Download Answer Explanations for: SAT January 2011 Section 5

Answer Explanations for: SAT January 2011
Section 5: Mathematics
1) C) By substitution, (x + y) = 8. Substitute 8 for z and for (x + y): x + y + z = 8 + 8 = 16.
2) A) Simply look at the circles. Only point A is within both circle X and circle Y. Point A is
also not within circle Z, so A is the correct answer.
3) B) Let s be the length of the shorter piece of rope, and set up an equation. 26 = 2s + 4
22 = 2s s = 11. See equation building.
4) D) If angles A and C are equal, then the triangle is isosceles, and AB and BC are equal, so
AB = 7. Note that the length of AC is unnecessary information. See plane geometry.
5) A) This equation only works for positive integer values of x and y when x = 1 and y = 3.
At first it appears that this question is unsolvable, since it asks you to solve a single
equation with two variables. The key to solving the equation is the fact that x and y
must both be positive integers. If y is anything greater than 3, any positive integer value
of x results in a value of 3x + 5y that is greater than 18, since when y = 4, 5y = 20.
Smaller values of y do not work either. If y = 2, 5y = 10. Because the difference
between 18 and 10 is 8, 3x would have to equal 8, which is impossible for an integer
value of x, since 8 is not a multiple of 3. Similarly, if y = 1, 5y = 5. Because the
difference between 18 and 5 is 13, 3x would have to equal 13, which is impossible for an
integer value of x, since 13 is not a multiple of 3. One other way of solving this problem
is to plug in answer choices for x and see which one results in a positive integer value for
y.
6) A) The brackets on the right side of the equation indicate an absolute value. The
absolute value of a number is equal to the distance of that number from zero. In other
words, the absolute value of a number is the “positive version” of that number. For
example, 8 = 8 and  8 = 8. Therefore, the only x-value that works for the equation in
the question is -1, since  1 = 1. See absolute values.
7) D) Referring to the graph, add the number of males in each of the three companies to
find the number of males in the new company: 50 + 75 + 75 = 200 males. Add the
number of females in each of the three companies to find the number of females in the
new company: 75 + 125 + 50 = 250 females. Therefore, the ratio of males to females in
the new company is 200:250, which reduces to 4:5 by dividing both sides by 50. If you
chose E, you answered the ratio of females to males rather than the ratio of males to
females. Be careful on ratio problems not to answer the ratio of b to a when the
problem asks you for the ratio of a to b. See ratios and proportions.
8) A) When the SAT asks you to solve for something like ab, there is a very good chance
that you will solve directly for ab as though it is a single variable without ever finding the
values of a or b individually. Therefore, your strategy on this problem should be to
manipulate the equation so as to isolate ab. Cross multiply the original equation to get
2a2b = 6a. Divide both sides by a to get 2ab = 6. Divide both sides by 2 to get ab = 3.
9) E) Per means divided by. Therefore, to find caffeine content per ounce, you must divide
caffeine content by ounces. For coffee, you have 60 mg/6 oz = 10 mg/oz. For tea, you
have 70 mg/8 oz = 8.75 mg/oz. For cola, you have 45 mg/12 oz = 3.75 mg/oz.
Therefore, E places the beverages in the correct order. If you answered A, you may
have done the calculations correctly but ordered the beverages from greatest to least
rather than from least to greatest. If you answered C, you may have ordered the
beverages in terms of caffeine content rather than caffeine content per ounce. If you
did not know how to do this question, answer choice E would have been an excellent
guess, as it is probably the common sense answer to anyone who drinks coffee, tea, and
cola.
10) E) Substitute 2 for y and 5 for x in the equation and solve for k. 2 = 5 + 2k -3 = 2k k =
-3/2. Therefore, the original equation can be rewritten as y = x + 2(-3/2), which
simplifies to y = x – 3. Plug 5 into this equation for y and solve for x to find the answer.
5 = x – 3 x = 8.
11) D) When the pipes are laid end to end, as in the top diagram, their total length is 6 • 5 =
30 inches. In order for the span of the pipes in the bottom diagram to be 22 inches,
there must be a total of 30 – 22 = 8 inches of overlapping pipes. Because the diagram
depicts four overlapping regions, if each overlapping region is equal in length, then each
overlapping region has a length of 8/4 = 2 inches.
12) B) The best way of doing this problem is graphically. Begin by adding 15 to both sides of
the inequality to get x2 + 8x + 15 0. On your graphing calculator, graph y = x2 + 8x +
15. Because you are trying to find the values at which x 2 + 8x + 15 0, you are looking
for the x-values at which the graph is at or below the x-axis, which are the x-values
where -5 x -3. B is the correct answer because -4 is the only answer choice that is
between -5 and -3. Another way to do this problem is simply to plug answer choices
into the initial inequality for x and see which one makes the inequality true. See
quadratics and inequalities and the number line.
13) C) To calculate how long it takes machine A to produce 32 chairs, set up a proportion.
To make sure you set it up right, make sure the units line up on both sides. In other
words, this/that = this/that. Here, you have (chairs)/(minutes) = (chairs)/(minutes); it is
wise to write the proportion in terms of minutes rather than hours since the question
asks for the answer in minutes. 6 chairs/60 minutes = 32 chairs/x minutes 6x = 1920
x = 320 minutes. To calculate how long it takes machine B to produce 32 chairs, set up a
similar proportion. 8 chairs/60 minutes = 32 chairs/x minutes 8x = 1920 x = 240
minutes. If machine A takes 320 minutes to produce 32 chairs and machine B takes 240
minutes to produce 32 chairs, then it takes machine A 320 – 240 = 80 minutes longer
than machine B. See ratios and proportions.
14) D) Since the angles of a triangle add to 180˚, you can write the following two equations,
in which l is the largest angle and s is the sum of the other two angles: l = 2s l + s =
180˚. Because l = 2s, you can substitute 2s for l in the second equation to get 2s + s =
180 3s = 180˚ s = 60˚. Plugging 60˚ back into the first equation for s, you get l = 2 •
60˚ = 120˚. Perhaps an easier way of approaching this problem is simply to solve it
intuitively without setting up any equations. If the large angle is double the sum of the
other two angles, then the large angle accounts for 2/3 of the total 180˚ in the triangle,
so the large angle is equal to 2/3 • 180˚ = 120˚. See plane geometry, equation building,
and linear systems.
15) D) .56m is just greater than .5m, which is equal to 1/2m or m/2. Therefore, you know
that D is pretty close to .56m. Observe the other answer choices, and if you feel that
any of them could potentially be closer to .56m than m/2 is, convert the fraction to a
decimal to compare. For instance, you could convert C to a decimal by entering 1/5 into
your calculator to get .2m, since m/5 is equal to (1/5)m. .2m is not closer to .56m than
m/2, so C cannot be the answer. It turns out that m/2 is in fact the closest of the
answer choices to .56m, so D is the correct answer.
16) C) The ratio of the length of an arc to the circumference of the entire circle is equal to
the ratio of the measure of the central angle to 360˚. The circumference of a circle is
equal to 2π times its radius, so the circumference of this circle is equal to 2π • 6 = 12π.
Because the central angle that intercepts arc ACB is 90˚, it is 1/4 of the total 360˚ of the
circle. Therefore, the length of arc ACB must be 1/4 of the total circumference of 12π,
so the length of arc ACB is 1/4 • 12π = 3π. Alternatively, you can set up a proportion.
x/12π = 90˚/360˚. Multiply both sides by 12π to get x = 3π. See circles.
17) E) The two diagonals of a square are the same length, and they are perpendicular
bisectors of each other, meaning that they intersect each other at the midpoint of both
segments and form four right angles at their intersection. The x-coordinate of the
midpoint of PR is found by averaging the x coordinates of P and R: (0 + 3)/2 = 1.5. The y
coordinate of this midpoint is 2, since PR is a horizontal line segment at y = 2.
Therefore, the midpoint of PR is (1.5, 2), so the other diagonal must intersect with PR at
(1.5, 2). Because PR is a horizontal line segment, the other diagonal must be a vertical
line segment since the two diagonals are perpendicular, so the x-coordinate of every
point along the other diagonal must be 1.5. To find the y-coordinate of the endpoints of
this diagonal, remember that it must be the same length as PR, which is 3 – 0 = 3.
Therefore, it must extend 1.5 units above and below PR, so the two endpoints are (1.5,
2 + 1.5) = (1.5, 3.5) and (1.5, 2 – 1.5) = (1.5, .5). See plane geometry and coordinate
geometry.
18) E) g(x) and f(x) refer to the y-values of the functions g and f, respectively. Therefore,
function g always has a y-value that is one greater than that of function f, the function
depicted in the graph. Because f has a y-intercept of 2, g must have a y-intercept of 2 +
1 = 3. See function notation and coordinate geometry.
19) B) Note that this sequence is neither arithmetic, which involves the addition of a
common difference, nor geometric, which involves multiplication by a common ratio.
Therefore, you cannot use any of the formulas you may have learned for arithmetic or
geometric sequences. The key to this problem lies in the second way the question
defines the nth term of the sequence: “1/n – 1/(n + 1).” Based on this definition, when
finding a sum of this sequence, the 1/n will always cancel with the 1/(n + 1) from the
previous term, since (n + 1) on the previous term is equal to n on the current term.
Therefore, all that will be left over when finding a sum of the sequence is the 1/n when
n = 1, which is equal to 1/1 = 1, and the -1/(n + 1) for the last term. In this case, the last
term is n = 50, so -1/(n + 1) = -1(50 + 1) = -1/51. If all that is left from the sum of the first
50 terms is 1 and -1/51, the sum of these 50 terms is equal to 1 – 1/51 = 50/51. See
series and sequences.
20) E) If AD = 1 and ABCD is a square, then BC = 1 and AB = 1. Since AB = 1 and E is the
midpoint of AB, EB = 1/2. Since EB = 1/2 and BC = 1, you can calculate the length of CE
using the Pythagorean Theorem: (1/2)2 + 12 = CE2 1/4 + 1 = CE2 5/4 = CE2 CE =
= √ /2. Because FE and CE are equal, FE = √ /2 as well. Since FE = √ /2 and EB =
√
1/2, FB = √ /2 – 1/2 = (√ – 1)/2. Since FB and BH are two sides of a square, they are
the same length, so BH = (√ – 1)/2. Because the problem does not specify that the
figure is not drawn to scale, you can assume that it is in fact drawn to scale. If you did
not see how to solve it geometrically, you could have taken a good guess by estimating
the length of BH visually given that AD = 1 and comparing your estimate to the decimal
equivalents of the answer choices, found by plugging them into your calculator. See
plane geometry and exponents and radicals.