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HSPT Math : HSPT Mathematics

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6 Diagnostic Tests 130 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1761 : Hspt Mathematics

Define an operation $\bigstar$ $\displaystyle \bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$.

If $a \bigstar \frac{1}{2} = -1$ $\displaystyle a \bigstar \frac{1}{2} = -1$, evaluate $\displaystyle a$.

Possible Answers:

a = 3 $\displaystyle a = 3$

$a = - \frac{1}{3}$ $\displaystyle a = - \frac{1}{3}$

$a = \frac{1}{3}$ $\displaystyle a = \frac{1}{3}$

a = -3 $\displaystyle a = -3$

Correct answer:

$a = - \frac{1}{3}$ $\displaystyle a = - \frac{1}{3}$

Explanation:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$

$a \bigstar \frac{1}{2} = \frac{a+ \frac{1}{2} }{a \cdot \frac{1}{2} } = \frac{\frac{2a}{2}+ \frac{1}{2} }{ \frac{a}{2} } = \frac{\frac{2a+1}{2} }{ \frac{a}{2} }$ $\displaystyle a \bigstar \frac{1}{2} = \frac{a+ \frac{1}{2} }{a \cdot \frac{1}{2} } = \frac{\frac{2a}{2}+ \frac{1}{2} }{ \frac{a}{2} } = \frac{\frac{2a+1}{2} }{ \frac{a}{2} }$

Equivalently

$a \bigstar \frac{1}{2} = \frac{2a+1}{2} \div \frac{a}{2} = \frac{2a+1}{2} \cdot \frac{2}{a} = \frac{2a+1}{a}$ $\displaystyle a \bigstar \frac{1}{2} = \frac{2a+1}{2} \div \frac{a}{2} = \frac{2a+1}{2} \cdot \frac{2}{a} = \frac{2a+1}{a}$

$a \bigstar \frac{1}{2} = -1$ $\displaystyle a \bigstar \frac{1}{2} = -1$, so set this equal to $\displaystyle -1$:

$\frac{2a+1}{a} = -1$ $\displaystyle \frac{2a+1}{a} = -1$

$\frac{2a+1}{a} \cdot a = -1 \cdot a$ $\displaystyle \frac{2a+1}{a} \cdot a = -1 \cdot a$

$\displaystyle 2a+1 = -a$

$\displaystyle 2a+1 + a - 1 = -a + a - 1$

3a= - 1 $\displaystyle 3a= - 1$

$3a\div 3 = - 1 \div 3$ $\displaystyle 3a\div 3 = - 1 \div 3$

$a = - \frac{1}{3}$ $\displaystyle a = - \frac{1}{3}$

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Example Question #1762 : Hspt Mathematics

Define an operation $\bigstar$ $\displaystyle \bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$.

If $a \bigstar 0.5 = 0.1$ $\displaystyle a \bigstar 0.5 = 0.1$, then evaluate $\displaystyle a$.

Possible Answers:

$a = - \frac{ 19} {10}$ $\displaystyle a = - \frac{ 19} {10}$

$a = \frac{10}{19}$ $\displaystyle a = \frac{10}{19}$

$a = \frac{ 19} {10}$ $\displaystyle a = \frac{ 19} {10}$

$a = - \frac{10}{19}$ $\displaystyle a = - \frac{10}{19}$

Correct answer:

$a = - \frac{10}{19}$ $\displaystyle a = - \frac{10}{19}$

Explanation:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$,

$a \bigstar 0.5 = \frac{a+0.5 }{a \cdot 0.5 } = \frac{a+0.5 }{0.5a }$ $\displaystyle a \bigstar 0.5 = \frac{a+0.5 }{a \cdot 0.5 } = \frac{a+0.5 }{0.5a }$

Set this equal to 0.1:

$\frac{a+0.5 }{0.5a } = 0.1$ $\displaystyle \frac{a+0.5 }{0.5a } = 0.1$

$\frac{a+0.5 }{0.5a } \cdot 0.5 a = 0.1 \cdot 0.5 a$ $\displaystyle \frac{a+0.5 }{0.5a } \cdot 0.5 a = 0.1 \cdot 0.5 a$

$\displaystyle a+0.5 = 0.05 a$

$(a+0.5) \cdot 100 = 0.05 a \cdot 100$ $\displaystyle (a+0.5) \cdot 100 = 0.05 a \cdot 100$

$a \cdot 100 +0.5 \cdot 100 = 5 a$ $\displaystyle a \cdot 100 +0.5 \cdot 100 = 5 a$

$\displaystyle 100a +5 0= 5 a$

$\displaystyle 100a +50 - 100a = 5 a - 100a$

$\displaystyle 50 = -95a$

$50 \div (-95) = -95a \div (-95)$ $\displaystyle 50 \div (-95) = -95a \div (-95)$

$a = -\frac{50}{95} = - \frac{10}{19}$ $\displaystyle a = -\frac{50}{95} = - \frac{10}{19}$

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Example Question #554 : Algebra

$\displaystyle K$ is a number less than 36.

Which of the following gives a solution of the equation

$\displaystyle |x- 17 | + K = 36$

in terms of $\displaystyle K$ ?

Possible Answers:

$\displaystyle x = 53 - K$

$\displaystyle x = -53 - K$

$\displaystyle x = -19 - K$

$\displaystyle x = 19 - K$

Correct answer:

$\displaystyle x = 53 - K$

Explanation:

$\displaystyle |x- 17 | + K = 36$,

$\displaystyle |x- 17 | + K - K = 36 - K$

$\displaystyle |x- 17 | = 36 - K$

Since K < 36 $\displaystyle K < 36$, $\displaystyle 36 - K > 0$, so solutions exist. One of the following two situations occurs:

$\displaystyle x- 17 = 36 - K$

$\displaystyle x- 17 + 17 = 36 - K + 17$

$\displaystyle x = 53 - K$

$\displaystyle -(x- 17 )= 36 - K$

$\displaystyle -x+ 17 = 36 - K$

$\displaystyle -x+ 17 - 17 = 36 - K - 17$

$\displaystyle -x = 19 - K$

$\displaystyle -(-x )= -(19 - K)$

$\displaystyle x = K - 19$

Of these two, only $\displaystyle x = 53 - K$ is among the choices, so it is the correct choice.

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Example Question #1763 : Hspt Mathematics

If L < 32 $\displaystyle L < 32$, then which of the following is equivalent to the expression $\displaystyle | L - 32 | + 12$ ?

Possible Answers:

44 + L $\displaystyle 44 + L$

44 - L $\displaystyle 44 - L$

20-L $\displaystyle 20-L$

20+L $\displaystyle 20+L$

Correct answer:

44 - L $\displaystyle 44 - L$

Explanation:

If L < 32 $\displaystyle L < 32$, then $\displaystyle L - 32 < 0$, so

$\displaystyle | L - 32 | + 12$

$\displaystyle = -(L - 32 )+ 12$

$\displaystyle = - L + 32 + 12$

$\displaystyle = - L + 44$

$\displaystyle = 44 - L$

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Example Question #1171 : Concepts

If L > 47 $\displaystyle L > 47$, then which of the following is equivalent to the expression $\displaystyle | L - 47 | - 32$ ?

Possible Answers:

$\displaystyle - L + 79$

L - 15 $\displaystyle L - 15$

$\displaystyle - L + 15$

L - 79 $\displaystyle L - 79$

Correct answer:

L - 79 $\displaystyle L - 79$

Explanation:

If L > 47 $\displaystyle L > 47$, then $\displaystyle L - 47 > 0$, so

$\displaystyle | L - 47 | - 32$

$\displaystyle = L - 47 - 32$

$\displaystyle = L - 79$

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Example Question #1172 : Concepts

If $\displaystyle 4< X< 12$, then

$\displaystyle |X - 12| + |X- 4|$

can be rewritten as which of the following?

Possible Answers:

$\displaystyle 16$

2X-16 $\displaystyle 2X-16$

$\displaystyle -2X + 16$

$\displaystyle 8$

Correct answer:

$\displaystyle 8$

Explanation:

If X > 4 $\displaystyle X > 4$, then

$\displaystyle X - 4 >0$, so

$\displaystyle |X- 4| = X-4$.

If X< 12 $\displaystyle X< 12$, then

$\displaystyle X - 12 < 0$, so

$\displaystyle |X- 12| = -(X-12)= -X + 12$

$\displaystyle |X - 12| + |X- 4|$

$\displaystyle = X-4 + (-X + 12)$

$\displaystyle = X-X -4 + 12$

= 8 $\displaystyle = 8$

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Example Question #1173 : Concepts

If X< 4 $\displaystyle X< 4$, then

$\displaystyle |X - 12| + |X- 4|$

can be rewritten as which of the following?

Possible Answers:

$\displaystyle 16$

$\displaystyle -2X + 16$

2X-16 $\displaystyle 2X-16$

$\displaystyle 8$

Correct answer:

$\displaystyle -2X + 16$

Explanation:

If X < 4 $\displaystyle X < 4$, then

$\displaystyle X - 4 < 0$, so

$\displaystyle |X- 4| = -(X-4)= -X + 4$.

Also,

$\displaystyle X - 12 < 4 - 12 = -8 < 0$, so

$\displaystyle |X- 12| = -(X-12)= -X + 12$

Therefore,

$\displaystyle |X - 12| + |X- 4|$

$\displaystyle = (-X + 4)+ (-X + 12)$

$\displaystyle = -2X + 16$

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Example Question #1764 : Hspt Mathematics

Define an operation $\bigstar$ $\displaystyle \bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$.

$a \bigstar (-1) = 1$ $\displaystyle a \bigstar (-1) = 1$. What is the value of $\displaystyle a$ ?

Possible Answers:

$a = \frac{1}{2}$ $\displaystyle a = \frac{1}{2}$

a= 2 $\displaystyle a= 2$

$a = - \frac{1}{2}$ $\displaystyle a = - \frac{1}{2}$

a= -2 $\displaystyle a= -2$

Correct answer:

$a = \frac{1}{2}$ $\displaystyle a = \frac{1}{2}$

Explanation:

$a \bigstar b = \frac{a+b}{ab}$ $\displaystyle a \bigstar b = \frac{a+b}{ab}$

and

$a \bigstar (-1) = 1$ $\displaystyle a \bigstar (-1) = 1$

$\frac{a+(-1)}{a(-1)}=1$ $\displaystyle \frac{a+(-1)}{a(-1)}=1$

$\frac{a-1}{-a} = 1$ $\displaystyle \frac{a-1}{-a} = 1$

$\frac{a-1}{-a} \cdot (-a) = 1 \cdot (-a)$ $\displaystyle \frac{a-1}{-a} \cdot (-a) = 1 \cdot (-a)$

$\displaystyle a- 1 = -a$

$\displaystyle a- 1 + a +1 = -a + a +1$

2a= 1 $\displaystyle 2a= 1$

$2a \div 2 = 1 \div 2$ $\displaystyle 2a \div 2 = 1 \div 2$

$a = \frac{1}{2}$ $\displaystyle a = \frac{1}{2}$

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Example Question #1172 : Concepts

Solve for $\displaystyle x$:

4x+6=30 $\displaystyle 4x+6=30$

Possible Answers:

x=20 $\displaystyle x=20$

x=9 $\displaystyle x=9$

x=6 $\displaystyle x=6$

x=30 $\displaystyle x=30$

Correct answer:

x=6 $\displaystyle x=6$

Explanation:

The first step to solve for $\displaystyle x$ is to undue the $\displaystyle 6$ by subtracting it from both sides of the equation.

This results in having

4x=24 $\displaystyle 4x=24$.

Then you must undue 4*x $\displaystyle 4*x$ by dividing by $\displaystyle 4$ on each side resulting in an answer of

24/4=6 $\displaystyle 24/4=6$.

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Example Question #1765 : Hspt Mathematics

Define an operation $\bigstar$ $\displaystyle \bigstar$ on the set of real numbers as follows:

$a \bigstar b = \frac{ab}{a+b}$ $\displaystyle a \bigstar b = \frac{ab}{a+b}$

$a \bigstar 1= -1$ $\displaystyle a \bigstar 1= -1$. What is the value of $\displaystyle a$ ?

Possible Answers:

a = -2 $\displaystyle a = -2$

$\displaystyle a = -0.5$

a = -1 $\displaystyle a = -1$

No real value of $\displaystyle a$ makes this true.

Correct answer:

$\displaystyle a = -0.5$

Explanation:

$a \bigstar b = \frac{ab}{a+b}$ $\displaystyle a \bigstar b = \frac{ab}{a+b}$, so

$a \bigstar 1 = \frac{a \cdot 1 }{a+1} = \frac{a }{a+1}$ $\displaystyle a \bigstar 1 = \frac{a \cdot 1 }{a+1} = \frac{a }{a+1}$

Set this equal to $\displaystyle -1$:

$\frac{a }{a+1} = -1$ $\displaystyle \frac{a }{a+1} = -1$

$\frac{a }{a+1} \cdot (a+1) = -1 \cdot (a+1)$ $\displaystyle \frac{a }{a+1} \cdot (a+1) = -1 \cdot (a+1)$

$\displaystyle a = -a - 1$

$\displaystyle a + a = -a - 1 + a$

$\displaystyle 2a = - 1$

$2a \div 2= - 1 \div 2$ $\displaystyle 2a \div 2= - 1 \div 2$

$\displaystyle a = -0.5$

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HSPT Math : HSPT Mathematics

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1761 : Hspt Mathematics

Example Question #1762 : Hspt Mathematics

Example Question #554 : Algebra

Example Question #1763 : Hspt Mathematics

Example Question #1171 : Concepts

Example Question #1172 : Concepts

Example Question #1173 : Concepts

Example Question #1764 : Hspt Mathematics

Example Question #1172 : Concepts

Example Question #1765 : Hspt Mathematics

All HSPT Math Resources

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