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ISEE Upper Level Math : Variables

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Example Questions

Example Question #1 : Operations

Machine A can produce 4 buttons in 2 days. Machine B can produce 20 buttons in 4 days. How many buttons can Machine A and B, working together, produce in 40 days?

Possible Answers:

$\dpi{100} 280$

$\dpi{100} 7$

$\dpi{100} 200$

$\dpi{100} 70$

Correct answer:

$\dpi{100} 280$

Explanation:

First determine how many buttons each machine can produce in a day.

Machine A: $\dpi{100} \frac{4}{2}=2\ buttons\ per\ day$

Machine B: $\dpi{100} \frac{20}{4}=5\ buttons\ per\ day$

Machine A and B can produce 7 buttons per day when they are working together.

$\dpi{100} 7\times 40=280\ buttons$

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Example Question #1 : Variables

Brian has 3 siblings. When his family orders pizza, each of the 4 children is given $\dpi{100} \frac{1}{4}$ of the pizza. Brian does not feel well so he only finishes $\dpi{100} \frac{1}{3}$ of his pizza. If the original pizza consisted of 12 slices of pizza, how many slices did Brian eat?

Possible Answers:

$\dpi{100} 1$

$\dpi{100} 3$

$\dpi{100} 4$

$\dpi{100} 6$

Correct answer:

$\dpi{100} 1$

Explanation:

Brian eats $\dpi{100} \frac{1}{3}\times \frac{1}{4}\times 12\ slices\ of\ pizza$ .

So Brian eats

$\dpi{100} \frac{1}{12}\cdot 12=1\ slice\ of\ pizza.$

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Example Question #3 : Operations

Solve for $\dpi{100} x$ :

$\dpi{100} 4x^{2}=256$

Possible Answers:

$\dpi{100} \pm 8$

$\dpi{100} \pm 2$

$\dpi{100} 4$

$\dpi{100} \pm 4$

Correct answer:

$\dpi{100} \pm 8$

Explanation:

$\dpi{100} 4x^{2}=256$

$\dpi{100} \frac{4x^{2}}{4}=\frac{256}{4}$

$\dpi{100} x^{2}=64$

$\dpi{100} \sqrt{x^{2}}=\sqrt{64}$

$\dpi{100} x=\pm 8$

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Example Question #4 : Operations

Simplify:

$(0.6x+0.4y) ^{2}$

Possible Answers:

$0.36x^{2} + 0.16 y ^{2}$

$0.36x^{2} + 0.24 xy + 0.16 y ^{2}$

$0.36x^{2} + 0.48 x^{2}y^{2} + 0.16 y ^{2}$

$0.36x^{2} + 0.24 x^{2}y^{2} + 0.16 y ^{2}$

$0.36x^{2} + 0.48 xy + 0.16 y ^{2}$

Correct answer:

$0.36x^{2} + 0.48 xy + 0.16 y ^{2}$

Explanation:

This can be solved using the pattern for the square of a sum:

$(0.6x+0.4y) ^{2}$

$=\left ( 0.6x \right ) ^{2} +2\cdot 0.6x \cdot 0.4y + \left ( 0.4y \right ) ^{2}$

$= 0.6^{2}x^{2} +2\cdot 0.6 \cdot 0.4 \cdot xy + 0.4^{2} y ^{2}$

$= 0.36x^{2} + 0.48 xy + 0.16 y ^{2}$

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Example Question #5 : Operations

Simplify:

$(0.6x-0.4y) ^{2}$

Possible Answers:

$0.36x^{2}-0.24xy+ 0.16y^{2}$

$0.36x^{2}- 0.16y^{2}$

$0.36x^{2}-0.48xy+ 0.16y^{2}$

$0.36x^{2}+0.16y^{2}$

$0.36x^{2}-0.48xy- 0.16y^{2}$

Correct answer:

$0.36x^{2}-0.48xy+ 0.16y^{2}$

Explanation:

This can be solved using the pattern for the square of a difference:

$(0.6x-0.4y) ^{2}$

$=\left ( 0.6x \right ) ^{2} - 2\cdot 0.6x \cdot 0.4y + \left ( 0.4y \right ) ^{2}$

$= 0.6^{2}x^{2} - 2\cdot 0.6 \cdot 0.4 \cdot xy + 0.4^{2} y ^{2}$

$= 0.36x^{2} - 0.48 xy + 0.16 y ^{2}$

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Example Question #6 : Operations

Simplify:

$\left (\frac{4}{3}x + \frac{5}{3} \right )^{2}$

Possible Answers:

$\frac{16}{9}x^{2} + \frac{80}{9}x +\frac{25}{9}$

$\frac{16}{9}x^{2} +\frac{25}{9} x$

$\frac{16}{9}x^{2} +\frac{25}{9}$

$\frac{16}{9}x^{2} + \frac{40}{9}x +\frac{25}{9}$

$\frac{16}{9}x^{2} + \frac{20}{9}x +\frac{25}{9}$

Correct answer:

$\frac{16}{9}x^{2} + \frac{40}{9}x +\frac{25}{9}$

Explanation:

This can be solved using the pattern for the square of a sum:

$\left (\frac{4}{3}x + \frac{5}{3} \right )^{2}$

$=\left (\frac{4}{3}x \right ) ^{2} +2 \cdot \frac{4}{3}x \cdot \frac{5}{3} + \left ( \frac{5}{3} \right ) ^{2}$

$=\frac{4^{2}}{3^{2}}\cdot x^{2} +2 \cdot \frac{4}{3}\cdot \frac{5}{3} \cdot x +\frac{5^{2}}{3^{2}}$

$=\frac{16}{9}x^{2} + \frac{40}{9}x +\frac{25}{9}$

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Example Question #2 : Variables

Multiply:

(5x - 3y) (3x + 8y)

Possible Answers:

$15 x ^{2}+31xy -24 y^{2}$

$15x^{2} -31xy+ 24y^{2}$

$15x^{2} - 24y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

Correct answer:

$15 x ^{2}+31xy -24 y^{2}$

Explanation:

(5x - 3y) (3x + 8y)

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

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Example Question #8 : Operations

Multiply:

(5x - 3y) (3x - 8y)

Possible Answers:

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} -31xy+ 24y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} +31xy - 24y^{2}$

Correct answer:

$15 x ^{2}-49xy +24 y^{2}$

Explanation:

Use the FOIL method:

(5x - 3y) (3x - 8y)

$= 5x \cdot 3x-5x \cdot 8y - 3y \cdot 3x + 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x-5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x + 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}-40xy - 9xy +24 y^{2}$

$= 15 x ^{2}-49xy +24 y^{2}$

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Example Question #9 : How To Multiply Variables

Define f(x) = 5 - 2x

What is f(3-2x) ?

Possible Answers:

4x-11

4x -1

Correct answer:

4x -1

Explanation:

Substitute 3-2x for :

f(x) = 5 - 2x

$f(3-2x) = 5 - 2 \left ( 3-2x \right )$

$f(3-2x) = 5 - 2 \cdot 3- \left ( - 2 \right )\cdot 2x \right )$

f(3-2x) = 5 - 6 +4x

f(3-2x) =4x -1

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Example Question #10 : How To Multiply Variables

Simplify:

$(\frac{4x^2}{81x^6})^{-\frac{1}{2}}$

Possible Answers:

$\frac{2}{9x^2}$

x^4

18x^2

$\frac{9x^2}{2}$

$\frac{4}{9x^2}$

Correct answer:

$\frac{9x^2}{2}$

Explanation:

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

$(\frac{4x^2}{81x^6})^{-\frac{1}{2}}=(\frac{81x^6}{4x^2})^{\frac{1}{2}}$

Apply the exponent within the parentheses and simplify. Remember that fractional exponents can be written as roots.

$\frac{(81x^6)^{\frac{1}{2}}}{(4x^2)^{\frac{1}{2}}}=\frac{\sqrt{81x^6}}{\sqrt{4x^2}}$

Simplify by taking the roots and canceling common factors.

$\frac{\sqrt{81x^6}}{\sqrt{4x^2}}=\frac{9x^3}{2x}=\frac{9x^2}{2}$

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ISEE Upper Level Math : Variables

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Operations

Example Question #1 : Variables

Example Question #3 : Operations

Example Question #4 : Operations

Example Question #5 : Operations

Example Question #6 : Operations

Example Question #2 : Variables

Example Question #8 : Operations

Example Question #9 : How To Multiply Variables

Example Question #10 : How To Multiply Variables

All ISEE Upper Level Math Resources

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