}$

which is impossible for any real value of , or

7x-1 = 0

7x = 1

$x = \frac{1}{7}$ .

$y^{3} + 49 y = 7y^{2} + 343$

$y^{3} + 49 y - 7y^{2} - 343 = 7y^{2} + 343 - 7y^{2} - 343$

$y^{3} - 7y^{2} + 49 y - 343 = 0$

$(y^{3} - 7y^{2}) + (49 y - 343 )= 0$

$y^{2} (y - 7) + 49 (y - 7) = 0$

$(y^{2} + 49) (y - 7) = 0$

By the Zero Product Principle, one of the factors is equal to 0:

$y^{2} + 49 = 0$

$y^{2} =- 49$

which is impossible for any real value of , or

y - 7 = 0

y = 7

Since $x = \frac{1}{7}$ and y = 7 , it can be determined that y > x .

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Example Question #152 : How To Find The Solution To An Equation

$x + \frac{3}{5} = \frac{7}{9}$

$y+ \frac{7}{9} = \frac{11}{13}$

Which is the greater quantity?

(a)

(b)

Possible Answers:

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

$x + \frac{3}{5} = \frac{7}{9}$

$x = \frac{7}{9} - \frac{3}{5}$

$= \frac{7\cdot 5 }{9 \cdot 5 } - \frac{3 \cdot 9}{5 \cdot 9}$

$= \frac{35 }{45 } - \frac{27}{45}$

$= \frac{8}{45 }$

$y+ \frac{7}{9} = \frac{11}{13}$

$y= \frac{11}{13} - \frac{7}{9}$

$= \frac{11 \cdot 9 }{13 \cdot 9 } - \frac{7 \cdot 13}{9 \cdot 13}$

$= \frac{99 }{117} - \frac{91}{117}$

$= \frac{8}{117}$

Between two fractions with the same numerator, the one with the lesser denominator is the greater, so

$\frac{8}{45 } > \frac{8}{117}$

and x > y .

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Example Question #152 : Equations

, , and all stand for positive quantities.

$t = \frac{1}{7} u + 63$

$t = \frac{1}{9} v+63$

Which is the greater quantity?

Possible Answers:

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Solve the equations for and in terms of :

$\frac{1}{7} u + 63 = t$

$\frac{1}{7} u + 63 - 63 = t - 63$

$\frac{1}{7} u = t - 63$

$7 \cdot \frac{1}{7} u =7 \cdot ( t - 63)$

u = 7t - 441

$\frac{1}{9} v+63 = t$

$\frac{1}{9} v + 63 - 63 = t - 63$

$\frac{1}{9} v= t - 63$

$9 \cdot \frac{1}{9} v =9 \cdot ( t - 63)$

v = 9t - 567

Therefore, we seek to determine which of 7t - 441 and 9t - 567 is greater, bearing in mind that both of these quantities, as well as , must be positive.

We can make the following observation:

Suppose

$9t - 567 \le 7t - 441$

Then

$9t - 567 - 7t + 567 \le 7t - 441 - 7t + 567$

$2t \le 126$

$2t \div 2 \le 126 \div 2$

$t \le 63$

But if $t \le 63$ , then

$9t \le 9 \cdot 63 = 567$

and

$9t - 567 \le 567 - 567 = 0$

$v \le 0$ , a contradiction.

Therefore, it must hold that 9t - 567 > 7t - 441 , and v> u .

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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 #145 : How To Find The Solution To An Equation

Example Question #141 : Equations

Example Question #146 : How To Find The Solution To An Equation

Example Question #147 : How To Find The Solution To An Equation

Example Question #148 : How To Find The Solution To An Equation

Example Question #149 : How To Find The Solution To An Equation

Example Question #151 : How To Find The Solution To An Equation

Example Question #151 : How To Find The Solution To An Equation

Example Question #152 : How To Find The Solution To An Equation

Example Question #152 : Equations

All ISEE Upper Level Quantitative Resources

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