math courts

(1) How many miles can a car travel in 20 minutes if it travels 34as fast as a

train going 80 miles per hour?

(2) Two boxes contain a total of 150 coins. 17 coins are moved from the first

box to the second. The first box now contains half as many coins as the second. How

many coins were originally in the first box?

(3) Find the sum of the 60 positive odd whole numbers beginning with 5.

(4) If (x, y) = (3, 9), what is y 2 − 3xy + 8?

(5) Find the value of the complex fraction below. Express your answer as a

common fraction.1

4−1

1−1

4

(6) Evaluate: ( 3√0.000027 · 3

√0.064)2

(7) A square is divided into nine smaller squares of equal area. The center

square is then divided into nine smaller squares of equal area and the pattern continues

indefinitely. What fractional part of the figure is shaded?

(8) At Pizza Perfect, Ron and Harold make pizza crusts. When they work

separately Ron finishes the job of making 100 crusts 1.2 hours before Harold finishes the

same job. When they work together they finish making 100 crusts in 1.8 hours. How many

hours, to the nearest tenth of an hour, does it take Ron working alone to make 100 crusts?

(9) If $19, 600 is divided among three people so that their shares are in the ratio

2: 3: 4, how many dollars are received by the person who gets the least amount?

(10) Max bought a new dirt bike and paid 10% down, which was $150. What

was the price of the bike?

(11) Find the difference between the mean and median in this set of data:

{4.8, 3.7, 5.2, 4.1, 4.7, 3.6, 4.8, 3.5}.

(12) If p dollars are invested at a rate of r% compounded annually, the total

value V of the investment after n years is given by V = p(1 + .1r)n. How many years does

it take a 6% investment to double its value?

(13) What is 18% of 320?

(14)√50 +

√75 lies between two consecutive integers, a and b, where a < b.

What is the value of b?

(15) Pat has $1.13 in pennies, nickels, and quarters. If she has no more than 4

coins of each type, how many coins does she have altogether?

(16) Jane’s mother asked her to drain and refill the fish pond in the backyard. It

takes five hours for the pond to empty. The hose will refill it in three hours. Unfortunately,

Jane did not close the drain when she turned on the hose. Assuming the pond fills and

drains at a constant rate, how many hours did it take to refill the pond? Express your

answer as a decimal.

(17) The graph of the equation y = |x | − 3 is translated two units to the left andthree units down. What are the coordinates of the minimum point of the new graph?

(18) Find the value of x on the number line shown, given the dots are evenly

spaced:

3

7

x 4

7

(19) The sum of three integers is 193. The smaller two are consecutive integers

and the larger two are consecutive even integers. What is the largest of these three

integers?

(20) Evaluate: (130)(130) + (131)(129).

(21) Estimate the following to a nearest ten:

0.5× 0.12× 250031× 0.09

(22) A state income tax rate is 2% of the first $12,000 of taxable income, 3% of

the next $16,000, and 6% of any amount beyond this. What percent of taxable income of

$48,000 would be paid in state tax?

(23) If 1x+ 1y= 3 and 1

x− 1y= −7 what is the value of x + y? Express your

answer as a common fraction.

(24) Find the value of (652 − 252)− 592 without using a calculator.

(25) If a half of a number, a third of the number, a fourth of the number and

two-sevenths of the number are added together, the sum is 575. What is the number?

(26) Alice leaves Denver, driving at a constant speed. After a while, she passes a

mile marker displaying a two-digit number. One hour later, she passes a second marker with

the same two digits in reverse order. In another hour she passes a third marker with the

same two digits separated by a zero. What is the rate of Alice’s car in miles per hour?

(27) Find the coordinates of the point halfway between the points (3, 7) and

(5, 1).

(28) The ratio of boys to girls in a school is 4 to 5. How many girls are in the

school if a total of 945 students are enrolled?

(29) Find the value of : 15· 37· 59· 711· . . . · 99

103· 101105

(30) A book is opened and the product of the two page numbers that appear is

3906. What is the larger of the two page numbers?

Copyright MATHCOUNTS Inc. All rights reserved

Answer Sheet

Number Answer Problem ID

1 20 miles 4A041

2 67 B5AA1

3 3,840 503B

4 8 CC2A1

5 3/8 DC412

6 0.000144 15DA1

7 1/2 513B

8 3.1 5C1C

9 4,355.56 $ 011C

10 1500 dollars D1012

11 .1 4B3A1

12 12 224A1

13 .4 0B2A1

14 16 55AA1

15 9 D3DA1

16 7.5 DAAA1

17 (−2,−6) B02C

18 1328

5CCA

19 66 C03A1

20 33,799 ACCA

21 50 1B3B

22 4 percent 5CD22

23 -3/10 A4D41

24 119 CA2A1

25 420 302B

26 45 miles per hour 045A1

27 (4,4) 04D12

28 525 3CB02

29 1/3605 4CC51

30 63 13DA1


Solutions

(1) 20 miles ID: [4A041]

The car travels at a speed of

3

4× 80 miles per hour = 3× 20 = 60 miles per hour.

In 20 minutes, the car travels

60 miles

60 minutes× 20 minutes = 1× 20 = 20 miles .

(2) 67 ID: [B5AA1]

No solution is available at this time.

(3) 3,840 ID: [503B]


(4) 8 ID: [CC2A1]

We have y 2 − 3xy + 8 = 92 − 3(3)(9) + 8 = 81− 81 + 8 = 8 .

(5) 3/8 ID: [DC412]

We have:1

4−1

1− 14

=1

4−134

=1

4− 43

=183

=3

8

(6) 0.000144 ID: [15DA1]


(7) 1/2 ID: [513B]

The desired area is the infinite series 49

(

1 + 19+ 192+ · · ·

)

.

Simplifying, we have 49

(

1

1− 19

)

=1

2.

(8) 3.1 ID: [5C1C]

Each of Ron and Harold make crusts at a certain rate so let’s call these rates r and h,

respectively. We can also say that

number of crusts made = rate × time.Thus if t is the amount of time it takes Ron to make 100 pizza crusts, we have that

100 = r × t = h × (t + 1.2).Additionally we have that they make them together in 1.8 hours, i.e.

100 = (r + h)× 1.8⇒ r + h = 1001.8.

We want to find t, so knowing that 100 = r · t we get t = 100r. Using 100

1.8= r + h gives us

r = 1001.8− h. We want to get rid of this h, so we use 100 = h × (t + 1.2) to get h = 100

t+1.2.

Substituting this in gives r = 1001.8− 100t+1.2. Substituting back into the even earlier equation

gives

t =100

1001.8− 100t+1.2

⇒ t =1

11.8− 1t+1.2

⇒ 1 =t

1.8− t

t + 1.2⇒ t(t + 1.2) − 1.8t = 1.8(t + 1.2)

⇒ 0 = t2 − 2.4t − 2.16Applying the quadratic formula gives

t =2.4±

√

2.42 − 4(1)(−2.16)2

=2.4±

√14.4

2

= 1.2±√3.6

Now, 3.6 is very close to 1.92 = 3.61. Thus, within 0.1, we have the answer to be

1.2 + 1.9 = 3.1 .

(9) 4,355.56 $ ID: [011C]


(10) 1500 dollars ID: [D1012]

If 10% of the price of the dirt bike was $150, then 100% of the price must have been ten

times the amount that Max paid down. Therefore, the price of the bike must have been

10× $150 = $1500 .

(11) .1 ID: [4B3A1]


(12) 12 ID: [224A1]


(13) .4 ID: [0B2A1]


(14) 16 ID: [55AA1]


(15) 9 ID: [D3DA1]


(16) 7.5 ID: [DAAA1]


(17) (−2,−6) ID: [B02C]

Since |x | is nonnegative, it is minimized when it equals 0, which occurs when x = 0. So,the minimum point of the graph of y = |x | − 3 is (0,−3). When we translate this to theleft two units and three units down, we get the point (−2,−6) .

(18) 1328

ID: [5CCA]

The dots at 37and 4

7are 4

7− 37= 17apart. There are four equal length segments between

these two dots, so each of these small segments has length 14· 17= 128. Therefore, we have

x =3

7+1

28=12

28+1

28=13

28.

(19) 66 ID: [C03A1]


(20) 33,799 ID: [ACCA]

Note that we have a difference of squares: (131)(129) = (130− 1)(130 + 1) = 1302 − 12.Thus, this expression evaluates to

(130)(130) + (131)(129) = 1302 + (1302 − 12) = 2 · 1302 − 1 = 33800 − 1 = 33799 .

(21) 50 ID: [1B3B]


(22) 4 percent ID: [5CD22]


(23) -3/10 ID: [A4D41]

First, we find 1x= 12

(

( 1x+ 1y) + ( 1

x− 1y))

= 12(3 + (−7)) = −2. Therefore, x = −1

2.

Similarly, we find 1y= 12

(

( 1x+ 1y)− ( 1

x− 1y))

= 12(3− (−7)) = 5. Therefore, y = 1

5. Our

desired sum is x + y = −12+ 15= − 3

10.

(24) 119 ID: [CA2A1]


(25) 420 ID: [302B]


(26) 45 miles per hour ID: [045A1]


(27) (4,4) ID: [04D12]

If the coordinates of the point halfway between the two points are (x, y), then x must be

the average of the x-coordinates 3 and 5 and y must be the average of the y -coordinates 7

and 1. The average of 3 and 5 is 3+52= 4 and the average of 7 and 1 is 7+1

2= 4, so the

(x, y) = (4, 4) .

(28) 525 ID: [3CB02]

Since the ratio of boys to girls is 4 : 5, the fraction of the students who are girls is 54+5= 59.

Therefore, the number of girls is 59· 945 = 525 .

(29) 1/3605 ID: [4CC51]

If we write out more terms of the product we see a pattern:

1

5· 37· 59· 711· 913· . . . · 95

99· 97101· 99103· 101105

The numerator of each fraction is the same as the denominator of the fraction two to the

left! We can cancel all these matching numbers, giving us 1·3103·105

. We could multiply out

from here, but first let’s simplify: we know 105 is divisible by 3 since the sum of the digits

is 1 + 0 + 5 = 6 which is divisible by 3. So we divide the top and bottom by 3 to get 1103·35

.

Performing the multiplication gives the value as1

3605.

(30) 63 ID: [13DA1]



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