A proofreader can read 40 pages in one hour. How many pages can this proofreader read in 90 minutes?
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Solution
Since 90 minutes is equal to 1.5 hours, a proofreader who can read 40 pages in one hour can read (1.5)(40) or 60 pages in 1.5 hours.
When n = 1⁄4, what is the value of \(\frac{2n − 5}{n}\)?
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Solution
Which of the following expressions is a polynomial factor of a16 − 16?
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Solution
Remember that a difference of squares factors easily, such as: a2 − b2 = (a + b)(a − b).
Using the same technique, you can factor a16 − 16 into (a8 + 4)(a8 − 4).
The factor (a8 − 4) is another difference of squares, so it can be factored further into itself: (a8 − 4) = (a4 + 2)(a4 − 2).
Of these factors, only (a4 + 2) is an answer choice
What is the product of the 2 solutions of the equation x2 + 3x − 21 = 0?
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Solution
The easiest way to solve this problem is to remember that when two binomial expressions are multiplied, there is a predictable result.
Take the following generalized example: (x + a)(x − b) = x2 − bx + ax − ab. If x2 − bx + ax − ab = 0, then the solutions to the equation are x = −a and x = b.
The product of the solutions is −ab. With this expression, x2+3x−21 = 0, the product of the solutions (−ab) is −21.
The perimeter of a square is 48 centimeters. What is its area, in square centimeters?
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Solution
If a square has side x, then its perimeter is 4x; this is because a square is defined as a rectangle where all four sides are of equal length.
Since the perimeter of the square is 48, then 48 = 4x and x = \(\frac{48}{4}\) = 12.
Thus, the length of one side of the square is 12.
The area of a square is defined as (side)2; therefore the area of this square is 122 or 144.
What is the 217th digit after the decimal point in the repeating decimal 0.3456?
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Solution
To solve this problem, recognize that the repeating decimal has four places (0.3456), and that the fourth place is occupied by the number 6.
Therefore, every place that is a multiple of 4 will be represented by the number 6. Since 217 is not divisible by 4, you know that the 217th digit cannot be 6;
eliminate answer choice E. Because 216 is a multiple of 4, the 216th digit will be 6.
Therefore, the 217th digit must be 3, the next digit in the repeating decimal.
(n7)11 is equivalent to:
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Solution
Remember that the rule for exponents states that for base number b and exponents x and y, (bx)y = bxy.
Thus, when you apply the numbers from this problem, you find that (n7)11 = n(7)(11) = n77.
For all x, 13 − 2(x + 5) = ?
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Solution
To solve this problem, you must distribute and add like terms, as follows:
13 − 2(x + 5) =
13 − 2x − 10 = 2x + 3
For all m and n, (3m + n)(m2 − n) = ?
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Solution
Since this problem requires you to multiply two binomials, you can utilize the FOIL (First, Outside, Inside, Last) method to multiply the expressions.
First: (3m)(m2) = 3m3
Outside: (3m)(−n) = −3mn
Inside: (n)(m2) = m2n
Last: (n)(−n) = −n2
Finally, add all these terms up to come up with your final answer. (3m + n)(m2 − n) = 3m3 − 3mn + m2n − n2.
What is the value of |5 − a| if a = 9?
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Solution
The absolute value of a number is its distance from zero, regardless of whether it is positive or negative.
Therefore, the value of |5 − 9|=|− 4| = 4.