Mathematics: Section 4
Question Number: 21 - 22 - 23 - 24 - 25 - 26 - 27 - 28 - 29 - 30 - 31 - 32 - 33 - 34 - 35 - 36 - 37 - 38
21.
Choice (A) is correct. Since 2x + 4 = 10, it follows that 2(2x + 4) = 2(10), or 4x + 8 = 20. You are trying to find the value of 4x – 20, so subtract 28 from both sides of the equation yielding 4x + 8 – 28 = 20 – 28, so 4x – 20 = –8.
You can also find the answer by solving the original equation for x. Since 2x + 4 = 10, it follows that 2x = 6, and x = 3. Substituting 3 for x in the expression 4x – 20 yields 4(3) – 20 = –8.
22.
Choice (E) is correct. Since sides 
and 
are equal in length, the angles opposite these sides have equal measure. Therefore, the measure of 
BAD is equal to y°. Since x = 25, it follows that 25 + 90 + y = 180, or y = 65.
23.
Choice (D) is correct. The consecutive integers u, v, w, x, and y can be written as u, u + 1, u + 2, u + 3, and u + 4, respectively. Since u + y = 10, it follows that u + (u + 4) = 10, so u = 3. The value of w is u + 2, which is 3 + 2, or 5.
24.
Choice (D) is correct. Of the cards in the box, 
are marked r, are marked s, and are marked t. Therefore 
are either marked r or marked s, so the probability of drawing a card marked either r or s is .
25.
Choice (A) is correct. The temperatures in the table, given in degrees Celsius, are to be compared with 125° Fahrenheit. This comparison will be easier if you use the formula to convert 125°F to degrees Celsius. Using the formula, 125 – 32 = 
C. It follows that 93 = C; therefore C = 93
= 51. There are four continents with temperatures above 51° Celsius. Therefore, four of the continents have had temperatures above 125° Fahrenheit.
26.
Choice (B) is correct. Segment is the hypotenuse of right triangle OBC, whose legs each have length 2, so = 2
. The perimeter of the square is 4 times the length of , so it equals 4 × 2, which is 8.
27.
Choice (B) is correct. Since f(c) = 4, you know by the definition of f that 
= 4. Solving for c gives 2c = 16, so c = 8.
28.
Choice (C) is correct. In this question, you are given three statements numbered I, II, and III and asked which statement or statements must be true. You need to consider each statement separately. The statements are about the numbers a, b, and c, which are the degree measures of the angles of a triangle. Since a, b, and c are angle measures of a triangle, each one is between 0 and 180, their sum is 180, and their average is 
, which is 60.
You are also given that a > b > c. The largest number, a, must be above the average, so a > 60; and the smallest number, c, must be below the average, so c < 60. Statements I and III deal with these conclusions. Statements I and III must be true.
Statement II is an inequality involving b. It is clearly possible for b to be between 45 and 90; for example, the values a = 100, b = 50, and c = 30, satisfy the conditions of the problem. But does b always lie between 45 and 90? No; for example, the values a = 105, b = 40, and c = 35 also satisfy the conditions of the problem. Since statement II is true for some values and not true for other values of a, b, and c, it is not a statement that must be true. Since the question asks which statement or statements must be true, the correct answer is I and III only.
29.
The correct answer is 6.6 or 
. It may be helpful to draw the line with points A, B, C, and D on it. The question tells you that they lie on a line in that order, so you know that B is between A and C, and that C is between B and D.
When you mark the three given lengths on your figure (as shown), you will see that BC = 8 – 3.4 = 4.6. BD is the sum of BC and CD, so it equals 4.6 + 2 = 6.6.
30.
The correct answer is 165. In the expression for target heart rate, a is the person's age in years, so when you apply the expression to a 14-year-old, you would substitute 14 for a. The heart rate is 0.8(220 – 14), which equals 0.8(206) = 164.8. Rounding this to the nearest integer gives 165.
31.
The correct answer is any number between 10 and 14. To find the possible lengths of the third side, the triangle inequality must be used. Since two of the sides have lengths 12 and 2, according to the triangle inequality the length of the third side must be less than the sum of the lengths of these two sides. In other words, the length of the third side must be less than 12 + 2 or less than 14.
If you let the length of the third side be x, then 12 must be less than the sum of x and 2, also by the triangle inequality. Therefore, 12 < x + 2, or 10 < x. Hence, 10 < x < 14. Since the question asked for one possible length of the third side, you should choose one number between 10 and 14 that will fit in the grid, and enter that number. Some answers that will be counted as correct are 11, 12.3, 13.8, and 13.
32.
The correct answer is 2500. Both 4 rods and 100 links are equal to 1 chain, so 4 rods must equal 100 links. If you multiply both quantities by 25, you get that 100 rods is equal to 2500 links.
33.
The correct answer is 
or .5. The first equation tells you that yz is equal to 10. Substituting 10 for yz in the second equation gives (2x)(10) = 5. You can solve for 2x by dividing both sides of the equation by 10 to get 2x = . (Note that it was not necessary to solve for x.)
34.
The correct answer is 4. In each trade, Paul will give 1 candy and receive 3 candies, so his number of candies will increase by 2. In each trade, Kate will give 3 candies and receive 1, so her number of candies will decrease by 2. If there are x trades, then Paul will end up with 24 + 2x candies and Kate will end up with 40 – 2x candies. If at that point they have an equal number of candies, then 24 + 2x = 40 – 2x, which gives 4x = 16, so x = 4. The number of trades that results in their having the same number of candies is 4.
35.
The correct answer is 36. Because the pentagon has all five sides congruent and all five angles congruent, 
ABC is congruent to AED. Since the measure of DAE was given as x°, the measure of CAB is also x°. Also, since triangles ABC and AED are isosceles, the measures of BCA and ADE are also x°. It is helpful if you label the figure with this information, as shown below.
Because ABC is congruent to AED, it follows that AC = AD and ACD is isosceles. Since the measure of ACD was given to equal 72°, the measure of ADC is also 72° You can also add this information to the figure, as shown.
Since two of the angles in ACD have measures of 72°, the third angle, CAD, has a measure of 36° because 72 + 72 + 36 = 180 in ACD. The five marked angles have equal measures, so x + 72 (BCD) must equal x + 36 + x (BAE). Therefore, x + 72 = x + 36 + x. Solving this gives x = 36.
36.
The correct answer is 10. Following the description of the sequence, you can extend the sequence: 4, 7, 3, 4, 1, 3, 2, 1, 1, 0,... . Note that the fifth term is found by taking the non-negative difference between the third and fourth terms (4 – 3 = 1), the sixth term is found by taking the non-negative difference between the fourth and fifth terms (4 – 1 = 3), etc. Once you have zero as a term, you should count to see what term this is. Since the 10th term is 0, the value of n is 10.
37.
The correct answer is 14. If the average of four numbers is 10, then the sum of the four numbers is 4 × 10 = 40. So q + r + s + t = 40. The average you are asked to compute is 
, which is equivalent to 
. Substituting 40 for q + r + s + t gives 
, which equals 14.
38.
The correct answer is 1111. It is useful to observe that x10 can be written as 
(xy), so (xy) = x10. Substituting 5 for and 5555 for x10 gives 5(xy) = 5555. Therefore, xy = 5555 ÷ 5 = 1111.
Another way to approach this problem is to multiply both numerator and denominator in the fraction by x, yielding 
, which is equivalent to 
. Therefore, = = . Substituting 5 for and 5555 for x10 gives 5 = 
. Therefore, xy = 1111.
