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SAT MATH PRACTICE TEST 1
MULTIPLE-CHOICE QUESTIONS (25 items)
1. The figure below, not drawn to scale, shows two identical semi-circles of
radius 20 cm and a right triangle.
Find the area of the shaded part . [ Use 14.3 ]
a.2828 cm b.
21056 cm c.2456 cm d.
2172 cm
2. The length of the bases of an isosceles trapezoid are 9 and 25. Find theperimeter if the measure of each of the lower base angles is 30.
a. 34216 c.3
102364
b. 66 d.3
102332
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3. The graphs of 032 byx and 163 22 yx are drawn on the sameset of axes. Find the value of b if the graph of the straight line passes
through the center of the circle.
a. 6 b. 6 c. 8 d. 8
4. Perform the indicated operation:
x
x
xx
x
4
2
2
22
a. 22
x
x b.
2x
xc.
22x
xd.
x
x2
2
5. If M and N are positive integers and .000,000,12852 NM What is the value
of NM ?
a. 20 b. 19 c. 18 d. 17
6. Vivien traveled by boat 5 miles upstream to fish in her favorite spot.
Because of the 4-mph current, it took her 20 minutes longer to get there
than to return. How fast will her boat go in still water?
a. mph342 b. mph23 c. mph42571
d. mph24115
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7. Eugeneis going to make an open-top box by cutting equal squares from the
four corners of an11 inch by 14 inch sheet of cardboard and folding up the
sides. If the area of the base is80 square inches, then
what is the length of one side of the square?
a. in11 b. in3
30520
c. in
3
49730
d. in
4
32925
8. Which isnota factor of mymmyymmyy 22 4213 ?
a. my b. my 6 c. my 7 d. my 14
9. A line has non zero slope equal to twice the X-intercept .The slope of the
line is also equal to one-half of the Y-intercept. What is the equation of the
line in slope- intercept form?
a. 18 xy b. 88 xy c. 84 xy d. 24 xy
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10. A chemist wants to make a 10% solution of fertilizer. How much water and
how much of 30% solution should he mix to get 30 Liter of a 10 % solution
of fertilizer?
a. 20 L of water and 10 L of 30% solution of fertilizerb. 15 L of water and 15 L of 30% solution of fertilizerc. 17 L of water and 13 L of 30% solution of fertilizerd. 12 L of water and 18 L of 30% solution of fertilizer
11. Given:a
ax1
and 33 1
aaV . Find the largest possible value of a
for which xV 4 .
a. 1 b.2
53 c.2
53 d.2
53
12. Simplify:2
22
2
2
)2(
)4(
22
)2(
ba
bab
aba
b
ba
baa
a. 1 b. ba 2 c. ba 2 d. b
13. A man engages in a shooting contest. Each time he hits the target, hereceives $2 and each time he misses, he pays $1. If after 20 shots the manhas lost $2, how many times did he hit the target?
a. 7 b. 15 c. 6 d. 5
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14. Each of the following statements is true. For which statement is the
converse also true?
a. If triangles X and Y are congruent, then X and Y have the same area.
b. If triangles X and Y are similar, then the corresponding sides of X and Yare proportional.
c. If quadrilateral S is a square, then S is a rectangle
d. If quadrilateral R is a rectangle, then the diagonals of R are equal
15. The ratio of the volumes of two similar cones is 8 to 27. The ratio of their
total surface areas is
a. 2 to 3 b. 4 to 9 c. 8 to 27 d. 22 to 33
16. Given that 4 nm and that 1222 nm , calculate the value of
2nm .
a. 7 b. 6 c. 8 d. 10
17. In the figure below, two 128 rectangles share a common corner and
overlap. What is the area of the shaded part?
a. 36 b. 45 c. 48 d. 54
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18. A volcanic crater has a flattened circular top of radius 100m. As a man
walks up the side of a volcanic crater from A to B, the height above sea
level is inversely proportional to the square of the distance from the center.
Given that 500 m from the center, the man is 200 m above sea level, findthe height of the volcanic crater.
a. 800 mb. 5000 mc. No possible answerd. The given information is not sufficient to solve the problem
19. Which regular polygon has three times as many diagonals as sides?
a. octagon b. nonagon c. decagon d. dodecagon
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23. Let x denote the length of a wooden stick. Find all possible lengths of the
stick if it is known that 753 x .
a. 4
3
2 x b. 4x c. 40 x d. 0x
24.Which of the following is the solution set of the inequality 84
434
x?
a. ]7,8( b.
4
19,
4
29c.
4
19,
4
29d. none of these
25. What term must be added to xyx 304
9 2 to make it a perfect square
trinomial?
a.281y b. 121 c.
2100y d. y64
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1 c 6 a 11 d 16 c 21 b
2 d 7 d 12 a 17 d 22 d
3 a 8 d 13 c 18 c 23 c
4 c 9 c 14 b 19 b 24 c
5 b 10 a 15 b 20 c 25 c
SOLUTION NEXT PAGE
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1. The figure below, not drawn to scale, shows two identical semi-circles of
radius 20 cm and a right triangle.
Find the area of the shaded part . [Use 14.3 ]
a.2828 cm b.
21056 cm c.2456 cm d.
2172 cm
Solut ion:
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To get the area of the shaded region, we get the difference between the areaof the circle with radius 20 cm and the area of an isosceles right triangle
whose leg measures 20 cm.
Area of circle = 2r
2
2014.3
22014.3
1256
Area of right triangle =2
21 legleg
=2
4040
800
Therefore the area of the shaded part is 1256 800=456.
The area of the shaded region is 456 cm2
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2. The length of the bases of an isosceles trapezoid are 9 and 25. Find theperimeter if the measure of each of the lower base angles is 30.
a. 34216 c. 3
102364
b. 66 d. 3
102332
Solut ion:
Total perimeter is 2593
3162
343
332
3
102332
Recall:
Theorem. In a 30O
60O
90O
triangle the sides are in the
ratio 3:2:1 .
To get the length of this, divide
8 by 3 , then multiply by 2.
The result is
3
3162
3
8
This is also
3
316
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3. The graphs of 032 byx and 163 22 yx are drawn on the sameset of axes. Find the value of b if the graph of the straight line passes
through the center of the circle.
a. 6 b. 6 c. 8 d. 8
Solut ion:
The center of the circle 163 22 yx is 0,3 .
We substitute the point 0,3 in the equation 032
byx
and solve for b.
032 byx
00332 b
6b
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4. Perform the indicated operation:
xx
xx
x
4
2
2
22
a. 22
x
x b.
2x
xc.
22x
xd.
x
x2
2
Solut ion:
x
x
xx
x
x
x
xx
x
42
2
242
2
222
x
x
x
x
42
2
2
42
2
2
x
x
x
x
2222
xx
x
x
x
22
x
x
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5. If M and N are positive integers and .000,000,12852 NM What is the value
of NM ?
a. 20 b. 19 c. 18 d. 17
Solut ion:
610128000,000,128
67 522
667 522
613 52
Therefore 6,13 NM
19 NM
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6. Vivien traveled by boat 5 miles upstream to fish in her favorite spot.
Because of the 4-mph current, it took her 20 minutes longer to get there
than to return. How fast will her boat go in still water?
a. mph342 b. mph23 c. mph4
2571d. mph24115
Solut ion:
Let x=speed of the boat in still water
So x+4=speed of the boat downstream [ because the speed of the current is 4 mph]
x-4=speed of the boat upstream
Use rdtr
td
3
1
downstreamupstreamtt [we convert 20 minutes to hours, 20 minutes = 1/3 hours]
3
1
4
5
4
5
xx, Multiply both sides by 3(x-4)(x+4)
16415415 2 xxx
1660156015 2 xxx
1362 x
mph342136 x
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7. Eugeneis going to make an open-top boxby cutting equal squares from the
four corners of an 11 inch by14 inch sheet of cardboard and folding up the
sides. If the area of the base is 80 square inches, then
what is the length of one side of the square?
a. in11 b. in3
30520
c. in
3
49730
d. in
4
32925
Solut ion:
Area of base = 80214211 xx
80214211 xx
407211 xx
037252 2 xx
Using quadratic formula
2237242525
2
x , we disregard the
negative value
in4
32925 x
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8. Which isnota factor of mymmyymmyy 22
4213 ?
a. my b. my 6 c. my 7 d. my 14
Solut ion:
There is a common binomial factor my , factor it out, the result is
22
4213 mymymy
The complete factored form of mymmyymmyy 22 4213 is
mymymy 76 .
my 14 isnota factor of mymmyymmyy 22 4213 .
mymy 76
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9. A line has non zero slope equal to twice the X-intercept .The slope of the
line is also equal to one-half of the Y-intercept. What is the equation of the
line in slope- intercept form?
a. 18 xy b. 88 xy c. 84 xy d. 24 xy
Solut ion:
bmxy , m=slope and b=y-intercept
x-intercept of y=mx+b ism
b , why? Because when y=0 then
m
bx
m
bm 2 slope equal to twice the X-intercept
2
bm slope equal half the Y-intercept
mb 2 , substitute in the first equation
m
mm
22
4
m
mb 2
8 b
So the answer is
84 xy
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10. A chemist wants to make a 10% solut ion of fertilizer. How much water and
how much of 30% solution should he mix to get 30 Liter of a 10 % solution
of fertilizer?
a. 20 L of water and 10 L of 30% solution of fertilizerb. 15 L of water and 15 L of 30% solution of fertilizerc. 17 L of water and 13 L of 30% solution of fertilizerd. 12 L of water and 18 L of 30% solution of fertilizer
Solution:
Let x = amount of water
30 x = amount of 30% solution
Equation: 301.0303.00 xx
33.09 x
63.0 x , divide both sides by 0.3
20 x , and 1030 x
The chemist needs 20 L of water and 10 L of 30% solution of fertilizer to get 30 Lof a 10% solution of fertilizer.
0% +30% = 10%
x 30-x 30
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11. Given:a
ax1
and 33 1
aaV . Find the largest possible value of a
for which xV 4 .
a. 1 b.2
53c.
2
53 d.
2
53
Solution:
xV 4
a
aa
a1
41
3
3
, factor the left side
aa
aa
aa
14
11
12
2
01
41
11
2
2
aa
aa
aa
04
11
12
2
aa
aa
0
1
a
aor
041
12
2
a
a
Recall:
2233 babababa
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First equation:
01
aa
012 a
12 a
1a
Second equation:
041
12
2
a
a
041 224 aaa
013 24 aa
013 24 aa
Let 2ay
0132 yy
Using quadratic formula
2
53y
2
532 a
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2
53a
The values of a are 1 ,2
53
, 2
53
The largest possible value of a is2
53
12. Simplify:2
22
2
2
)2(
)4(
22
)2(
ba
bab
aba
b
ba
baa
a. 1 b. ba 2 c. ba 2 d. b
Solution:
bababbaba
baa
b
ba
babaa
22
22
22
2)2(
1
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13. A man engages in a shooting contest. Each time he hits the target, hereceives $2 and each time he misses, he pays $1. If after 20 shots the manhas lost $2, how many times did he hit the target?
a. 7 b. 15 c. 6 d. 5
Solut ion:
Let x= number of times a man hits the target
After 20 shots the man receives/lost
2202 xx
2203 x
6x
14.Each of the following statements is true. For which statement is the
converse also true?
a. If triangles X and Y are congruent, then X and Y have the same area.
b. If triangles X and Y are similar, then the corresponding sides of X and
Y are proport ional.
c. If quadrilateral S is a square, then S is a rectangle
d. If quadrilateral R is a rectangle, then the diagonals of R are equal
Solut ion:
Recall:
The converse of the statement If p then q is If q then p.
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6
8
1012
4
104
a. If triangles X and Y are congruent, then X and Yhave the same area.
The converse of statement a is
If triangles X and Y have the same area then X and Y are congruent .
Counterexample:
Area= 24 Area= 24 Two triangles have the same area but are not congruent. Therefore the
converse of statement a is false.
b. If triangles X and Y are similar, then the corresponding sides of X andY are proport ional.
The converse of statement b is
If the corresponding sidesof triangles X and Y are proportional then X
and Y are similar.
Which is also true
Answer is b.
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We will show that the converse of c and d are false.
c. If quadrilateral S is a square, then S is a rectangle
The converse of statement c is
If quadrilateral S is a rectangle then S is a square.
Counterexample:
The figure above is a rectangle, but it is not a square. Therefore the
converse of statement C is false.
d. If quadrilateral R is a rectangle, then the diagonals of R are equal
The converse of statement d is
If the diagonals of quadrilateral R are congruent, then r is a rectangle
Counterexample:
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The diagonals of the quadrilateral (isosceles trapezoid) above are
congruent, but it is not a rectangle. Therefore the converse of
statement d is false.
15.The ratio of the volumes of two similar cones is 8 to 27. The ratio of their
total surface areas is
a. 2 to 3 b. 4 to 9 c. 8 to 27 d. 22 to 33
Solut ion:
27
8
2
1 V
V, therefore the ratio of the corresponding sides [circumference
of base, height, etc] is
3
2
27
83
2
1 s
s, and the ratio of their total surface areas is
94
32
2
2
1
A
A
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16. Given that 4 nm and that 1222 nm , calculate the value of
2nm .
a. 7 b. 6 c. 8 d. 10
Solut ion:
4 nm , get the square of both sides
162 22 nmnm
16222 mnnm
16212 mn
42 mn
Therefore
222 2 nmnmnm
mnnm 222
412
8
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17.In the figure below, two 128 rectangles share a common corner and
overlap. What is the area of the shaded part?
a. 36 b. 45 c. 48 d. 54
Solut ion:
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Area of shaded part = Area of 8x12 rectangle Area of the two right triangles
formed by connecting one corner of the rectangle to the intersection point
along the right edge of that rectangle
2
1122
98128
63696
54
18. A volcanic crater has a flattened circular top of radius 100m. As a man
walks up the side of a volcanic crater from A to B, the height above sea
level is inversely proportional to the square of the distance from thecenter. Given that 500 m from the center, the man is 200 m above sea
level, find the height of the volcanic crater.
a. 800 mb. 5000 mc. No possible answerd. The given information is not sufficient to solve the problem
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Solut ion:
Let levelseaaboveheighth and centerthefromdistanced
2dkh , where k is a constant of variation
When 500d , 200h
2500200
k 7105 k
To get the height of the volcanic crater we let 100d , which is the radiusof the top radius and solve for h .
So 2
7
100
105h
mh 5000
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19. Which regular polygon has three times as many diagonals as sides?
a. octagon b. nonagon c. decagon d. dodecagon
Recall:
If a polygon has n sides then the number of diagonals is equal to
2
3nn
nnn
32
3
nnn 632
092 nn
09
nn
9n disregard 0n
Answer is nonagon
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20. Find the equation of the circle with center in the first quadrant and is
tangent to the lines 7y , 9y , and 4x .
a. 6441 22 yx
b. 161422
yxc. 6414 22 yx
d. 10022 22 yx
Solut ion:
First we graph the tangent lines.
Based on the graph of the tangent lines , the y-coordinate of the center
of the circle is
12
97
,
So the radius is equal to 87119
Recall:
The equation of the
circle with center at
kh, and radius ris
222 rkyhx
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The x-coordinate of the center is 8 units to te right
of 4x or 4.
Therefore the center of the circle is 1,4 and the equation is 6414 22 yx
The graph of the circle is
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21. Simplify:
2
1
2
1
2
4
1
2
1
23
432
a. 16 b.3
32c.
3
16d.
3
8
Solut ion:
2
1
2
1
2
1
2
1
2
1
2
1
2
4
1
2
1
23
1632
23
432
2
1
2
1
2
1
2
1
33
1622
3162
3
32
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22.Which of the following statements isALWAYStrue for any real numbers
a,b,c,d?
a. If ba , then bcac
b. If ba , thenba
11
c. If ba and dc , then bdac .
d. If ba and dc , then dbca
Solut ion:
a. If ba , then bcac is only true when c is a posit ive number.
b. If ba , thenba
11
is false when a or b is zero.
c. If ba and dc , then bdac .
5 -2 -1 -7 -5 14
false
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d. If ba and dc , then dbca is always true. why? Seeproof below
ba given
cbca * add c to both sides
dc given
dbcb * * add b to both sides
dbcbca * and **
dbca
23. Let x denote the length of a wooden stick. Find all possible lengths of
the stick if it is known that 753 x .
a. 43
2 x b. 4x c. 40 x d. 0x
Solut ion:
7537753 xx
, add 5 to both sides
1232 x
, divide both sides by 3
4
3
2
x
The length of the wooden stick cannot be negative
Therefore40 x
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24.Which of the following is the solution set of the inequality 84
434
x?
a. ]7,8( b.
4
19,
4
29 c.
4
19,
4
29 d. none of these
Solut ion:
44
438
4
434
xx
and8
4
43
x
First inequality:
4
4
43
x
1643 x
194 x
4
19 x
We get the intersection of 4
19x
and 4
29x
, the result in set intervalnotation is
4
19,
4
29
Second inequality:
8
4
43
x
3243 x
294 x
4
29 x
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25.What term must be added to xyx 304
9 2 to make it a perfect square
trinomial?
a. 281y b. 121 c. 2100y d. y64
Solut ion:
We get the square root of the first term
xx2
3
4
9 2
?
2
3230
xxy
Therefore we should add 210y
or
2100yto make
xyx 304
9 2
a perfect square trinomial.
2
22 102
310030
4
9
yxyxyx
y10