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Grades 7 / 8 Tests and Answer Keys

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Student Name

1.

Solve for $x$ . Express $x$ as an integer or a fraction where the numerator and denominator have no common factors.

$1-\dfrac{3x}{2}=\dfrac{5}{4}$

(2 pts) 1.
2.

Three coins are tossed, each of which are equally likely to come up heads or tails. What is the probability that exactly 2 of the coins come up heads?

(3 pts) 2.
3.

Reduce the following expression to a simple fraction where the numerator and denominator have no common factors.

$\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2}}}$

(3 pts) 3.
4.

A circle is inscribed in a square as shown. The square has perimeter 64 cm. How many square centimeters are in the square but not in the circle (the shaded region of the figure)? Use 3.14 for $\pi$ and express your answer to the nearest hundredth of a square centimeter

(3 pts) 4.
5.

What is the largest prime factor of $2025$ ?

(3 pts) 5.
6.

Four students take a quiz, and each receives a score which is a whole number. The highest score is 10, and the lowest is 2. If the mean (average) score is 8, what is the mode of the scores?

(3 pts) 6.
7.

The degree measures of the angles of a triangle equal $2x$ , $3x$ , and $4x$ for some $x$ . What is the measure (in degrees) of the greatest angle?

(3 pts) 7.

TOTAL POINTS

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Key Student Name

1.

Solve for $x$ . Express $x$ as an integer or a fraction where the numerator and denominator have no common factors.

$1-\dfrac{3x}{2}=\dfrac{5}{4}$

(2 pts) 1. $-\dfrac{1}{6}$
2.

Three coins are tossed, each of which are equally likely to come up heads or tails. What is the probability that exactly 2 of the coins come up heads?

(3 pts) 2. $\dfrac{3}{8}$ or $0.375$ or $37.5\%$
3.

Reduce the following expression to a simple fraction where the numerator and denominator have no common factors.

$\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2}}}$

(3 pts) 3. $\dfrac{3}{4}$
4.

A circle is inscribed in a square as shown. The square has perimeter 64 cm. How many square centimeters are in the square but not in the circle (the shaded region of the figure)? Use 3.14 for $\pi$ and express your answer to the nearest hundredth of a square centimeter

(3 pts) 4. $55.04$
5.

What is the largest prime factor of $2025$ ?

(3 pts) 5. $5$
6.

Four students take a quiz, and each receives a score which is a whole number. The highest score is 10, and the lowest is 2. If the mean (average) score is 8, what is the mode of the scores?

(3 pts) 6. $10$
7.

The degree measures of the angles of a triangle equal $2x$ , $3x$ , and $4x$ for some $x$ . What is the measure (in degrees) of the greatest angle?

(3 pts) 7. $80$

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Solutions Student Name

1.

Solve for $x$ . Express $x$ as an integer or a fraction where the numerator and denominator have no common factors.

$1-\dfrac{3x}{2}=\dfrac{5}{4}$

(2 pts) 1. $-\dfrac{1}{6}$

Solution:

$4-6x=5\Rightarrow-6x=1\Rightarrow x=-1/6.$
2.

Three coins are tossed, each of which are equally likely to come up heads or tails. What is the probability that exactly 2 of the coins come up heads?

(3 pts) 2. $\dfrac{3}{8}$ or $0.375$ or $37.5\%$

Solution: There are 8 possible outcomes, all equally likely: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Three (HHT, HTH, THH) have exactly 2 heads. So the probability is 3/8.
3.

Reduce the following expression to a simple fraction where the numerator and denominator have no common factors.

$\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2}}}$

(3 pts) 3. $\dfrac{3}{4}$

Solution:

$\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2}}}=\dfrac{1}{2-\dfrac{1}{3/2}}=\dfrac{1}{2% -\dfrac{2}{3}}=\dfrac{1}{4/3}=\dfrac{3}{4}$
4.

A circle is inscribed in a square as shown. The square has perimeter 64 cm. How many square centimeters are in the square but not in the circle (the shaded region of the figure)? Use 3.14 for $\pi$ and express your answer to the nearest hundredth of a square centimeter

(3 pts) 4. $55.04$

Solution: The sides of the square must be 64/4=16 cm. Therefore the area of the square is $16^{2}=256cm^{2}$ . The diameter of the circle must also be 16 cm, so the radius is 8cm and the area is $3.14\cdot 8^{2}=200.96cm^{2}$ . The difference is $55.04cm^{2}$ .
5.

What is the largest prime factor of $2025$ ?

(3 pts) 5. $5$

Solution: $2025=3^{4}5^{2}$
6.

Four students take a quiz, and each receives a score which is a whole number. The highest score is 10, and the lowest is 2. If the mean (average) score is 8, what is the mode of the scores?

(3 pts) 6. $10$

Solution: The mean is 8, so the total of the 4 scores must be $8\cdot 4=32$ . Since the highest and the lowest scores sum to 12, the middle 2 scores must total $32-10-2=20$ . Since they must be whole numbers no greater than 10, they must both be 10 as well, and so the three scores of 10 make 10 the mode.
7.

The degree measures of the angles of a triangle equal $2x$ , $3x$ , and $4x$ for some $x$ . What is the measure (in degrees) of the greatest angle?

(3 pts) 7. $80$

Solution: The sum of the three angles is $2x+3x+4x=9x=180$ . So $x=20$ , and $4x=80$ .

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #2

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed. Student Name

1.

If the diameter of a tire on a vehicle is 16 inches, approximately how far does the vehicle travel in one revolution of the tire?

A. 20 in. B. 30 in. C. 50 in. D. 60 in.

(2 pts) 1.
2.

Evaluate the expression: $9+4^{2}-4\sqrt{16}+4\div 2$

(3 pts) 2.
3.

Find all solutions for $x$ :

$x^{2}+70=15x^{2}+10$

(3 pts) 3.
4.

Given values $a$ and $b$ on a real number line, which expression will always give the distance between $a$ and $b$ ?

A. $a-b$ B. $b-a$ C. $|a-b|$ D. $|a+b|$

(3 pts) 4.
5.

The volume of a cubic form is 125 cubic inches. Find the surface area of the cube.

(3 pts) 5.
6.

Factor completely over reals: $x^{4}-81$

(3 pts) 6.
7.

Find the one solution $(x,y)$ to the following pair of equations:

$\displaystyle x-y$ $\displaystyle=1$

$\displaystyle 2x+y$ $\displaystyle=5$

(3 pts) 7.

TOTAL POINTS

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #2

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed. Key Student Name

1.

If the diameter of a tire on a vehicle is 16 inches, approximately how far does the vehicle travel in one revolution of the tire?

A. 20 in. B. 30 in. C. 50 in. D. 60 in.

(2 pts) 1. C
2.

Evaluate the expression: $9+4^{2}-4\sqrt{16}+4\div 2$

(3 pts) 2. 11
3.

Find all solutions for $x$ :

$x^{2}+70=15x^{2}+10$

(3 pts) 3. $x=\pm\dfrac{\sqrt{210}}{7}$
4.

Given values $a$ and $b$ on a real number line, which expression will always give the distance between $a$ and $b$ ?

A. $a-b$ B. $b-a$ C. $|a-b|$ D. $|a+b|$

(3 pts) 4. C.
5.

The volume of a cubic form is 125 cubic inches. Find the surface area of the cube.

(3 pts) 5. $150$ in²
6.

Factor completely over reals: $x^{4}-81$

(3 pts) 6. $(x^{2}+9)(x-3)(x+3)$
7.

Find the one solution $(x,y)$ to the following pair of equations:

$\displaystyle x-y$ $\displaystyle=1$

$\displaystyle 2x+y$ $\displaystyle=5$

(3 pts) 7. $(2,1)$

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #3

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Student Name

1.

Today is a Monday. 150 days from today what day of the week will it be?

(2 pts) 1.
2.

The square in the figure below has an area of 4 cm². What is the area in cm² of the shaded region? Round your answer to the nearest hundredths.

(3 pts) 2.
3.

Two fair six-sided dice are rolled and the numbers on their face are recorded. What is the probability (expressed as a fraction in simplest form) that the sum of the two numbers is a prime?

(3 pts) 3.
4.

You are given that $\divideontimes$ is an arithmetic operator such that

$5\divideontimes 4=\dfrac{5}{16},\quad 2\divideontimes 10=\dfrac{1}{50},\quad 5% \divideontimes 10=\dfrac{1}{20}.$

What is the value of $2\divideontimes 4$ expressed as a fraction in simplest form?

(3 pts) 4.
5.

How much tax (in dollars and cents) is charged on a purchase of $952.18 if the tax rate is 7.75% for the first $500 and 3% for the amount over $500?

(3 pts) 5.
6.

In a chemistry experiment, the liquid in a container is doubling every 5 minutes. At 60 minutes the container is full. At what time will the container be half full?

(3 pts) 6.
7.

Ann and Sue buy identical stationary packs with $S$ sheets of paper and $E$ envelopes each. Ann uses her pack to write 1-sheet letters and Sue uses hers to write 3-sheet letters. Ann used all her envelopes and had 50 sheets of paper left, while Sue used all of her sheets of paper and had 50 envelopes left. Find the number of sheets of paper in the stationary pack.

(3 pts) 7.

TOTAL POINTS

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #3

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Key Student Name

1.

Today is a Monday. 150 days from today what day of the week will it be?

(2 pts) 1. Thursday
2.

The square in the figure below has an area of 4 cm². What is the area in cm² of the shaded region? Round your answer to the nearest hundredths.

(3 pts) 2. $4-\pi=0.86$
3.

Two fair six-sided dice are rolled and the numbers on their face are recorded. What is the probability (expressed as a fraction in simplest form) that the sum of the two numbers is a prime?

(3 pts) 3. $\dfrac{5}{12}$
4.

You are given that $\divideontimes$ is an arithmetic operator such that

$5\divideontimes 4=\dfrac{5}{16},\quad 2\divideontimes 10=\dfrac{1}{50},\quad 5% \divideontimes 10=\dfrac{1}{20}.$

What is the value of $2\divideontimes 4$ expressed as a fraction in simplest form?

(3 pts) 4. $\dfrac{1}{8}$
5.

How much tax (in dollars and cents) is charged on a purchase of $952.18 if the tax rate is 7.75% for the first $500 and 3% for the amount over $500?

(3 pts) 5. $52.32$
6.

In a chemistry experiment, the liquid in a container is doubling every 5 minutes. At 60 minutes the container is full. At what time will the container be half full?

(3 pts) 6. $55$ minutes
7.

Ann and Sue buy identical stationary packs with $S$ sheets of paper and $E$ envelopes each. Ann uses her pack to write 1-page letters and Sue uses hers to write 3-page letters. Ann used all her envelopes and had 50 sheets of paper left, while Sue used all of her sheets of paper and had 50 envelopes left. Find the number of sheets of paper in the stationary pack.

(3 pts) 7. $150$

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #3

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Solutions Student Name

1.

Today is a Monday. 150 days from today what day of the week will it be?

(2 pts) 1. Thursday

Solution: 147 is the nearest multiple of 7, so 147 days from now will be a Monday. So 150 days from now will be a Thursday.
2.

The square in the figure below has an area of 4 cm². What is the area in cm² of the shaded region? Round your answer to the nearest hundredths.

(3 pts) 2. $4-\pi=0.86$

Solution: The side of the square, which is also the diameter of the circle is 2 cm. The shaded area is the area of the square minus the area of the circle. That is, $4-\pi(1)^{2}=4-\pi=\boxed{0.86}$
3.

Two fair six-sided dice are rolled and the numbers on their face are recorded. What is the probability (expressed as a fraction in simplest form) that the sum of the two numbers is a prime?

(3 pts) 3. $\dfrac{5}{12}$

Solution: The possibilities for the sum of two numbers being prime are
$(1,1),(2,1),(1,2),(3,2),(2,3),(4,1),(1,4),(3,4),(4,3),(5,1),(1,5),(2,5),(5,2),% (5,6),(6,5)$ . Therefore the probability is $\dfrac{15}{36}=\boxed{\dfrac{5}{12}}$ .
4.

You are given that $\divideontimes$ is an arithmetic operator such that

$5\divideontimes 4=\dfrac{5}{16},\quad 2\divideontimes 10=\dfrac{1}{50},\quad 5% \divideontimes 10=\dfrac{1}{20}.$

What is the value of $2\divideontimes 4$ expressed as a fraction in simplest form?

(3 pts) 4. $\dfrac{1}{8}$

Solution: The operation is $a\divideontimes b=\dfrac{a}{b^{2}}$ , therefore $2\divideontimes 4=\dfrac{2}{4^{2}}=\dfrac{2}{16}=\boxed{\dfrac{1}{8}}$
5.

How much tax (in dollars and cents) is charged on a purchase of $952.18 if the tax rate is 7.75% for the first $500 and 3% for the amount over $500?

(3 pts) 5. $52.32$

Solution: The answer is

$500(0.0775)+(952.18-500)(0.03)=52.3154\approx\boxed{52.32}$
6.

In a chemistry experiment, the liquid in a container is doubling every 5 minutes. At 60 minutes the container is full. At what time will the container be half full?

(3 pts) 6. $55$ minutes

Solution: Since liquid is doubling every 5 minutes and the container is full at 60 minutes, the container will be half full 5 minutes before 60 i.e. at $55$ minutes.
7.

Ann and Sue buy identical stationary packs with $S$ sheets of paper and $E$ envelopes each. Ann uses her pack to write 1-sheet letters and Sue uses hers to write 3-sheet letters. Ann used all her envelopes and had 50 sheets of paper left, while Sue used all of her sheets of paper and had 50 envelopes left. Find the number of sheets of paper in the stationary pack.

(3 pts) 7. $150$

Solution: Since Ann writes one sheet letters, the number of letters she writes is $S-50$ . The number of envelopes she uses is $E$ and so $E=S-50$ . Sue writes 3-sheet letters and so she writes $S/3$ letters and uses $E-50$ envelopes. Therefore $S/3=E-50$ . Solving for $S$ gives $S/3=S-100\Rightarrow 2S=300\Rightarrow\boxed{S=150}$ .

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #4

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed. Student Name

1.

Evaluate

$(-100)^{0}+3\big{|}-4^{2}-(-8)\big{|}+2\big{|}\sqrt{5^{2}-3^{2}}+(-5)^{2}\big{|}$

(2 pts) 1.
2.

A drama club at Riverdale High School consists only of eleventh grade and twelfth-grade students. If $\dfrac{2}{5}$ the students in the drama club are from eleventh grade and there are 40 more twelfth-grade students than eleventh-grade students, how many twelfth-grade students are there in the drama club?

(3 pts) 2.
3.

Two fair six-sided dice are rolled. What is the probability that the sum of the two numbers rolled is at least 8?

(3 pts) 3.
4.

How many gallons of paint would be needed to paint the walls and the bottom of a rectangular swimming pool that is 40 feet long, 20 feet wide, and 10 feet deep if one gallon of paint covers 250 square feet?

(3 pts) 4.
5.

A factory makes toy cars using 6 assembly machines. Each machine assembles 15 cars every $\dfrac{2}{3}$ minutes. How many cars can all 6 machines assemble in two minutes?

(3 pts) 5.
6.

Find the positive $x$ -intercept, if any, for

$f(x)=(x+3)^{2}-4$

(3 pts) 6.
7.

Simplify the expression

$\dfrac{3^{4}\times 9^{2}}{27^{3}}\times\left(\dfrac{8^{2}\times 16}{4^{3}}% \right)^{\frac{1}{2}}$

(3 pts) 7.

TOTAL POINTS

UND MATHEMATICS TRACK MEET INDIVIDUAL TEST #4

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed. Key Student Name

1.

Evaluate

$(-100)^{0}+3\big{|}-4^{2}-(-8)\big{|}+2\big{|}\sqrt{5^{2}-3^{2}}+(-5)^{2}\big{|}$

(2 pts) 1. $83$
2.

A drama club at Riverdale High School consists only of eleventh grade and twelfth-grade students. If $\dfrac{2}{5}$ the students in the drama club are from eleventh grade and there are 40 more twelfth-grade students than eleventh-grade students, how many twelfth-grade students are there in the drama club?

(3 pts) 2. $120$ students
3.

Two fair six-sided dice are rolled. What is the probability that the sum of the two numbers rolled is at least 8?

(3 pts) 3. $\dfrac{15}{36}=\dfrac{5}{12}$
4.

How many gallons of paint would be needed to paint the walls and the bottom of a rectangular swimming pool that is 40 feet long, 20 feet wide, and 10 feet deep if one gallon of paint covers 250 square feet?

(3 pts) 4. $8$ gallons
5.

A factory makes toy cars using 6 assembly machines. Each machine assembles 15 cars every $\dfrac{2}{3}$ minutes. How many cars can all 6 machines assemble in two minutes?

(3 pts) 5. $270$ cars
6.

Find the positive $x$ -intercept, if any, for

$f(x)=(x+3)^{2}-4$

(3 pts) 6. None
7.

Simplify the expression

$\dfrac{3^{4}\times 9^{2}}{27^{3}}\times\left(\dfrac{8^{2}\times 16}{4^{3}}% \right)^{\frac{1}{2}}$

(3 pts) 7. $\dfrac{4}{3}$

UND MATHEMATICS TRACK MEET TEAM TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed.

1.

What is the radius of a circle having the same area as a rectangle with sides of length $11$ and $13$ ?

(20 pts) 1.
2.

The area of a triangle is $150$ . If the base of the triangle is three times the height, what is the height?

(20 pts) 2.
3.

Find positive numbers $r$ and $s$ with $rs=81$ and $r+s=20$ .

(20 pts) 3.
4.

In a class your grade is determined by the average of five exam scores. If the average of your first two exams is 77%, what must the average of your last three exams be to get an 85% in the class?

(20 pts) 4.
5.

A bacterial culture doubles every $12$ minutes. There are $30,720$ bacteria at noon. How many bacteria were there at 10 am earlier that same day?

(20 pts) 5.
6.

What is the greatest common divisor of $360$ and $525$ ?

(20 pts) 6.
7.

Find the positive solution to $2^{x^{2}-3x}=16\cdot 8^{x+5}$ .

(20 pts) 7.
8.

Suppose the price of an item is increased by $20$ % and then later the new price is decreased by $10$ %. Compared to the original price, what percentage increase does the final price represent?

(20 pts) 8.
9.

Solve the system of equations

$\begin{cases}\hphantom{3}x+2y=2025\\ 3x+5y=1776\end{cases}$

(20 pts) 9.
10.

Suppose $a$ and $b$ are positive real numbers such that $a-b=5$ and $ab=2$ . What is $a+b$ ?

(20 pts) 10.

TOTAL POINTS

UND MATHEMATICS TRACK MEET TEAM TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Key

1.

What is the radius of a circle having the same area as a rectangle with sides of length $11$ and $13$ ?

(20 pts) 1. $6.75$
2.

The area of a triangle is $150$ . If the base of the triangle is three times the height, what is the height?

(20 pts) 2. $10$
3.

Find positive numbers $r$ and $s$ with $rs=81$ and $r+s=20$ .

(20 pts) 3. $\{5.64,14.36\}$
4.

In a class your grade is determined by the average of five exam scores. If the average of your first two exams is 77%, what must the average of your last three exams be to get an 85% in the class?

(20 pts) 4. $90\frac{1}{3}$ %
5.

A bacterial culture doubles every $12$ minutes. There are $30,720$ bacteria at noon. How many bacteria were there at 10 am earlier that same day?

(20 pts) 5. $30$
6.

What is the greatest common divisor of $360$ and $525$ ?

(20 pts) 6. $15$
7.

Find the positive solution to $2^{x^{2}-3x}=16\cdot 8^{x+5}$ .

(20 pts) 7. $3+2\sqrt{7}\approx 8.29$
8.

Suppose the price of an item is increased by $20$ % and then later the new price is decreased by $10$ %. Compared to the original price, what percentage increase does the final price represent?

(20 pts) 8. $8$ %
9.

Solve the system of equations

$\begin{cases}\hphantom{3}x+2y=2025\\ 3x+5y=1776\end{cases}$

(20 pts) 9. $x=-6573$ , $y=4299$
10.

Suppose $a$ and $b$ are positive real numbers such that $a-b=5$ and $ab=2$ . What is $a+b$ ?

(20 pts) 10. $\sqrt{33}\approx 5.74$

UND MATHEMATICS TRACK MEET TEAM TEST #1

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are allowed. Solutions

1.

What is the radius of a circle having the same area as a rectangle with sides of length $11$ and $13$ ?

[Sketch of solution: We must have $\pi r^{2}=11\cdot 13$ and so $r=\sqrt{143/\pi}\approx 6.75$ .]
2.

The area of a triangle is $150$ . If the base of the triangle is three times the height, what is the height?

[Sketch: We have $A=\frac{1}{2}bh$ and $b=3h$ . So solve $150=\frac{1}{2}\cdot 3h\cdot h$ to find $h=10$ .]
3.

Find positive numbers $r$ and $s$ with $rs=81$ and $r+s=20$ .

[Sketch: Since $(x-r)(x-s)=x^{2}-20x+81$ , the quadratic formula gives $\{r,s\}\approx\{5.64,14.36\}$ .]
4.

In a class your grade is determined by the average of five exam scores. If the average of your first two exams is 77%, what must the average of your last three exams be to get an 85% in the class?

[Sketch: If $a$ , $b$ are the first two exam scores and $c$ , $d$ , $e$ are the last three, we have $(a+b)/2=77$ and $(a+b+c+d+e)/5=85$ . Solve to get $c+d+e=271$ , and so $(c+d+e)/3=90\frac{1}{3}$ .]
5.

A bacterial culture doubles every $12$ minutes. There are $30,720$ bacteria at noon. How many bacteria were there at 10 am earlier that same day?

[Sketch: The population doubles $10$ times between 10 am and noon so the number of bacteria at 10 am is $30720/2^{10}=30$ .]
6.

What is the greatest common divisor of $360$ and $525$ ?

[Sketch: We have $360=2^{3}\cdot 3^{2}\cdot 5$ and $525=3\cdot 5^{2}\cdot 7$ so the GCD is $3\cdot 5=15$ .]
7.

Find the positive solution to $2^{x^{2}-3x}=16\cdot 8^{x+5}$ .

[Sketch: We have $2^{x^{2}-3x}=2^{4+3x+15}$ , and so $x^{2}-6x-19=0$ . The solutions are $3\pm 2\sqrt{7}$ . The positive solution is $3+2\sqrt{7}\approx 8.29$ .]
8.

Suppose the price of an item is increased by $20$ % and then later the new price is decreased by $10$ %. Compared to the original price, what percentage increase does the final price represent?

[Sketch: If $P$ is the original price, the final price is $(.9)(1.20)P=(1.08)P$ . So the price has increased by $8$ %.]
9.

Solve the system of equations

$\begin{cases}\hphantom{3}x+2y=2025\\ 3x+5y=1776\end{cases}$

[Sketch: Use Gaussian elimination or substitution to find $x=-6573$ , $y=4299$ .]
10.

Suppose $a$ and $b$ are positive real numbers such that $a-b=5$ and $ab=2$ . What is $a+b$ ?

[Sketch: Since $(a+b)^{2}=(a-b)^{2}+4ab=33$ we have $a+b=\sqrt{33}\approx 5.74$ .]

UND MATHEMATICS TRACK MEET TEAM TEST #2

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed.

1.

What is 2025% of 2025?

(20 pts) 1.
2.

In the diagram to the right, the four segments: $\overline{MQ},\overline{NR},\overline{OS},\overline{PT}$ all intersect at the same point creating 8 angles labeled with either their angle measures or variable names. What is the value of $a+b+c+d+g$ ?

(20 pts) 2.
3.

A game consists of black and white pieces. The number of black pieces is 4 more than 3 times the number of white pieces. Seven white and 15 black pieces are removed each round. After several rounds, there are 3 white and 55 black pieces left. How many pieces were there in the beginning?

(20 pts) 3.
4.

A tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games, and Lara won 2 games, how many games did Monica win?

(20 pts) 4.
5.

I have a container with $1\frac{5}{6}$ cups of flour. The container is $\frac{1}{3}$ full. How much flour does the container have when it is full?

(20 pts) 5.
6.

The average of two whole numbers is 18 and their product is 308. What is the positive difference between the two numbers?

(20 pts) 6.
7.

In the rectangle $A B C D$ , the measure of $\overline{AB}$ is 24 inches, the measure of $\overline{AD}$ is 11 inches, and the measure of $\overline{EF}$ is 10 inches. If $\overline{ED}$ is parallel to $\overline{FG}$ , how many square inches are in the combined area of the shaded regions.

(20 pts) 7.
8.

All of the shirts in Joel’s closet are either gray t-shirts or plaid flannels. Two-fifths of his shirts are flannels. If he were to buy another flannel, then $\dfrac{3}{7}$ of his shirts would be flannels. How many t-shirts are there in Joel’s closet?

(20 pts) 8.
9.

The length of each side of a parallelogram is multiplied by 6 to create a similar parallelogram whose area is 612 ft². How many square feet are in the area of the smaller parallelogram?

(20 pts) 9.
10.

Five marbles are in a bowl. Two are green and 3 are black. What is the probability of picking a green marble out of the bowl on the first draw (and not returning it to the bowl) and a black marble on the second draw?

(20 pts) 10.

TOTAL POINTS

UND MATHEMATICS TRACK MEET TEAM TEST #2

University of North Dakota Grades 7/8

January 13, 2025

School Team Name

Calculators are NOT allowed. Key

1.

What is 2025% of 2025?

(20 pts) 1. $41,006.25$
2.

In the diagram to the right, the four segments: $\overline{MQ},\overline{NR},\overline{OS},\overline{PT}$ all intersect at the same point creating 8 angles labeled with either their angle measures or variable names. What is the value of $a+b+c+d+g$ ?

(20 pts) 2. $216$
3.

A game consists of black and white pieces. The number of black pieces is 4 more than 3 times the number of white pieces. Seven white and 15 black pieces are removed each round. After several rounds, there are 3 white and 55 black pieces left. How many pieces were there in the beginning?

(20 pts) 3. $212$
4.

A tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games, and Lara won 2 games, how many games did Monica win?

(20 pts) 4. $2$
5.

I have a container with $1\frac{5}{6}$ cups of flour. The container is $\frac{1}{3}$ full. How much flour does the container have when it is full?

(20 pts) 5. $5\frac{1}{2}$ cups
6.

The average of two whole numbers is 18 and their product is 308. What is the positive difference between the two numbers?

(20 pts) 6. $8$
7.

In the rectangle $A B C D$ , the measure of $\overline{AB}$ is 24 inches, the measure of $\overline{AD}$ is 11 inches, and the measure of $\overline{EF}$ is 10 inches. If $\overline{ED}$ is parallel to $\overline{FG}$ , how many square inches are in the combined area of the shaded regions.

(20 pts) 7. $154$ in²
8.

All of the shirts in Joel’s closet are either gray t-shirts or plaid flannels. Two-fifths of his shirts are flannels. If he were to buy another flannel, then $\dfrac{3}{7}$ of his shirts would be flannels. How many t-shirts are there in Joel’s closet?

(20 pts) 8. $12$
9.

The length of each side of a parallelogram is multiplied by 6 to create a similar parallelogram whose area is 612 ft². How many square feet are in the area of the smaller parallelogram?

(20 pts) 9. $17$
10.

Five marbles are in a bowl. Two are green and 3 are black. What is the probability of picking a green marble out of the bowl on the first draw (and not returning it to the bowl) and a black marble on the second draw?

(20 pts) 10. $\dfrac{3}{10}$

	$\displaystyle x-y$	$\displaystyle=1$
	$\displaystyle 2x+y$	$\displaystyle=5$