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PSAT Math : Arithmetic

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Example Questions

Example Question #1 : How To Divide Negative Numbers

What is $\frac{-36}{-9}$ ?

Possible Answers:

-(+4)

Correct answer:

Explanation:

A negative number divided by a negative number always results in a positive number. divided by equals . Since the answer is positive, the answer cannot be or any other negative number.

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Example Question #5 : Negative Numbers

Solve for :

16-4x=x+6

Possible Answers:

$\frac{1}{2}$

$-\frac{18}{5}$

$-\frac{1}{2}$

Correct answer:

Explanation:

Begin by isolating your variable.

Subtract from both sides:

16-4x-x=6 , or 16-5x=6

Next, subtract from both sides:

-5x=6-16 , or -5x=-10

Then, divide both sides by :

$x=\frac{-10}{-5}$

Recall that division of a negative by a negative gives you a positive, therefore:

$x=\frac{10}{5}$ or x=2

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Example Question #1561 : Sat Mathematics

If is a positive number, and -3b is also a positive number, what is a possible value for ?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

Because $\dpi{100} \small -3b$ is positive, $\dpi{100} \small b$ must be negative since the product of two negative numbers is positive.

Because $\dpi{100} \small ab$ is also positive, $\dpi{100} \small a$ must also be negative in order to produce a prositive product.

To check you answer, you can try plugging in any negative number for $\dpi{100} \small a$ .

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Example Question #1 : Negative Numbers

, , and are all negative odd integers. Which of the following three expressions must be positive?

I) $a^{b+c}$

II) $a^{b-c}$

III) $a^{c-b}$

Possible Answers:

II only

III only

I only

None of these

All of these

Correct answer:

All of these

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.

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Example Question #1791 : Psat Mathematics

is a positive integer; and are negative integers. Which of the following three expressions must be negative?

I) $b^{a+c}$

II) $b^{a-c}$

III) $b^{c-a}$

Possible Answers:

None of I, II or III

I and II only

II and III only

I and III only

I, II and III

Correct answer:

None of I, II or III

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of and , which are not given in the problem.

The correct response is none of these.

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Example Question #3 : How To Multiply Negative Numbers

, , and are all negative numbers. Which of the following must be positive?

Possible Answers:

$\frac{ b^{5}c^{4}}{a^{12}}$

$\frac{ a^{6}b^{4}}{c^{9}}$

$\frac{a^{4}b^{7}}{c^{5}}$

$\frac{ a^{8}c^{3}}{b^{6}}$

$\frac{ a^{7}c^{5}}{b^{3}}$

Correct answer:

$\frac{a^{4}b^{7}}{c^{5}}$

Explanation:

The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.

$\frac{ a^{6}b^{4}}{c^{9}}$ : $a^{6}$ and $b^{4}$ are positive, yielding a positive dividend; $c^{9}$ is a negative divisor; this result is negative.

$\frac{ a^{7}c^{5}}{b^{3}}$ : $a^{7}$ and $c^{5}$ are negative, yielding a positive dividend; $b^{3}$ is a negative divisor; this result is negative.

$\frac{ a^{8}c^{3}}{b^{6}}$ : $a^{8}$ is positive and $c^{3}$ is negative, yielding a negative dividend; $b^{3}$ is a positive divisor; this result is negative.

$\frac{ b^{5}c^{4}}{a^{12}}$ : $b^{5}$ is negative and $c^{4}$ is positive, yielding a negative dividend; $a^{12}$ is a positive divisor; this result is negative.

$\frac{a^{4}b^{7}}{c^{5}}$ : $a^{4}$ is positive and $b^{7}$ is negative, yielding a negative dividend; $c^{5}$ is a negative divisor; this result is positive.

The correct choice is $\frac{a^{4}b^{7}}{c^{5}}$ .

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Example Question #6 : Negative Numbers

and are positive numbers; is a negative number. All of the following must be positive except:

Possible Answers:

$a+ b^{2} +c^{6}$

$a^{3}+ b^{4} +c$

$a^{2} -b^{7} +c$

$a^{4}+ b^{5} +c^{3}$

$a^{2}-b^{3} +c$

Correct answer:

$a^{4}+ b^{5} +c^{3}$

Explanation:

Since and are positive, all powers of and will be positive; also, in each of the expressions, the powers of and are being added. The clue to look for is the power of and the sign before it.

In the cases of $a^{3}+ b^{4} +c$ and $a+ b^{2} +c^{6}$ , since the negative number is being raised to an even power, each expression amounts to the sum of three positive numbers, which is positive.

In the cases of $a^{2} -b^{7} +c$ and $a^{2}-b^{3} +c$ , since the negative number is being raised to an odd power, the middle power is negative - but since it is being subtracted, it is the same as if a positive number is being added. Therefore, each is essentially the sum of three positive numbers, which, again, is positive.

In the case of $a^{4}+ b^{5} +c^{3}$ , however, since the negative number is being raised to an odd power, the middle power is again negative. This time, it is basically the same as subtracting a positive number. As can be seen in this example, it is possible to have this be equal to a negative number:

a = 1, b = -2, c= 1 :

$a^{4}+ b^{5} +c^{3} = 1^{4}+ (-2)^{5} +1^{3} = 1 + (-32)+ 1 = -30$

Therefore, $a^{4}+ b^{5} +c^{3}$ is the correct choice.

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Example Question #2 : How To Multiply Negative Numbers

Let be a negative integer and be a nonzero integer. Which of the following must be negative regardless of whether is positive or negative?

Possible Answers:

$a^{2}+ b$

$a+b^{2}$

None of the other answers is correct.

$a^{2}b$

$ab^{2}$

Correct answer:

$ab^{2}$

Explanation:

Since $b^{2}$ is positive, $ab^{2}$ , the product of a negative number and a positive number, must be negative also.

Of the others:

$a^{2}b$ is incorrect; if is negative, then $a^{2}$ is positive, and $a^{2}b$ assumes the sign of .

$a^{2}+ b$ is incorrect; again, $a^{2}$ is positive, and if is a positive number, $a^{2}+ b$ is positive.

$a+b^{2}$ is incorrect; regardless of the sign of , $b^{2}$ is positive, and if its absolute value is greater than that of , $a+b^{2}$ is positive.

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Example Question #3 : How To Multiply Negative Numbers

Given that a, b are both integers, $-10 \leq a \leq -1$ , and $-10 \leq b \leq -1$ , which of the following is correct about the sign of the expression (a -10)(b +10) ?

Possible Answers:

The expression can be positive, negative, or zero.

The expression must be negative.

The expression must be positive.

The expression must be negative or zero.

The expression must be positive or zero.

Correct answer:

The expression must be negative or zero.

Explanation:

If $-10 \leq a \leq -1$ , then we know that is any number between or equal to and . Therefore must be a negative number.

Also, if $-10 \leq b \leq -1$ , then we know that is any number between or equal to and . Therefore must be a negative number.

Now looking the expression (a -10)(b +10) we can find the sign of each component in the expression.

Since is negative, we know that a negative number minus another number is still a negative number.

Therefore, (a-10) is a negative number.

Since is between or equal to and we can plug in these end values in to determine the sign of (b +10) .

(-10+10)=0

(-1+10)=9

Therefore, (b +10) is either zero or a positive number.

Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that (a -10)(b +10) is negative or zero.

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Example Question #231 : Arithmetic

Find the product.

$(-270) \times (-131)$

Possible Answers:

35370

35100

Correct answer:

35370

Explanation:

When multiplying together two negatives, our value for the product become positive. $(-270) \times (-131) = 270 \times 131 = 35370$

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PSAT Math : Arithmetic

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Divide Negative Numbers

Example Question #5 : Negative Numbers

Example Question #1561 : Sat Mathematics

Example Question #1 : Negative Numbers

Example Question #1791 : Psat Mathematics

Example Question #3 : How To Multiply Negative Numbers

Example Question #6 : Negative Numbers

Example Question #2 : How To Multiply Negative Numbers

Example Question #3 : How To Multiply Negative Numbers

Example Question #231 : Arithmetic

All PSAT Math Resources

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