math courts alg

(1) Jasmine wishes to purchase some trading cards. She has $7.50 and the

cards each cost $0.85, tax included. What is the most number of cards she can buy?

(2) Solve for x : 2x =(116

)1

(3) Express as a mixed number:

0.64 30.008 + 4

4

20.25+

5

16 1

2

(4) A thief stole 57of Brendons money and spent 5

7of the amount stolen. The

thief was then caught, and the remaining money was returned to Brendon. The remaining

amount was $40 less than the amount Brendon had after being robbed. How many dollars

did Brendon have before the theft?

(5) What is the maximum value that can be attained from the following

expression when grouping symbols are added?

4 + 5 8 + 4 2 3

(6) A restaurant makes 7 burgers from one pound of ground beef and three

quarter pounds of filler. One pound of ground beef costs $3.50, the buns, filler and other

extras cost a combined $0.15 per burger, and the restauraunt must make a 55% profit on

each sale. How many dollars to the nearest nickel should be charged per burger?

(7) What is the maximum number of square inches in the area of a rectangle

with a perimeter of 12 inches?

(8) At a fishing tournament, you are awarded $1.00 for each fish weighing 2

pounds or over, and your are assessed $2.00 for each fish weighing less than 2 pounds.

Given that Sue caught 12 fish, but neither made nor lost money, how many of her fish

weighed less than 2 pounds?

(9) Evaluate: 1022 982

(10) The director of the school cafeteria reported that 70% of the girls and 89%

of the boys in the school ate the school lunch on Friday. This represents 80% of the

students in the school. What is the ratio of girls to boys in the school? Express your

answer as a common fraction.

(11) Simplify: 3703715+370379370373

(12) What is the sum of all two-digit numbers that have a units digit that is more

than twice the tens digit?

(13) The sum of three different positive unit fractions is 67. What is the least

number that could be the sum of the denominators of these fractions?

(14) Kesha caught 40 fish, marked them, and returned them to the pond. Later,

she randomly netted 60 fish and found 4 marked fish in this group. What is the expected

number of fish in the pond?

(15) Evaluate: 10250

103

(16) A US size 712shoe is equivalent to a European size 402

3, a US size 8 equals a

European size 4113, and each time the US size increases by 1

2size, the European size

increases by 23size. Shaquille ONeal wears a US size 23. What size would he need if he

bought European shoes?

(17) What is the sum of all positive odd multiples of 3 that are less than 100?

(18) The side length of a square is 16 cm long. The midpoints of each side are

joined to form a second square. The process of joining the midpoints of the sides of the

innermost square is repeated. What is the number of centimeters in the perimeter of the

fifth square?

(19) In an open book, the product of the two facing page numbers is 1190. What

is the sum of the page numbers on the two facing pages?

(20) Find the arithmetic mean of all solutions to |x 2| = 2

(21) Kieshas average for ten tests in math class this term is 92. When Mrs.

Foley ignores the lowest score, Kieshas average is 94. What was Kieshas lowest math test

score this term?

(22) A typical compact disc plays 60 minutes of music and costs $15.00. A

typical long-playing record plays 40 minutes of music. Assuming that the money spent of a

recording is paying only for the amount of music, how many dollars should a record cost?

(23) The product of two consecutive odd positive integers is 23 more than their

sum. What is the product of these two integers?

(24) Given that 2x 3y = 8 and 4x + 3y = 2, what is the product of x and y?

(25) The ratio of the sum and difference between two positive integers a and b is73. What is the greatest possible product of the two integers such that the product is less

than 1000?

(26) Two triangles have sides with unit lengths 3, 7, and 9 and 4.5, 10.5, and

13.5, respectively. What is the ratio of the number of square units in the area of the

smaller triangle to the number of square units in the area of the larger triangle?

(27) The formula F = 95C + 32 is used to convert temperatures from degrees

Celsius (C) to degrees Fahrenheit (F). What is the number of degrees in the Fahrenheit

equivalent to 20C?

(28) The gasoline-oil mixture recommended for a motorized weed trimmer is 39 :

1. What percent of each gallon of fuel mixture is oil? Express your answer as a decimal to

the nearest tenth of a percent.

(29) How many integers n satisfy the inequality 8pi n 10pi?

(30) If f (x) = 4x+13

what is the value of [f 1(1)]1?

Copyright MATHCOUNTS Inc. All rights reserved

Answer Sheet

Number Answer Problem ID

1 8 C51A

2 4 005B

3 3 2/3 DC2C

4 490 dollars 22B11

5 318 D5122

6 1.45 0C111

7 9 square inches 5C341

8 4 3A011

9 800 D12C

10 9/10 DA3D

11 8 55DD

12 410 C24D

13 47 B5531

14 600 3B3D

15 10 01111

16 61 1/3 C3DD

17 867 C5C5

18 16 CD331

19 69 23A11

20 2 AA1A

21 74 A3021

22 10 CD1A

23 35 C2A11

24 2 22B2225 810 45DD

26 4/9 00421

27 68 F 151B

28 2.5 153D

29 57 CB011

30 2 ACD01


Solutions

(1) 8 ID: [C51A]

The cost of n cards is (0.85)n dollars. Jasmine can buy n cards only if (0.85)n 7.5.Rewriting this inequality in terms of fractions, we have

17

20n 15

2.

Multiplying both sides by 2017

gives

n 15017

,

and converting to mixed numbers gives

n 81417

.

Since Jasmine must buy a whole number of trading cards, the largest number she can

afford is 8 .

(2) 4 ID: [005B]

First, we notice that 16 = 24, so we have

2x =

(1

24

)1

= (24)1 = 2(4)(1) = 24,

so x = 4 .

(3) 3 2/3 ID: [DC2C]

We can take care of the roots of decimals by writing the decimals as fractions:

0.64 =

64

100=

64100

=8

10=

4

5,

30.008 = 3

81000

=3

2

3

103= 2

10= 1

5,

4

4

20.25= 4

4

81/4=

4

4 4

81=

4

16

81=

4

24

34=

2

3.

We also have 516 1

2= 516 2 = 5

24 2 = 5

25 = 2, so

0.64 30.008 + 4

4

20.25+

5

16 1

2=

4

5(15

)+

2

3+ 2 = 3

2

3.

(4) 490 dollars ID: [22B11]

No solution is available at this time.

(5) 318 ID: [D5122]


(6) 1.45 ID: [0C111]


(7) 9 square inches ID: [5C341]


(8) 4 ID: [3A011]


(9) 800 ID: [D12C]

This factors as a difference of squares into (102 98)(102 + 98) = 4 200 = 800 .

(10) 9/10 ID: [DA3D]


(11) 8 ID: [55DD]


(12) 410 ID: [C24D]


(13) 47 ID: [B5531]


(14) 600 ID: [3B3D]


(15) 10 ID: [01111]

We have 102 = 1102

and 1103

= 103, so

10250

103=

10350

102= 103250 = (10)(1) = 10 .

(16) 61 1/3 ID: [C3DD]


(17) 867 ID: [C5C5]


(18) 16 ID: [CD331]


(19) 69 ID: [23A11]


(20) 2 ID: [AA1A]


(21) 74 ID: [A3021]


(22) 10 ID: [CD1A]


(23) 35 ID: [C2A11]


(24) 2 ID: [22B22]Adding the two equations gives 6x = 6, so x = 1. Substituting this into the first equation

gives 2 3y = 8. Solving for y gives y = 2, so xy = 2 .

(25) 810 ID: [45DD]


(26) 4/9 ID: [00421]


(27) 68 F ID: [151B]


(28) 2.5 ID: [153D]


(29) 57 ID: [CB011]

The number pi is between 3.14 and 3.15, so 8pi is between 8(3.15) = 25.2 and8(3.14) = 25.12. Likewise, 10pi is between 31.4 and 31.5. This suffices to establish thatthe integers n between 8pi and 10pi are precisely

25,24,23,22, . . . , 28, 29, 30, 31.There are 25 negative integers in this list, 31 positive integers, and one more integer (0),

making 57 integers in total.

(30) 2 ID: [ACD01]

Substituting f 1(x) into our expression for f , we find

f (f 1(x)) =4f 1(x) + 1

3.

Since f (f 1(x)) = x for all x in the domain of f 1, we have

x =4f 1(x) + 1

3.

Solving for f 1(x), we obtain f 1(x) = 3x14

. In particular, f 1(1) = 3114

= 1/2, so

[f 1(1)]1 = 2 .


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math courts alg