Download Answer Explanations for: ACT April 2012, Form 70G Mathematics

Answer Explanations for: ACT April 2012,
Form 70G
Mathematics
1) B) Subtract 16 from both sides to get 3x/2 = -6. Multiply both sides by 2/3 to get x = -4.
2) J) Probability can be defined as (number of desired outcomes)/(number of possible
outcomes). Here, there are 100 possible outcomes and 36 + 22 = 58 possible outcomes.
Therefore, the probability of not selecting a red jelly bean is 58/100 = 29/50. If you
answered H, you likely missed the “NOT,” since 21/50 is the probability of selecting a
red. See counting and probability.
3) C) Add 2.4x to both sides to get 6x + 7.99 = 1.03. Then subtract 7.99 from both sides to
get 6x = -6.96. Divide both sides by 6 to get x = -1.16.
4) G) If  YCB = 81˚, then  BCA = 180˚ – 81˚ = 99˚, since the two angles form a linear pair
and are therefore supplementary. Based on the same reasoning, if  BAX = 81˚, then 
BAC = 180˚ – 150˚ = 30˚. If  BAC = 30˚ and  BCA = 99˚, then  B = 180˚ – 30˚ – 99˚ =
51˚ since the angles of a triangle must add to 180˚. If you did not know how to do this
problem, you could have taken a good guess by estimating the angle visually.
Remember that despite what is stated in the directions, all geometric figures are drawn
to scale. See plane geometry.
5) E) To solve a quadratic, set it equal to zero by getting all the terms on the left side of the
equation. In this case, you have x2 – 6x – 16 = 0. Then solve the equation by graphing it
on your calculator and finding the x-intercepts, using the quadratic formula, or factoring
it. To factor, find two numbers that add to -6 and multiply to -16: -8 and 2. Therefore,
you have (x – 8)(x + 2) = 0, so x = 8 and x = -2, so E is the correct answer. If you
answered A or B, you may have factored the quadratic correctly but forgotten that the
solutions to a quadratic have the opposite signs of its factors. See quadratics.
6) G) The two sets of bars that enclose the expressions are absolute value bars. To solve
an absolute value, perform all the calculations within the absolute value. Then, if the
result is positive, keep it as is, but if it is negative, make it positive. Do not make the
common mistake of making the individual numbers inside the absolute value negative –
do all the calculations within each absolute value first, and then make the result positive
if it is not positive already. Here, the expression in the question simplifies to 2 –  4 =
2 – 4 = -2. You also could have done this problem on your calculator. On your TI
calculator, go to MATH num abs. Put whatever is enclosed in the absolute value
bars inside the parentheses after the “abs,” and hit enter to solve. See absolute values.
7) C) First, decrease the price by 20%. Remember that the most efficient way to decrease
something by 20% is to multiply it by .8, since when you take 20% off, the new price is
80% of the original price. $70 • .8 = $56. Now increase this price by 6% by multiplying it
by 1.06 (the 1 represents the original $56 and the .06 represents the 6% being added to
it). $56 • 1.06 = $59.36. If you answered D, you likely tried to combine the 20%
reduction and the 6% increase into a single reduction of 14%. This method does not
work; you must always do multi-step percentage problems in multiple steps. See
percentages.
8) K) Plug in -3 for x and 2 for y and evaluate the expression. Be careful with the negatives.
(-3)2 • 23 – (-3)•22 + (-3) = 9•8 – (-3)•4 – 3 = 72 + 12 – 3 = 81. If you answered F, you
mistakenly calculated (-3)2 as -9 rather than 9. This may be because you entered it on
your calculator and forgot to put parentheses around the -3 before squaring it. If you
answered H, you mistakenly subtracted 12 for the middle term instead of adding 12.
See exponents and radicals.
9) C) First, determine how many inches the car travels in 2 minutes. Because there are 60
seconds in a minute, there are 120 seconds in 2 minutes. Therefore, you can set up a
proportion. When doing so, make sure your units line up on both sides. In other words,
this/that = this/that. In this case, (inches)/(seconds) = (inches)/(seconds), so 11 inches/5
seconds = x inches/120 seconds. Cross multiply to get 11 • 120 = 5x 1320/5 = x x =
264 inches. Now that you know the car travels 264 inches in 120 seconds, or two
minutes, you must convert this value to feet by dividing by 12, since 12 inches make up
a foot. 264/12 = 22 feet. See ratios and proportions.
10) H) Since a full case contains 24 cans, the 1/2 full case contains 1/2 • 24 = 12 cans. The
2/3 full case contains 2/3 • 24 = 16 cans. Each of the two 5/6 full cases contain 5/6 • 24
= 20 cans, for a total of 20 •2 = 40 cans between the two cases. Add these three
numbers together to get a total of 12 + 16 + 40 = 68 cans. If you answered F, you may
have misread the problem and thought there was only one 5/6 full case.
11) A) Any point that is 3 units from (6, 4) could be an adjacent vertex to the vertex at (6, 4).
The most obvious points that are 3 units from (6, 4) will share either the x or the y
coordinate with (6, 4) and have the other coordinate 3 units greater than or less than
that of (6, 4). A is the correct answer because it shares the y coordinate with (6,4) and
has an x coordinate that is 3 greater than that of (6, 4), so it is a distance of 3 units from
(6, 4). See coordinate geometry.
12) K) Add the two quantities of vanilla extract together to find the total amount needed for
one cake: 1/8 + 3/4 = 7/8 teaspoon. Then multiply this by 2 to get the amount needed
for two cakes: 7/8 • 2 = 7/4 = 1¾. If you can perform these operations in your head, do
so. If not, do them on your TI calculator and hit MATH
a fraction.
frac to convert your answer to
13) C) To find a midpoint, average the x coordinates of the two endpoints to get the xcoordinate of the midpoint: (1 + 7)/2 = 4. Then average the y coordinates of the two
endpoints to get the y-coordinate of the midpoint: (9 + -3)/2 = 3. Therefore, the
midpoint is (4, 3). See coordinate geometry.
14) G) F proves the two lines are parallel based on corresponding angles. H and J prove the
two lines are parallel based on alternating exterior angles. K proves the two lines are
parallel based on alternating interior angles. G would not prove the two lines are
parallel. Indeed, lines m and n appear to be parallel in the diagram, and  1 and  7 do
not at all appear to be congruent. See plane geometry.
15) C) The area of a triangle is equal to ½bh where b is the base and h is the height, which
by definition is measured at a right angle to the base. On a right triangle, such as the
one in this problem, the two legs of the triangle can be considered the base and the
height, respectively. Therefore, the area of this triangle is equal to ½ (x + 4)(2x + 2). To
simply this expression, FOIL the two expressions in the parentheses and then multiply
every term by ½. To expand via FOIL, multiply the First terms of each, the Outer terms
of each, the Inner terms of each, and the Last terms of each. (x + 4)(2x + 2) = x • 2x + x •
2 + 4 • 2x + 4 • 2 = 2x2 + 2x + 8x + 8 = 2x2 + 10x + 8. Now multiply each term by ½ to get
x2 + 5x + 4. An easier way of doing this problem would be to first multiply ½ by (2x + 2)
to get (x + 1) and multiply this term by (x + 4), but this way is slightly trickier to see. See
quadratics and plane geometry.
16) K) When given problems in function notation, substitute whatever value is inside the
parentheses for every x in the equation. Therefore, f(-3) = (-3)2 – (-3) + 1 = 9 + 3 + 1 =
13. If you answered G, you mistakenly calculated (-3)2 as -9 rather than 9. This may be
because you entered it on your calculator and forgot to put parentheses around the -3
before squaring it. If you answered H, you mistakenly subtracted 3 for the middle term
instead of adding 3. See function notation.
17) D) The area of a rectangle is equal to lw, where l is its length and w is its width.
Therefore, the area of this rectangle is equal to 25 • 16 = 400. The area of a square is
equal to s2 where s is the length of one of its sides, so s2 = 400 s = √
= 20 feet. See
plane geometry.
18) G) The point on the curve that lines up with September 2005 on the x-axis lines up with
a value of 20 boats sold on the y-axis. If you answered H, you looked at the point
representing the actual number sold in September 2005 rather than the value on the
model curve.
19) D) When calculating this average, be sure you use the actual numbers sold rather than
values from the model curve. Based on the graph, in January 2006, 6 boats were sold.
In February 2006, 15 boats were sold. In March 2006, 20 boats were sold. In April 2006,
30 boats were sold. Add the number of boats sold in each month and divide by the
number of months (4) to get the average number of boats sold per month. (6 + 15 + 20
+ 30)/4 = 71/4 = 17 . See averages.
20) F) In July 2005, 33 boats were sold at $30,000 each, for a total revenue of 33 • $30,000 =
$990,000. A sales tax rate of 6% would produce a total sales tax of .06 • $990,000 =
$59,400. In July 2006, 30 boats were sold at $30,000 each, for a total revenue of 30 •
$30,000 = $90,000. A sales tax rate of 7% would produce a total sales tax of .07 •
$900,000 = $63,000. Therefore, the total sales tax from July 2006 was $63,000 –
$59,400 = $3,600 greater than the total sales tax from July 2005. See percentages.
21) A) Remember that you can simplify like terms between the numerator and the
denominator of a fraction by subtracting the exponent of the denominator from the
exponent of the numerator. In other words, xa/xb = xa – b. Therefore, A is the correct
answer. If you chose B, you forgot to subtract 1 from the exponent of c 4. The c in the
denominator can be thought of as c1, so it is necessary to subtract 1 from the exponent
of c4 when simplifying. If you choose C, you incorrectly attempted to combine the
exponents of different variables, which is never mathematically valid. If you chose D,
you added the exponents instead of subtracting them. See exponents and radicals.
22) K) If her heart beats 70 times per minute, then it beats 70 • 60 = 4200 times per hour,
since there are 60 minutes in an hour. Therefore, it beats 4200 • 24 = 100,800 times in
24 hours. To put this number into scientific notation, you must move the decimal 5
places to the left, so you must multiply 1.008 by 105. You can also put this number into
scientific notation on your TI calculator by putting the calculator into scientific mode,
typing the number, and hitting enter. Note that the answer will appear as 1.008 E 5.
23) D) Add the three numbers in the ratio to find a number that represents the perimeter: 5
+ 6 + 9 = 20. Therefore, the longest side of the triangle accounts for 9/20 of the
triangle’s total perimeter. Now you can set up a proportion. When doing so, make sure
your units line up on both sides. In other words, this/that = this/that. In this case, (long
side)/(perimeter) = (long side)/(perimeter), so 9/20 = x/100. x = (9/20) • 100 = 45. See
ratios and proportions.
24) H) Draw a right triangle and label one of the non-right angles A. Since sin =
opposite/hypotenuse, label the side opposite  A 20 and the hypotenuse 25. Using the
Pythagorean Theorem, find the side adjacent to  A. 252 – 202 = x2 x2 = 225 x = 15.
You can also found the missing side without using the Pythagorean Theorem if you
recognize that you have a 3-4-5 Pythagorean Triple. Either way, because the side
adjacent to  A is 15 and side opposite  A is 20, tan A = 20/15, since tan =
opposite/adjacent. You could also do this problem on your calculator by taking the
inverse sin (sin-1) of  A to find the measure of  A. Then use your calculator to find the
tan of this angle. See basic trigonometry.
25) E) You could do this problem by trial and error, checking the different answer choices
using the Pythagorean Theorem until you get one for which a2 + b2 = c2. However, this
problem can be done much more quickly if you recognize that answer choice E is a 3-4-5
Pythagorean Triple and can therefore work as the side lengths of a right triangle. See
right triangles.
26) G) To find the slope of this graph, use algebra to isolate the y, thereby putting the
equation into slope-intercept (y = mx + b) form, in which m represents the slope. 3y = 2x + 1 y = -2/3x + 1/3. Therefore, the graph has a slope of -2/3. See coordinate
geometry.
27) C) Simply combine like terms. Begin with the x2. Since 2x2 is the only x2 term, there
must be a 2x2 in the answer, so D and E can be eliminated. Then calculate the x term. x
+ 2x + 3x – (2x + 2x + 2x) = x + 2x + 3x – 6x = 0. Therefore, there should be no x term,
which eliminates A and B, leaving C as the only possible answer. You did not even need
to worry about adding the constants in order to figure out the answer.
28) K) 6 inches is equal to .5 feet, since there are 12 inches in a foot. Since the volume of a
rectangular solid is equal to the surface area of its base multiplied by its height, the
volume of this patio is equal to 270 • .5 = 135 cubic feet.
29) C) Since the area of the patio is 270 square feet, as stated in the text above question 28,
substitute 270 for A in the equation to calculate C. C = 3.5(270) + 120 = 945 + 120 =
$1065.
30) J) Since the text above question 28 tells you that the length of the patio is 18 feet but
does not tell you the width, you must calculate the width in order to answer this
question. Since the area of a rectangle is equal to length time width and the patio has
an area of 270 square feet, you can calculate the width of the patio as follows: 18w =
270 w = 15 feet. Since the perimeter of the patio not against the house consists of
two lengths of the patio and one width of the patio, this portion of the perimeter is
equal to 2(18) + 15 = 51 feet. If you answered K, you solved for the entire perimeter of
the patio. See plane geometry.
31) C) Before solving, convert the 8 miles to feet so that both distances are in the same
units. Since 5280 feet are in a mile, 8 miles is equal to 8(5280) feet. (It is unnecessary to
actually calculate this number, since it is not calculated in any of the answer choices.)
Relative to the angle of ascent, you are given the opposite side and the hypotenuse.
Therefore, you can set up an equation using sin, since sin = opposite/hypotenuse.
Letting x be the angle of ascent, sin x = 10,000/(8(5280)). Arcsin (also called inverse sin
or sin-1) is the inverse function of sin, so you can isolate x by taking the arcsin of both
sides: x = arcsin 10,000/(8(5280)). If you answered A, you forgot to convert the 8 miles
to feet. See basic trigonometry.
32) F) Enter the three numbers involving fractions or roots into your calculator to get
decimal equivalents in order to more easily put the numbers in order. √
1.73, 1
1.83, and 5/3 1.67. Therefore, F places the numbers in the correct order from
greatest to least.
33) E) In a geometric sequence, each term is multiplied by the same number, known as the
common ratio, to get the next term. In this sequence, the common ratio is equal to -2,
since each term is multiplied by -2 to get the next term. Therefore, the fifth term is
equal to -8 • -2 = 16, the sixth term is equal to 16 • -2 = -32, and the seventh term is
equal to -32 • -2 = 64. See series and sequences.
34) F) Draw the triangle if you are having trouble visualizing it. F could be true because MO
and OP are the two legs of the triangle if PM is the hypotenuse, and there is no reason
the two legs cannot be congruent. Indeed, if they were congruent, it would be a 45-4590 right triangle. G could not be true because the hypotenuse MP (which is the same as
PM) cannot be congruent to one of the legs. H cannot be true because the hypotenuse
of the triangle is PM, so  O must be the right angle, since the right angle is always
opposite the hypotenuse. J and K cannot be true because a right triangle can be neither
equilateral nor equiangular. See right triangles.
35) C) Set up a proportion to find the perimeter of DEF. When doing so, make sure your
units line up on both sides. In other words, this/that = this/that. In this case,
(ABC)/(DEF) = (ABC)/(DEF), so 2/3 = 20/x. If it is not obvious at this point that x = 30,
cross multiply to get 2x = 60 x = 30. See ratios and proportions.
36) G) 40% of net sales would be found by multiplying the net sales, S, by 40/100 (which is
equivalent to .4), so every answer except for G can be eliminated. G also correctly
subtracts the costs, C, from this value. F and H incorrectly take 40% of the costs, K
multiplies the sales by 40 rather than 40%, and J combines both of these errors. See
percentages and equation building.
37) B) Begin by setting up a trig equation featuring three parts: an angle and two sides. You
are given the side adjacent to the 35˚ angle, and you are trying to find the side opposite
the 35˚ angle, so you should set up a trig equation in terms of tangent, which is equal to
opposite/adjacent. tan35˚ = x/450 x = 450tan35˚. See basic trigonometry.
38) G) Use algebra to isolate x. (5 + 3x)/2 > 3
inequalities and the number line.
5 + 3x > 6
3x > 1
x > 1/3. See
39) E) Since lance contributes 1.5 times as much as Sophie, he contributes $1.5 to every $1
she contributes. Therefore, he contributes 1.5/(1.5 +1) = 1.5/2.5 of the total amount
that they contribute. Now that you have found the ratio of part to whole, you can set
up a proportion. When doing so, make sure your units line up on both sides. In other
words, this/that = this/that. In this case, (Lance’s portion)/(total contribution) = (Lance’s
portion)/(total contribution), so 1.5/2.5 = x/2500. At this point, it may be intuitive to
you that x = 1500. If not, multiply both sides of the proportion by 2500 to get x = 1500.
This $1500 represents Lance’s contribution in a single year, so multiply this amount by 4
to calculate his 4-year contribution: $1500 • 4 = $6000. See ratios and proportions.
40) J) The answer choices to this problem provide a big hint that you want to factor the
denominators of both fractions in order to determine the least common denominator.
In factored form, the initial equation can be written as
+
. The least
–
–
common denominator of two fractions can be found by breaking the denominators of
both fractions into their prime factorization, and multiplying all the prime factors
together but only counting the prime factors that appear in both denominators once. In
this case, (x – 2) appears in both denominators, so it only has to be counted once.
Therefore, the least common denominator is equal to 4(x – 2)(x + 2). If you answered K,
you simply multiplied the denominators of both, forgetting that it was unnecessary to
count the (x – 2) twice. Although K would be a common denominator, it would not be
the least common denominator.
41) C) The area of a circle is equal to πr2. Since the diameter of a circle is equal to twice its
radius, the radius of this circle is equal to 3. Therefore the area of this circle is equal to
π • 32 = 9π. If you answered E, you used the diameter instead of the radius in the area
formula. If you answered B, you found the circle’s circumference instead of its area.
See circles.
42) F) Solve this problem by using the slope formula to find the slope. Slope = rise/run =
(change in y)/(change in x) = (y2 – y1)/(x2 – x1). Plugging the two points into this
equation, you get (-2 – -5)/(10 – -2) = 3/12 = 1/4. Plug this slope in for m in y = mx + b
(slope intercept form, in which m is the slope and b is the y-intercept), and plug in the x
and y coordinates from one of the two points into the equation for x and y, respectively:
-2 = (1/4)(10) + b. Solve algebraically for b. -2 = 2.5 + b b = -2.5 = -9/2. Now that you
know both m and b, you can write the equation of the line in y-intercept form: y = 1/4x
– 9/2. See coordinate geometry.
43) E) The 53 people who used both treadmills and bikes are counted among the 117 who
used treadmills and among the 89 who used bikes. Therefore, if you add the 117 who
used treadmills to the 89 who used bikes, you would be counting the 53 who used both
twice, so it is necessary to subtract 53 from this sum. Therefore, 117 + 89 – 53 = 153
people used treadmills, bikes, or both.
44) H) Stating that something is divisible by a number and stating that it is a multiple of a
number mean the same thing. Therefore, the number in the question is a multiple of
both 7 and 3. Since the least common multiple of 7 and 3 is 21, the number must also
be a multiple of 21. The largest two digit multiple of 21 is equal to 21 • 4 = 84. Once
you figured out that you needed a number that was a multiple of both 7 and 3, you also
could have solved the problem simply by guess and check, starting with the largest twodigit number offered among the answer choices and working your way down until you
found one that was a multiple of both 7 and 3. See number properties.
45) A) Perpendicular lines have negative reciprocal slopes. For instance, a line with a slope
of 3/4 would be perpendicular to a line with a slope of -4/3, and a line with a slope of -2
would be perpendicular to a line with a slope of 1/2. Slope is equal to m in y = mx + b
(slope-intercept) form, so the slope of the line in the question is 1/4. Therefore, any line
perpendicular to this line must have a slope of -4, so A is the correct answer. D and E
are both parallel to the line in the question, since parallel lines have the same slopes.
See coordinate geometry.
46) J) Find the total north-south distance and the total east-west distance between the
helicopter and the emergency, and then use the Pythagorean Theorem to find the
straight-line distance between the two points. The helicopter is 2 + 22 = 24 miles north
and 4 + 6 = 10 miles east of the emergency. If you notice that you have a 5-12-13
Pythagorean Triple, you could immediately recognize that the distance between the two
points is 26 miles. If you did not recognize this triple, you could use the Pythagorean
Theorem to find this distance: 242 + 102 = x2 x2 = 676 x = 26. See coordinate
geometry.
47) D) Based on statement 1, x = 2, since 2 is the only even prime number. Based on
statement 3, z = 16, since 16 Is the only perfect square between 10 and 20. Therefore,
the expression in the question is equal to 16y/2, or 8y. Based on statement 2, y must
equal either 7 or 8, so the expression could equal 8 • 7 = 56 or 8 • 8 = 64, so the answer
is D. See number properties.
48) G) The shaded region involves y values that are greater than or equal to the y value of
the quadratic at any x value. Since the equation in the answer choices with the x2 term
is the quadratic, F, H, and J can be eliminated because they feature the incorrect
inequality sign for the quadratic. To decide between G and K, you must recognize that,
although they both get the direction of the inequality sign correct for the linear
function, K features the incorrect y-intercept (b in y = mx + b form) of -3, while G
features the correct y-intercept of 3. Furthermore, K features the wrong constant for
the quadratic, which needs a constant of -3 since it is shifted down 3 units from the x
axis. See coordinate geometry.
49) A) If you subtract both sides of the bottom equation from both sides of the top
equation, the x’s cancel, and you are left with 2y = -a – b. Therefore, y = (-a – b)/2 = -(a
+ b)/2. See linear systems.
50) G) The ratio of the length of an arc to the circumference of a circle is equal to the ratio
of the measure of the central angle that intercepts that arc to 360˚. Since the
circumference of a circle is equal to 2πr, you can set up the following proportion:
45/360 = π/2πr. This proportion simplifies to 1/8 = 1/2r 2r = 8 r = 4. See circles.
51) A) Substitute 3 for x in the first equation to get 3a = b. Substitute 8 for x in the other
equation to get 8a – 2 = b. If 3a and 8a – 2 are both equal to b, then we can set the two
expressions equal to each other. 8a – 2 = 3a 5a = 2 a = 2/5. See linear systems.
52) J) If a quadratic has one real solution, then it is a perfect square, so it is of the form x 2 +
2ax + a2 = 0. In this case, 2a = -6 so a = -3. Therefore, k = (-3)2 = 9. A less algebraic way
of thinking about it is that in perfect squares, the c term is equal to half the b term
squared (assuming the quadratic is written as ax2 + bx + c = 0). Another way you could
solve this problem is by graphing it on your calculator, trying different answer choices as
possible values of k until you get one that produces a graph that has exactly one xintercept, since each x-intercept represents a real solution. See quadratics.
53) D) A complete revolution is equivalent to 2π radians. Therefore, 3π/2 radians is equal to
3/4 of a complete revolution, since
= 3/4. Since the wheelbarrow wheel rotated ¾
of a complete revolution, it travelled a distance equal to 3/4 of its circumference. See
circles and advanced trigonometry.
54) K) The domain of a function represents all possible x-values for that function. In this
problem, the one factor that limits the domain is the possibility of the denominator
being equal to zero, since it is mathematically impossible to divide by zero. X values that
make the denominator equal zero are outside the domain of this function. These x
values would be the x values where the absolute value of x is equal to 100, so x = 100
and x = -100 are both outside the domain of this function. Another way of doing this
problem is to plug the answer choices into the equation for x and calculate f(x) using
your calculator. For values outside the domain of the function, the calculator will give
you an error.
55) A) The location of E on line segment BC is irrelevant and is simply included to confuse
you. The area of a rectangle is base times height, and the area of a triangle is 1/2 base
times height. Since the triangle and the rectangle share the same base (AD) and the
same height, the triangle must have an area 1/2 that of the rectangle. See plane
geometry.
56) K) It is a bit confusing based on the diagram, but the slant height (which has a length of
3√ ) is perpendicular line drawn from the middle of the edge of the base to the top of
the pyramid. In other words, it is not the edge of the pyramid that is labeled 3√ , but
rather the altitude of the triangle face that forms the side of the pyramid. Because
these triangles are equilateral, you know that each angle has a measure of 60˚.
Therefore, the two right triangles formed when the altitude to this triangle is drawn (as
shown in the diagram) are 30-60-90 triangles, which have side lengths of the ratio x-x√
-2x. Since the x√ side is labeled as 3√ , you know that the other leg must be 3 and the
hypotenuse must be 2 • 3 = 6. Therefore, each of the 8 edges of the pyramid must be 6
units in length, so the sum of these edges is 6 • 8 = 48. See right triangles.
57) D) The distance from any point on a circle to the center of the circle is equal to the
circle’s radius. Therefore, for a point to be a possible center point for the circle in the
question, it must be 5 units from both (-7, 0) and (1, 0). Use the Pythagorean Theorem
(or the distance formula, or standard form for a circle, since all three are really the same
thing) to see which of the options could work. (-3, 3) is 5 units from (-7, 0) because -3
and -7 are 4 units apart, and 3 and 0 are 3 units apart. You do not even need to use
they Pythagorean Theorem, since you have a 3-4-5 Pythagorean Triple. (-3, 3) is also 5
units from (1, 0) because -3 and 1 are 4 units apart, and 3 and 0 are 3 units apart, again
forming a 3-4-5 Pythagorean Triple. Because it is 5 units from both points on the circle, I
could be the center of the circle. III works as a possible center of the circle as well,
based on the same reasoning. Since -3 is 4 units from both -7 and 1 and 3 is 3 units from
0, (-3, 3) is 5 units from both (-7, 0) and (1, 0), as you again have 3-4-5 Pythagorean
Triples. II does not work since it is exactly 4 units from both (-7, 0) and (1, 0), since it
shares the same y-coordinate of both these points and has an x-coordinate that is 4
units from the x-coordinates of both points. Therefore, D is the correct answer. Had
you not recognized the 3-4-5 triangles in this problem, you could have used the
Pythagorean Theorem or distance formula to find the distances between the points, but
doing so is unnecessarily time consuming. See circles, right triangles, and coordinate
geometry.
58) J) First, you must determine how many even numbers there are between 2 and 50,
inclusive. If it is not obvious that there are 25, count that there are 5 even numbers
from 2 through 10. Therefore, there must also be 5 even numbers from 12 through 20,
from 22 through 30, from 32 through 40, and from 42 through 50, a total of 5 • 5 = 25
even numbers. Now you must determine how many of these 25 even numbers have a
units digit that is double the tens digit. (Many students are unfamiliar with the term
“units digit;” the units digit is also known as the ones digit, the right-most digit of any
integer.) Begin with a tens digit of 1 and create a number that meets this criteria by
doubling the tens digit to create the units digit. Do this for every tens digit less than or
equal to 4, since anything greater would be greater than 50. This method yields the
following numbers: 12, 24, 36, and 48. Therefore, 4 out of the 25 even numbers meet
the conditions of the problem, which is equal to 4/25 = .16 = 16%. If you answered H,
you may have found the percent of all the numbers from 2 through 50 that met the
conditions rather than just the percent of the even numbers. See number properties.
59) A) Because g(x) is shifted down 4 units from f(x) you must have a -4 outside the
parentheses, since f(x) – 4 represents a shift downward of 4 units from f(x). Therefore,
you can eliminate B and C. Because g(x) is shifted to the right 2 units from f(x), you must
have a -2 in parentheses with the x, since f(x – 2) represents a shift to the right of 2 units
from f(x). Therefore, you can eliminate E. The correct answer is A instead of D because
g(x) is less steep (specifically, half as steep) as f(x), and the multiplier of 1/2 is needed to
make the graph half as steep. In other words 1/2(f(x)) is always half as steep (has half
the slope) of f(x). See function notation.
60) K) To get this problem correct, it helps to understand a bit of basic logic. If you have an
original statement “If A, then B,” then its inverse is “If B, then A,” its converse is “If not
A, then not B,” and its contrapositive is “If not B, then not A.” If the original statement is
true, then its contrapositive is necessarily true, but you cannot draw any conclusions
about the truth or falsehood of the inverse or converse. Therefore, K is the correct
answer, since it represents the contrapositive of the original statement. F represents
the converse of the original statement, and J represents the inverse of the original
statement. Even if you have not studied logic, you can get this problem right just by
reasoning through it. F, G, and J are not necessarily true because it is possible that the
parade is cancelled for some reason other than rain, such as a high winds. The original
statement does not imply that rain the only thing that could cause the parade to be
cancelled. H is incorrect because it simply gets things backwards; the fact that it is not
raining is no reason the parade would be cancelled. If you think carefully about K, it is
apparent that it has the exact same meaning as the original statement.