Algebra - Level 2

(1) If the ratio of 2x � y to x + y is 2 to 3, what is the ratio of x to y? Express

your answer as a common fraction.

(2) If x

y= 2 and z

x= 4, what is the value of z

y?

(3) The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of

the coordinates of the midpoint?

(4) In the sequence 16, 80, 48, 64, A, B, C, D, each term beyond the second

term is the arithmetic mean (average) of the previous two terms. What is the value of D?

(5) What isp200 to the nearest tenth?

(6) Solve for n: 2n � 4n = 64n�36.

(7) What is the product of the coordinates of the midpoint of a line segment

with endpoints at (2; 3) and (�6; 5)?

(8) Evaluate: (22)3.

(9) What percent of 5x is equal to x=5?

(10) While watching a parade I saw some clowns and horses. I counted 30 legs

and 10 heads. How many horses did I see in the parade?

(11) Find the minimum value of the function f (x) = 3(x2 � 1) + 2.

(12) What is the value ofp1;000;000� 3

p1;000;000?

(13) A ball bounces back up 23of the height from which it falls. If the ball is

dropped from a height of 243 cm, after how many bounces does the ball �rst rise less than

30 cm?

(14) Jamie bought new jeans for $10 because the price marked was 75% o� the

original price. What was the original price?

(15) A line goes through point A(9; 1), point B(19; k) and point C(7; 0). What is

the value of k?

(16) How many square feet are in three square yards?

(17) If f (x) = 8x3 � 6x2 � 4x + 5, �nd the value of f (�2).

(18) The positive di�erence between two consecutive perfect squares is 35. What

is the greater of the two squares?

(19) In the arithmetic sequence 17; a; b; c; 41, what is the value of b?

(20) In this 2 by 2 grid of squares, the total length of all 12 of the line segments

is 12 units. In a similar 40 by 40 grid of squares that are the same size, what is the total

length of all of the line segments?

(21) Simplify: 9m7p12

12m5p15Express your answer as a common fraction with positive

exponents.

(22) Paul shared his baseball cards with three friends. He gave half his cards to

one friend, one-third of the cards that were left to his second friend, and the remaining 12

cards to a third friend. How many cards did Paul have at the beginning?

(23) Eighty percent of the students in a class (group A) share 40% of the candy

equally. The remaining 20% of the students (group B) share the other 60% of the candy

equally. The ratio of the amount of candy a student in group A has to the amount of

candy a student in group B has is equal to what common fraction?

(24) Mr. Sanchez's students were asked to add two positive integers. Juan

subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the

correct answer?

(25) If x2 = y � 3 and x = �5, what is the value of y?

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Answer Sheet

Number Answer Problem ID

1 5/4 14112

2 8 1DBC

3 14 DC53

4 59 AD22

5 14.1 511D1

6 72 2253

7 -8 A43D

8 64 C02B

9 4 percent CB5

10 5 horses 3C02

11 -1 C3241

12 900 B2C2

13 6 bounces 3D42

14 40 dollars 255

15 6 41BC

16 27 square feet B212

17 -75 CCB22

18 324 5BC

19 29 DA42

20 3280 units 3B31

21 3m2

4p35012

22 36 cards 25051

23 1/6 A02

24 22 01D3

25 28 2201

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Solutions

(1) 5/4 ID: [14112]

It is given that 2x�yx+y

= 23. Multiplying both sides by 3(x + y), we obtain

3(2x � y) = 2(x + y)

) 6x � 3y = 2x + 2y

) 4x = 5y

) 4 � xy= 5

) x

y=

5

4:

Thus, the ratio of x to y is5

4.

(2) 8 ID: [1DBC]z

y= z

x� xy= 4 � 2 = 8 .

(3) 14 ID: [DC53]

The midpoint of a line segment with endpoints (x1; y1); (x2; y2) is(x1+x22

; y1+y22

).

Thus, the midpoint of this line segment is(2+82; 3+15

2

), which simpli�es to (5; 9). Thus,

the sum of the coordinates of the midpoint is 14 .

(4) 59 ID: [AD22]

We have

A =64 + 48

2= 56

B =64 + 56

2= 60

C =60 + 56

2= 58

D =58 + 60

2= 59

So the answer is 59 .

(5) 14.1 ID: [511D1]

We know that 142 = 196, which is very close to 200. We now try to �nd 14:12. We know

that 14:1 = 14 + 0:1, so we have that

14:12 = (14 + 0:1)2 = 142 + 2(0:1)(14) + 0:12 = 196 + 2:8 + 0:01 = 198:81. Since 14:12

is roughly about 2:8 greater than 142, then 14:22 must be at least 2:8 greater than 14:12.

Therefore, our estimate for 14:22 is approximately 198:8 + 2:8 = 201:6. However, our

estimate for 14:12 is closer to 200 than that for 14:22, sop200 is approximately equal to

14:1 .

(6) 72 ID: [2253]

Since 4 = 22, 4n = 22n. Since 64 = 26, 64n�36 = 26(n�36). Thus,

2n+2n = 26(n�36) ) 3n = 6n � 216

So 3n = 216) n = 72 .

(7) -8 ID: [A43D]

Since the midpoint of a segment has coordinates that are the average of the endpoints', we

see that the midpoint has coordinates(2�62; 3+5

2

)= (�2; 4). Thus our desired answer is

�2 � 4 = �8 .

(8) 64 ID: [C02B]

We have (22)3 = 22�3 = 26 = 64 .

(9) 4 percent ID: [CB5]

We can see thatx=5

5x=

1

25:

This is equal to 4 percent.

(10) 5 horses ID: [3C02]

Let the number of clowns in the parade be c and the number of horses be h. We are

looking for the value of h. Assuming that each clown has 2 legs and 1 head, and that each

horse has 4 legs and 1 head, we can set up the following system of equations:

2c + 4h = 30

c + h = 10

To solve for h, we need to eliminate c from the equations above. We can rewrite the

second equation above as c = 10� h, and substituting this into the �rst equation to

eliminate c gives 2(10� h) + 4h = 30, or h = 5. Thus, there are 5 horses in the parade.

(11) -1 ID: [C3241]

Since x2 � 0, we have f (x) = 3x2 � 3 + 2 = 3x2 � 1 � �1, with equality when x = 0, so

�1 is the minimum value.

(12) 900 ID: [B2C2]

We havep1;000;000� 3

p1;000;000 =

p106 � 3

p106 = (106)

1

2 � (106)1

3

= 106�1

2 � 106�1

3 = 103 � 102 = 1000� 100 = 900 :

(13) 6 bounces ID: [3D42]

If you call the number of bounces b, then this problem can be phrased as: what is the

minimum b, such that 243 �(23

)by . We can write a system of equations to represent the

information given in the problem:

x � y = 2

x � y = 120

Solving for x in the �rst equation yields x = y + 2. Substituting into the second gives

(y + 2) � y = 120, or y 2 + 2y � 120 = 0. This quadratic equation factors into

(y + 12)(y � 10) = 0, so y = 10. Given y , we can solve for x to get x = 12, so

x + y = 22 .

(14) 28 ID: [2201]

Substituting �5 for x , we can rewrite our given equation as: (�5)2 = y � 3, so we have

25 = y � 3. Adding three to each side, we �nd y = 28 .

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