SECTION 5
1. C Translate the words into math: 7 × n = 84, and we want to know the value of 4n.
7n = 84, so n = 12, and 4n = 48, answer C.
2. E Out of the initial 16 gallons, 4 are removed, so 12 gallons remain. The fraction can
be reduced to , which is 75%, answer E.
3. D We can use PITA for this question. First use C, 0.33, for t. If we must add t to each of
the four values, we will have 4t. 4 × 0.33 = 1.32. When 1.32 is added to the sum in
the question of 2.74, the total is 4.06, which is less than the 4.22 we were looking for.
Eliminate A, B, and C and try a larger number. In D, 0.37 × 4 = 1.48, and 1.48 + 2.74
= 4.22, just what we were looking for.
4. B Because the triangle is equilateral, and each of the three sides is cut in half, all of the
short segments in the triangle are equal. This means that all four triangles within the
larger triangle are equilateral and congruent. Since one of the four triangles is
shaded, the shaded area is of the area of the larger triangle. × 24 = 6, answer B.
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5. E Only s is equal to a number without a variable, so plug that into the equations in the
question. The value of s is negative. The variable r is positive, because r = − , and
when − takes the place of s, the negative signs cancel out: and r = 4.
Because r is positive, q is negative: q = − becomes q = − . Finally, because q is
negative, p is positive: p = − , then and p = 4.
6. C First find the value of x: 27 divided by 3 is 7, with 2 left over, so x = 2. 2x = 4 so
divide 23 by 4: 4 goes into 23 evenly 5 times and there are 3 left over, so the answer
is C, 3.
7. A This is a great Plugging In question. Make d = 2, so the first 10 hours cost $2.00,
and e = 3, so service costs $3.00 an hour. Pick a number above 10 for h to make
your math easier: If h = 20, it can be split into two parts: The first 10 hours cost
$2.00, and the second 10 hours cost 10 × $3.00 = $30.00. Altogether that’s $32.00,
the number we’re looking for in the answers. Plug in the values for d, e, and h. The
expression in A becomes 2 + 3(20 – 10) = 2 + 3(10) = 32. A works, but always check
all the answers just in case: B gives 602, C gives 80, D gives –480, and E gives 640. A
is correct.
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, 8. E In the graph, y is always positive. E is correct because absolute value is always
positive. One way to answer this question is to plug in points from the graph into
the equations in the answers. Using point (1, 2), A works because 2 = 2(1). Eliminate
B because 2 does not equal 11; also, because of x2, this is a parabola, not a line.
Although the point works in C, 2x2 indicates a parabola, so eliminate C. Eliminate D
because 2 does not equal 1 – 2. In E 2 = 2 does work. Now try the other point from
the graph just on equations A and E. A is out because 2 does not equal –2.
9. 256 To solve this equation, get by itself. = 16, so square both sides: ( )2 = 162,
so x = 256.
10. 1 or 9 List the numbers described: All the odd integers from 0 to 10 are 1, 3, 5, 7, 9. Of
these numbers, 3, 5, and 7 are prime, leaving only 1 or 9.
11. 2 Because the two triangles have the same area and each have a base with length 5,
their heights must be equal as well. For a triangle, Area = bh. The bottom triangle
has a height of 5 because the base is on the x-axis and the tip is at –5. This means
that t = 7 – 5 = 2.
12. 52, 59, or 66
One way to solve this problem is to plug in numbers for j until you get a result that
is between 50 and 70. Another way is to list the multiples of 7 that are close to the
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range given, such as 49, 56, and 63, and add 3 to each one, which gives 52, 59, and
66 respectively. Any one of these three numbers will get you the credit for the
question. You need to come up with only one possible value.
13. 148 Use Fred’s Theorem: A line crossing two parallel lines creates big angles and small
angles. The big angle that matches y is split by a line perpendicular to d and e. The
big angle is 58 + 90 = 148, which is also the value for y. Another way to solve this is
to find the third angle of the triangle: 180 – 90 – 58 = 32. The 32° angle and the y°
angle make up a straight line, so 180 – 32 = 148.
14. 15 We need to find the cost of a single share of stock x. Of the nine people, 5 bought a
share each, so if cost = c, then that is 5c. There are also 3 people who each bought
of a share, so that’s 3( c). There is one more person, who bought of a share.
Since the club spent $114, we can create the equation .
This simplifies to . Multiply both sides by to
get c by itself, and c = 114( ) = 15.
15. 4 First simplify the equation 4x • n2 = 4x+1 • n, to 4x • n = 4x+1, then plug in. If x = 2,
then 42 • n = 42+1. Since 16n = 43, then 16n = 64 and n = 4.
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1. C Translate the words into math: 7 × n = 84, and we want to know the value of 4n.
7n = 84, so n = 12, and 4n = 48, answer C.
2. E Out of the initial 16 gallons, 4 are removed, so 12 gallons remain. The fraction can
be reduced to , which is 75%, answer E.
3. D We can use PITA for this question. First use C, 0.33, for t. If we must add t to each of
the four values, we will have 4t. 4 × 0.33 = 1.32. When 1.32 is added to the sum in
the question of 2.74, the total is 4.06, which is less than the 4.22 we were looking for.
Eliminate A, B, and C and try a larger number. In D, 0.37 × 4 = 1.48, and 1.48 + 2.74
= 4.22, just what we were looking for.
4. B Because the triangle is equilateral, and each of the three sides is cut in half, all of the
short segments in the triangle are equal. This means that all four triangles within the
larger triangle are equilateral and congruent. Since one of the four triangles is
shaded, the shaded area is of the area of the larger triangle. × 24 = 6, answer B.
www.cracksat.net
5. E Only s is equal to a number without a variable, so plug that into the equations in the
question. The value of s is negative. The variable r is positive, because r = − , and
when − takes the place of s, the negative signs cancel out: and r = 4.
Because r is positive, q is negative: q = − becomes q = − . Finally, because q is
negative, p is positive: p = − , then and p = 4.
6. C First find the value of x: 27 divided by 3 is 7, with 2 left over, so x = 2. 2x = 4 so
divide 23 by 4: 4 goes into 23 evenly 5 times and there are 3 left over, so the answer
is C, 3.
7. A This is a great Plugging In question. Make d = 2, so the first 10 hours cost $2.00,
and e = 3, so service costs $3.00 an hour. Pick a number above 10 for h to make
your math easier: If h = 20, it can be split into two parts: The first 10 hours cost
$2.00, and the second 10 hours cost 10 × $3.00 = $30.00. Altogether that’s $32.00,
the number we’re looking for in the answers. Plug in the values for d, e, and h. The
expression in A becomes 2 + 3(20 – 10) = 2 + 3(10) = 32. A works, but always check
all the answers just in case: B gives 602, C gives 80, D gives –480, and E gives 640. A
is correct.
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, 8. E In the graph, y is always positive. E is correct because absolute value is always
positive. One way to answer this question is to plug in points from the graph into
the equations in the answers. Using point (1, 2), A works because 2 = 2(1). Eliminate
B because 2 does not equal 11; also, because of x2, this is a parabola, not a line.
Although the point works in C, 2x2 indicates a parabola, so eliminate C. Eliminate D
because 2 does not equal 1 – 2. In E 2 = 2 does work. Now try the other point from
the graph just on equations A and E. A is out because 2 does not equal –2.
9. 256 To solve this equation, get by itself. = 16, so square both sides: ( )2 = 162,
so x = 256.
10. 1 or 9 List the numbers described: All the odd integers from 0 to 10 are 1, 3, 5, 7, 9. Of
these numbers, 3, 5, and 7 are prime, leaving only 1 or 9.
11. 2 Because the two triangles have the same area and each have a base with length 5,
their heights must be equal as well. For a triangle, Area = bh. The bottom triangle
has a height of 5 because the base is on the x-axis and the tip is at –5. This means
that t = 7 – 5 = 2.
12. 52, 59, or 66
One way to solve this problem is to plug in numbers for j until you get a result that
is between 50 and 70. Another way is to list the multiples of 7 that are close to the
www.cracksat.net
range given, such as 49, 56, and 63, and add 3 to each one, which gives 52, 59, and
66 respectively. Any one of these three numbers will get you the credit for the
question. You need to come up with only one possible value.
13. 148 Use Fred’s Theorem: A line crossing two parallel lines creates big angles and small
angles. The big angle that matches y is split by a line perpendicular to d and e. The
big angle is 58 + 90 = 148, which is also the value for y. Another way to solve this is
to find the third angle of the triangle: 180 – 90 – 58 = 32. The 32° angle and the y°
angle make up a straight line, so 180 – 32 = 148.
14. 15 We need to find the cost of a single share of stock x. Of the nine people, 5 bought a
share each, so if cost = c, then that is 5c. There are also 3 people who each bought
of a share, so that’s 3( c). There is one more person, who bought of a share.
Since the club spent $114, we can create the equation .
This simplifies to . Multiply both sides by to
get c by itself, and c = 114( ) = 15.
15. 4 First simplify the equation 4x • n2 = 4x+1 • n, to 4x • n = 4x+1, then plug in. If x = 2,
then 42 • n = 42+1. Since 16n = 43, then 16n = 64 and n = 4.
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