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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 #153 : Equations

, , and all stand for positive quantities.

$t = \frac{1}{5} u + 3$

$t = \frac{1}{3} v + 5$

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:

Solve the equations for and in terms of :

$\frac{1}{5} u + 3 = t$

$\frac{1}{5} u + 3 - 3 = t - 3$

$\frac{1}{5} u = t - 3$

$5 \cdot \frac{1}{5} u =5 \cdot ( t - 3)$

u = 5t - 15

$\frac{1}{3} v + 5 = t$

$\frac{1}{3} v + 5 - 5 = t- 5$

$\frac{1}{3} v = t- 5$

$3 \cdot \frac{1}{3} v =3 \cdot ( t- 5)$

v = 3t - 15

5 > 3 and is positive, so by the properties of inequality,

5 t > 3t

5 t -15 > 3t - 15

u > v

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Example Question #153 : How To Find The Solution To An Equation

Solve for :

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

Possible Answers:

$x = 6 +\frac { 3}{4} t$

$x =8+\frac {4}{3} t$

$x =8-\frac {4}{3} t$

$x = 6 -\frac { 3}{4} t$

Correct answer:

$x = 6 -\frac { 3}{4} t$

Explanation:

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

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

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

$3 \cdot \frac{1}{3} x = 3 \cdot \left ( 2 - \frac{1}{4} t \right )$

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

$x = 6 -\frac { 3}{4} t$

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Example Question #154 : How To Find The Solution To An Equation

m - (-46) = -73

n - (-73) = -46

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) and (b) are equal

Explanation:

m - (-46) = -73

m - (-46)+ (-46 ) = -73+ (-46 )

m = - (73 + 46 ) = -119

n - (-73) = -46

n - (-73) + (-73) = -46 + (-73)

n = - (46+73 ) = -119

m = n

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Example Question #831 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Untitled

Figure NOT drawn to scale

Above is a straight line on a graph. Which is the greater quantity?

(a)

(b) 18

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:

(a) is the greater quantity

Explanation:

If we go from the point (48, 60) to (24, 42), we see that if the first coordinate decreases by 24, the second decreases by 18. Going from (24, 42) to the point on the -axis, the first coordinate again decreases by 24, so the second coordinate again decreases by 18:

y = 42 - 18 = 24> 18 .

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Example Question #156 : How To Find The Solution To An Equation

$2 < \frac{1}{Y}< 4$

The reciprocal of is between 2 and 4. Which is the greater quantity?

(a)

(b) $\frac{1}{3}$

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

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

(a) is the greater quantity

Correct answer:

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

Explanation:

$2 < \frac{1}{Y}< 4$ , so

$\frac{1}{2} > \frac{1}{ \frac{1}{Y}}> \frac{1}{4}$

$\frac{1}{2} >Y> \frac{1}{4}$

Also,

2 < 3< 4 , so

$\frac{1}{2} > \frac{1}{3}> \frac{1}{4}$

Therefore, it possible for

$\frac{1}{2} > Y >\frac{1}{3}$ ,

$Y = \frac{1}{3}$ ,

$\frac{1}{3}>Y > \frac{1}{4}$ ,

making it inconclusive whether or $\frac{1}{3}$ is the greater.

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Example Question #157 : How To Find The Solution To An Equation

Solve for :

$t = \frac{1}{4} x + 3$

Possible Answers:

x = 4t+ 12

$x = \frac{1}{4 }t+\frac{3}{4 }$

$x = \frac{1}{4 }t- \frac{3}{4 }$

x = 4t- 12

Correct answer:

x = 4t- 12

Explanation:

$\frac{1}{4} x + 3 = t$

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

$\frac{1}{4} x = t - 3$

$4 \cdot \frac{1}{4} x =4 \cdot ( t - 3)$

$x = 4\cdot t - 4 \cdot 3$

x = 4t- 12

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Example Question #161 : How To Find The Solution To An Equation

, , and all stand for negative quantities.

$t = \frac{1}{4} u -6$

$t = \frac{1}{6} v - 4$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

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

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

Solve the first equation for in terms of , using the properties of equality to isolate the :

$\frac{1}{4} u -6 = t$

$\frac{1}{4} u -6 + 6 = t + 6$

$\frac{1}{4} u = t+ 6$

$4 \cdot \frac{1}{4} u =4 \cdot ( t+ 6)$

$u = 4 \cdot t+4 \cdot 6$

u = 4t+24

Solve for in the second equation similarly:

$\frac{1}{6} v - 4 = t$

$\frac{1}{6} v - 4 + 4 = t + 4$

$\frac{1}{6} v = t + 4$

$6 \cdot \frac{1}{6} v =6 \cdot ( t + 4)$

$v = 6t + 6 \cdot 4$

v = 6t + 24

t < 0 , so by the properties of inequality,

4 < 6

4t> 6t

4t + 24 > 6t + 24

u > v

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Example Question #162 : How To Find The Solution To An Equation

Define f(x) = (x+ 1) (x-1) and g(x) = 2x + 1 .

(f+g) (c) = 8

Which is the greater quantity?

(a)

(b) 2

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:

The definition f(x) can be rewritten by noting that this is the product of a sum and a difference of the same two terms, and that the product is the difference of their squares:

$f(x) = (x+ 1) (x-1) = x^{2} - 1 ^{2} = x^{2} - 1$

By definition,

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

Since (f+g) (c) = 8 , it holds that

$c^{2} + 2c = 8$ , or

$c^{2} + 2c - 8 = 0$

We can factor the trinomial using two integers whose sum is 2 and whose product is ; by a little trial and error we find 4 and , so

(c-2) (c+4) = 0 .

By the Zero Product Principle,

c- 2 = 0 , in which case c= 2 ; or,

c + 4 = 0 , in which case c= -4 < 2 .

It is therefore unclear whether is less than or equal to 2.

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Example Question #163 : How To Find The Solution To An Equation

Define $f(x) = x^{2}$ .

(f+g)(5) = 60 .

Which is the greater quantity?

(a) f(5)

(b) g(5)

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:

(b) is the greater quantity

Explanation:

$f(x) = x^{2}$ , so, setting x = 5 ,

$f(5) = 5^{2} = 25$

By definition,

(f+g) (5) = f(5)+ g(5)

so, by substitution,

60 = 25+ g(5)

60 -25 = 25+ g(5)- 25

g(5) = 35

Therefore, g(5) > f(5 ) .

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Example Question #164 : How To Find The Solution To An Equation

Define $f(x) = \left ( x+ 2\right )^{2}$ and $g(x) = x^{2}$ .

(f-g) (c) = 0

Which is the greater quantity?

(a) 0

(b)

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

Correct answer:

(b) is the greater quantity

Explanation:

$f(x) = \left ( x+ 2\right )^{2}$ can be rewritten using the square of a binomial pattern:

$f(x) = \left ( x+ 2\right )^{2} = x^{2} + 2 \cdot x \cdot 2 + 2 ^{2}= x^{2}+ 4x + 4$

By definition,

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

(f-g)(c) = 4c + 4

Since

(f-g)(c) = 0 , it holds that

4c + 4 = 0

Solving for :

4c + 4 - 4 = 0 - 4

4c= -4

$4c\div 4 = -4 \div 4$

c = -1 , which is less than 0.

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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 #153 : Equations

Example Question #153 : How To Find The Solution To An Equation

Example Question #154 : How To Find The Solution To An Equation

Example Question #831 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #156 : How To Find The Solution To An Equation

Example Question #157 : How To Find The Solution To An Equation

Example Question #161 : How To Find The Solution To An Equation

Example Question #162 : How To Find The Solution To An Equation

Example Question #163 : How To Find The Solution To An Equation

Example Question #164 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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