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ISEE Upper Level Quantitative : Algebraic Concepts

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 #91 : How To Find The Solution To An Equation

A hat costs $70.80 after a 20% discount. How much did the hat cost before the discount?

Possible Answers:

$\$90.80$

$\$56.64$

$\$84.96$

It is impossible to tell from the information given.

$\$88.50$

Correct answer:

$\$88.50$

Explanation:

Since $70.80 is the price after a 20% discount off the original price, it is 80% of that original price. The problem is equivalent to asking:

$70.80 is 80% of what amount?

Let be the price before discount.

0.80 x = 70.80

$0.80 x \div 0.80 = 70.80 \div 0.80$

$x = \$88.50$

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Example Question #92 : How To Find The Solution To An Equation

x + y = 14

2x + 3y = 36

Which is the greater quantity?

(A)

(B)

Possible Answers:

(B) is greater

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

(A) and (B) are equal

(A) is greater

Correct answer:

(B) is greater

Explanation:

x + y = 14 , so x = 14-y .

Substitute for in the other equation:

2x + 3y = 36

2 (14-y)+ 3y = 36

28-2y + 3y = 36

$28 +\left ( 3-2 \right ) y = 36$

28 + y = 36

28 + y -28 = 36-28

y = 8

x = 14-y = 14-8 = 6

y > x , so (B) is greater.

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Example Question #93 : How To Find The Solution To An Equation

A line includes the points (3,6) and (4,9) . Which is the greater quantity?

(A) The -coordinate of the -intercept.

(B) The -coordinate of the -intercept.

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:

(A) is greater

Explanation:

We can figure out the equation of the line as follows:

Set $x_{1} = 3, y_{1} = 6,x_{2} =4, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-6}{4-3} = \frac{3}{1} = 3$

In the slope-intercept formula, we set

x = 3, y = 6,m = 3 and solve for :

y = mx + b

$6 = 3 \cdot 3 + b$

6 = 9+ b

6 - 9 = 9+ b - 9

-3= b

The equation is y = 3x -3

The -intercept is (0, -3) . To find the -intercept, we substitute 0 for :

0 = 3x -3

0+ 3 = 3x -3 + 3

3 = 3x

$3 \div 3 = 3x \div 3$

x= 1

The -intercept is (1,0) .

This makes (A), the -coordinate of the -intercept, greater.

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Example Question #91 : How To Find The Solution To An Equation

Define $f (x) = \sqrt{4-x}$ and $g(x) = \sqrt{x+3}$ .

What is the domain of the function f + g ?

Possible Answers:

$\left [ - 4, 3\right ]$

$(- \infty, -3]$

$\left [ -3, 4\right ]$

$(- \infty,\infty )$

$[4,\infty )$

Correct answer:

$\left [ -3, 4\right ]$

Explanation:

$f (x) = \sqrt{4-x}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$4 - x \geq 0$

$4 - x + x \geq 0 + x$

$4 \geq x$ or $(- \infty, 4]$

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

$x+3 \geq 0$

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

$x \geq -3$ or $[-3,\infty )$

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

$(- \infty, 4] \cap [-3,\infty ) = \left [ -3, 4\right ]$

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Example Question #91 : Equations

Define $f (x) = \sqrt{4-x}$ and $g(x) = \sqrt{x+3}$ .

Which is the greater quantity?

(A) $\left (f+g \right ) (4)$

(B) $\left (f+g \right ) (-3)$

Possible Answers:

(A) and (B) are equal

(B) is greater

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

(A) is greater

Correct answer:

(A) and (B) are equal

Explanation:

$\left ( f + g \right ) (x) = f (x) + g(x) = \sqrt{4-x} + \sqrt{x+3}$

Substitute and to determine the values of the respective expressions:

$\left ( f + g \right ) (4) = \sqrt{4-4} + \sqrt{4+3}= \sqrt{0} + \sqrt{7} = \sqrt{7}$

$\left ( f + g \right ) (-3) = \sqrt{4-(-3)} + \sqrt{-3+3}= \sqrt{7} + \sqrt{0} = \sqrt{7}$

The expressions are equal.

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Example Question #93 : How To Find The Solution To An Equation

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

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

For which value of is the expression $N \Leftrightarrow 5$ undefined?

Possible Answers:

N = - 5

$N = - \frac{1}{5}$

$N = \frac{1}{5}$

N = 0

N = 5

Correct answer:

$N = - \frac{1}{5}$

Explanation:

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

$N\Leftrightarrow 5 = \frac{N + 5}{ N \cdot 5 +1} = \frac{N + 5}{5 N +1}$

This expression is undefined if and only if the denominator is equal to 0, so

5N + 1 = 0

5N + 1 -1 = 0 -1

5N = -1

$5N \div 5 = -1 \div 5$

$N = - \frac{1}{5}$

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Example Question #92 : How To Find The Solution To An Equation

A line includes the points (6, 3.1) and (2, -1.5) . What is the -intercept of this line (-coordinate rounded to the nearest tenth)?

Possible Answers:

(3.3,0)

(0.2,0)

(1.2,0)

(-2.1,0)

(-3.8,0)

Correct answer:

(3.3,0)

Explanation:

Let $x_{1} = 2, y_{1} = -1.5, x_{2} =6, y_{2} = 3.1$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{3.1-(-1.5)}{6-2} = \frac{4.6}{4} = 1.15$

Apply the point-slope formula setting

$x_{1} = 2, y_{1} = -1.5, m = 1.15$ :

$y - y_{1} = m (x - x_{1})$

y - (-1.5)= 1.15 (x - 2)

y+1.5= 1.15 (x - 2)

y+1.5= 1.15 x - 2.3

Set $y \, = 0$ to find the -coordinate of the -intercept:

0+1.5= 1.15 x - 2.3

1.5= 1.15 x - 2.3

1.5+ 2.3= 1.15 x - 2.3 + 2.3

3.8= 1.15 x

$3.8 \div 1.15 = 1.15 x \div 1.15$

$x \approx 3.3$

The -intercept is (approximately at) (3.3,0)

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Example Question #95 : How To Find The Solution To An Equation

A line includes the points (5, 3.1) and (3, -1.7) . What is the -intercept of this line?

Possible Answers:

(0,-4.8)

(0,3.7)

(0,-2)

(0,-8.9)

(0,-6.5)

Correct answer:

(0,-8.9)

Explanation:

Let $x_{1} = 3, y_{1} = -1.7, x_{2} = 5, y_{2} = 3.1$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{3.1-(-1.7)}{5-3} = \frac{4.8}{2} = 2.4$

Apply the point-slope formula setting

$x_{1} = 5, y_{1} = 3.1, m= 2.4$

$y - y_{1} = m (x - x_{1})$

y - 3.1= 2.4 (x - 5)

Set x = 0 to find the -coordinate of the -intercept:

y - 3.1= 2.4 (0- 5)

y - 3.1= 2.4 (- 5)

y - 3.1= -12

y - 3.1+ 3.1 = -12 + 3.1

y = -8.9

The -intercept is (0,-8.9) .

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Example Question #91 : How To Find The Solution To An Equation

A line includes the points (5, -5.2) and (3, -1.8) . Which of these is the slope of that line?

Possible Answers:

$-\frac{10}{17}$

$-\frac{8}{7}$

$-\frac{17}{10}$

$-\frac{7}{8}$

The correct answer is not among the other choices

Correct answer:

$-\frac{17}{10}$

Explanation:

Let $x_{1} = 3, y_{1} = -1.8 , x_{2} = 5, y_{2} = -5.2$

We calculate the slope as follows:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{-5.2-(-1.8)}{5-3} = \frac{-3.4}{2} = -1.7 = -\frac{17}{10}$

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Example Question #91 : Equations

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

$a \ast b = \frac{a + 2b }{ a + 3b}$

Let be a positive number. Then which is the greater quantity?

(A) $t \ast 4t$

(A) $4t \ast t$

Possible Answers:

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

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

(B) is greater

Explanation:

Substitute each pair of expressions:

$a \ast b = \frac{a + 2b }{ a + 3b}$

$t \ast 4t = \frac{t + 2 \cdot 4t }{ t + 3 \cdot 4t } = \frac{t + 8t }{ t + 12t } = \frac{9t }{ 13t } = \frac{9 }{ 13 }$

$4t \ast t = \frac{4t + 2 \cdot t }{4t + 3 \cdot t } = \frac{4t + 2t }{ 4t + 3t } = \frac{6t }{ 7t } = \frac{6 }{ 7 }$

We can compare these fractions by writing them with a common denominator:

$t \ast 4t = \frac{9 }{ 13 } = \frac{9 \times 7 }{ 13 \times 7 } = \frac{63}{91}$

$4t \ast t =\frac{6 }{ 7 }=\frac{6 \times 13 }{ 7 \times 13 } = \frac{78}{91}$

$4t \ast t > t \ast 4t$ regardless of the value of , making (B) greater.

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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 : Algebraic Concepts

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #91 : How To Find The Solution To An Equation

Example Question #92 : How To Find The Solution To An Equation

Example Question #93 : How To Find The Solution To An Equation

Example Question #91 : How To Find The Solution To An Equation

Example Question #91 : Equations

Example Question #93 : How To Find The Solution To An Equation

Example Question #92 : How To Find The Solution To An Equation

Example Question #95 : How To Find The Solution To An Equation

Example Question #91 : How To Find The Solution To An Equation

Example Question #91 : Equations

All ISEE Upper Level Quantitative Resources

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