Praxis I Math Review By: Professor Peter Eley

Question 1 Text answers to A scientist has a very sophisticated microscope and laser cutting tool. She begins with a sample that is inches long and cuts precisely three one-thousandths of an inch off the end. What is the size of the remaining piece? = inches = inches = 0.23 inches = inches = inches

Explanation This may seem complex, but it is a simple subtraction problem if you construct it properly. Convert three one- thousandths of an inch into a number form. 3 / 1000 = Then subtract the size of the cut piece from the whole sample inches inches = inches

Question 2 Text answers to A man flipped a coin four times and got "heads" each time. What is the probability that he will get "heads" on the fifth time he flips the coin? = 20% = 25% = 50% = 80% = More than 50% but less than 80%

Explanation This can be a tricky question. Each time you flip a coin there is a 50/50 or 50% chance of getting either "heads" or "tails." It does not matter how many times you flip the coin, the result of each flip has its own probability of 50%. What makes this question tricky is that people seem to think that if the coin lands on "heads" or "tails" many times in a row that somehow it is more likely to do the opposite on the next flip. This is not true, it is always 50%.

Question 3 Text answers to What is the value of x in this triangle? = = = = √ = √ 2

Explanation This triangle is a special kind of right triangle called a triangle. That means that the angles of it are 30, 60 and 90 degrees. This also means that the dimension of the sides are special as well. They follow the one-two-square root of three rule. This means that sides will be proportional to each other where the shortest side equals n times one, the hypotenuse equals n times two and the remaining side equals n times the square root of three. In the question, the shortest side is 1. Therefore, the shortest side can be thought of as being n times 1. Since the shortest side is 1, n must equal 1. Let’s check on the hypotenuse. n * 2 = 1 * 2 = 2. Now we have what we need for the answer. Since n = 1 then x = n * √ 3 = 1 * √ 3 = √ 3

Question 4 Text answers to Solve for x: x/3 * 4/7 = 12/ = = = = = 7

Explanation The numerators and denominators are multiplied separately in order to get the product, so in this problem: x/3 * 4/7 = 12/21 Can be thought of as two equations: Numerator: x * 4 = 12 and Denominator: 3 * 7 = 21 When expressed like this it is easy to see that x = 3.

Question 5 Text answers to What is the value of the angle x? = = = = = None of the above

Explanation If you add the internal angles of any triangle, no matter what shape, they always equal 180 degrees. We know from the square in the corner that this triangle is a right triangle, so one angle is 90°. Another angle of the triangle is given as 50°. Adding 90° + 50° yields 140°. The angle that remains must be 180° - 140° or 40°.

Question 6 Text answers to In a shuffled, standard deck of playing cards, what is the probability of drawing a red ace? = 1/ = ¼ = 1/ = 1/ = 2%

Explanation Probability of drawing a red ace = the number of red aces divided by the total number of cards. Since there are 2 red aces in a standard playing deck of 52 cards, the chance of drawing a red ace is 2 in 52 or 1 in 26. Therefore, 1/26.

Question 7 Text answers to If 4x 2 = y which of the following is true? = x = = x = √ (y/4) = x = (√y) / = x = √y * 4 = Both and

Explanation Solve the equation for x: 4x 2 = y Divide each side by 4: x 2 = (y/4) Then take the square root of both sides. x = √ (y/4) or Answer B. However, that equation can also be expressed as: x = (√y ) / (√4) OR x = (√y) / 2 which is Answer C. Since Answer E states that both B and C are correct, this is the best choice.

Question 8 Text answers to The ant wants to get back to his home, but there are several paths he can follow, as shown in the diagram. Whenever he reaches a junction, he is equally likely to choose any one path as he is to choose any other. He cannot turn back. What is the probability he reaches home? =1/ = 1/ = ¼ = 1/ =2/3

Explanation To reach home, he either follows route ABEGH or route ACEGH. P( He follows route ABEGH) = ½ × ½ × ½ = 1/8 P( He follows route ACEGH) = ½ × ½ × ½ = 1/8 Therefore, the probability he reaches home = 1/8 + 1/8 = 1/4.

Question 9 Text answers to From the above diagram, what is the value of a/x? = 2/ = 9/ = 4/ =3/ = 2/1

Explanation The sides x and a both face a 75° angle. Of the remaining sides in the two triangles, the two longer sides are in the ratio 24/16 = 3/2 and the two shorter sides are in the ratio 18/12 = 3/2. Therefore, triangles ABC and XYZ are similar with ratio 3/2. Therefore, a/x = 3/2 also.

Question 10 Text answers to Jordan's teacher asked her to solve the equation x/3 + 3/x = -2. This was Jordan's answer: x/3 + 3/x = -2 ⇒ (x + 3)/(3 + x) = -2 ⇒ x + 3 = -2(3 + x) ⇒ x + 3 = -6 -2x ⇒ 3x = -9 ⇒ x = -3 Jordan has got the right answer, but she made some mistakes. How many mistakes did Jordan make? = = = = = 4

Explanation Jordan made two mistakes: 1. In the first line she said x/3 + 3/x = (x + 3)/(3 + x). Fractions are not added in this way. It is not generally true that a/b + c/d = (a + c)/(b + d). Certainly in this case it is wrong because (unless x = -3) the left- hand side would simplify to 1 and 1 ≠ -2. If x = -3, then the left-hand side would be 0/0 and 0/0 is undefined. 2. In the second line she multiplied both sides by (3 + x) which would be alright except for the case where x = -3. In that case both sides would be multiplied by 0. And, of course, x does equal -3. After making two mistakes, Jordan managed to get the right answer!

Question 11 Text answers to The table above shows the x and y coordinates of a straight line. What is the equation of that line? = x = 5*y = y = x = y = 6*x = y = 5*x = y =5*x + 11

Explanation Recall that the equation of a straight line is y = mx + b. When x = 1, y = 11 and when x = 2, y = 16 Slope is rise over run. Y "rises" 5 while x "runs" 1 so slope is 5. Take any two points from the table and solve for b. y = 16, m = 5, x = 2 16 = 5 * 2 + b b = 6 so y = 5*x + 6

Question 12 Text answers to What percent of 80 is 20? = 20% = 25% = ¼% = 80% = 50%

Explanation First, convert the phrase into an equation where "of" is a multiplication sign and "is" is an equal sign. Use a variable (in this case x) to describe the number you are solving. x% * 80 = 20 Solve for x by dividing both sides by 80 x% = 20 / 80 x% = 1 / 4 x% = 0.25 The percent sign means we need to multiply by 100. x = 25%

Question 13 Text answers to Which of the following is equal to a quarter of a million? = 40, = 250, = 1/(4,000,000) = = 500,000

Explanation Since one million is 1,000,000, quarter of a million is ¼ * 1,000,000 or 250,000

Question 14 Text answers to Which of the following fractions is least? = 11/ = 99/ = 25/ = 3/ = 501/500

Explanation Of the five fractions given, for are greater than 1; that is the numerators are greater than the denominators. Only one of the fractions, 99/100, is less than 1, so it must be the least.

Question 15 Text answers to Which of the following sales commission is greatest? = 1% of $1, = 10% of $ = 12.5% of $ = 15% of $ = 25% of $40

Explanation The five sales commissions are as follows: 1 st choice: 1% of $1,000 is $10. 2 nd choice 10% of $200 is $20. 3 rd choice 12.5% of $100 is $ th choice 15% of $100 is $15. 5 th choice 25% of $40 is $10. You can rule out certain choices by comparing them to the other similar choices.

Question 16 Text answers to If P/5 = Q, then P/10= = 10Q = 2Q = Q/ = Q/ = Q/20

Explanation Several ways to solve this problem exists. p/5 = q Then p=5q, and so P/10 = 5q/10 = q/2

Question 17 Text answers to C is the center of the circle. Which of the following must be true? = QC and RC have the same length =QR and RC have the same length =QC is perpendicular to QR =QR is perpendicular to RC =ΔQRC is equilateral

Explanation Since “c” is the center of the circle, “QC” and “RC” are both radii of the circle and therefore have the same length.

Question 18 Text answers to In the Venn diagram above, black circle represent the integers 2 to 10, inclusive, and the blue circle represents 6 to 12, inclusive. How many integers are represented by the intersection? = one, =two, =three, =four, =five, =six

Explanation In the Venn diagram, the intersection region represents the integers that are in both the Black and Blue circles that is the integers that are among the integers 6 through 12, or 6, 7, 8, 9, and 10, which amount to five integers.