Grades 9 / 10 Tests and Answer Keys

The average of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed?

  (2 pts) 1.

In $\mathrm{△}ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, What is the measure of $\mathrm{\angle }ABC$ (in degrees)?

  (3 pts) 2.

A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the word MATH?

  (3 pts) 3.

There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses?

  (3 pts) 4.

A hiker is exploring a trail. The trail has three sections: the first 25% of the trail is along a river, the next 5/8 of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail?

  (3 pts) 5.

The first 2 hours of Melanie’s trip was spent traveling at 100 km/h. The remaining 200 km of Melanie’s trips was spent traveling at 80 km/h. What is Melanie’s average speed during this trip (rounded to 2 decimal places)?

  (3 pts) 6.

In the diagram, $\mathrm{△}ABC$ has $AB=BC=3x+4$ and $AC=2x$ and rectangle $DEFG$ has $EF=2x-2$ and $FG=3x-1$. The perimeter of $\mathrm{△}ABC$ is equal to the perimeter of rectangle $DEFG$. What is the area of $\mathrm{△}ABC$?

  (3 pts) 7.

The average of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed?

  (2 pts) 1. $26$

In $\mathrm{△}ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, What is the measure of $\mathrm{\angle }ABC$ (in degrees)?

  (3 pts) 2. $30$

A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the word MATH?

  (3 pts) 3. $10$

There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses?

  (3 pts) 4. $7$

A hiker is exploring a trail. The trail has three sections: the first 25% of the trail is along a river, the next 5/8 of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail?

  (3 pts) 5. $24$ km

The first 2 hours of Melanie’s trip was spent traveling at 100 km/h. The remaining 200 km of Melanie’s trips was spent traveling at 80 km/h. What is Melanie’s average speed during this trip (rounded to 2 decimal places)?

  (3 pts) 6. $88.89$ km/h

In the diagram, $\mathrm{△}ABC$ has $AB=BC=3x+4$ and $AC=2x$ and rectangle $DEFG$ has $EF=2x-2$ and $FG=3x-1$. The perimeter of $\mathrm{△}ABC$ is equal to the perimeter of rectangle $DEFG$. What is the area of $\mathrm{△}ABC$?

  (3 pts) 7. $168$

The average of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed?

  (2 pts) 1. $26$

Solution: When 10 numbers have an average of 17, their sum is $10\cdot 17=170$. When 9 numbers have an average of 16, their sum is $9\cdot 16=144$. Therefore, the number that was removed is $170-144=26$.

In $\mathrm{△}ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, What is the measure of $\mathrm{\angle }ABC$ (in degrees)?

  (3 pts) 2. $30$

Solution: Since $CD=DE=EC$, then $\mathrm{△}CDE$ is equilateral, which means that $\mathrm{\angle }DEC={60}^{\circ }$. Since $\mathrm{\angle }DEB$ is a straight line, then $\mathrm{\angle }CEB={180}^{\circ }-\mathrm{\angle }DEC={180}^{\circ }-{60}^{\circ }={120}^{\circ }$. Since $CE=EB$, then $\mathrm{△}CEB$ is isosceles with $\mathrm{\angle }ECB=\mathrm{\angle }EBC$. Since $\mathrm{\angle }ECB+\mathrm{\angle }CEB+\mathrm{\angle }EBC={180}^{\circ }$, then $2\times \mathrm{\angle }EBC+{120}^{\circ }={180}^{\circ }$, which means that $2\times \mathrm{\angle }EBC={60}^{\circ }$ or $\mathrm{\angle }EBC={30}^{\circ }$. Therefore, $\mathrm{\angle }ABC=\mathrm{\angle }EBC={30}^{\circ }$.

A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the word MATH?

  (3 pts) 3. $10$

Solution: Since TASTE is 3 and SET is 2, then TA=1. Since HAT is 7 and TA is 1, then H is 6. Since MAT is 4 and H is 7, then MATH is 10. (Note that S, E, T, and A do not have unique values.)

There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses?

  (3 pts) 4. $7$

Solution: Since 10 students have black hair and 3 students have black hair and wear glasses, then $10-3=7$ students have black hair but do not wear glasses.

A hiker is exploring a trail. The trail has three sections: the first 25% of the trail is along a river, the next 5/8 of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail?

  (3 pts) 5. $24$ km

Solution: Since 25% is equivalent to $\frac{1}{4}$, then the fraction of the trail along the river and through the forest is $\frac{1}{4}+\frac{5}{8}=\frac{2}{8}+\frac{5}{8}=\frac{7}{8}$. This means that the hill represents $\frac{1}{8}$ of the trail, so that the trail is 24 km long.

The first 2 hours of Melanie’s trip was spent traveling at 100 km/h. The remaining 200 km of Melanie’s trips was spent traveling at 80 km/h. What is Melanie’s average speed during this trip (rounded to 2 decimal places)?

  (3 pts) 6. $88.89$ km/h

Solution: In 2 hours traveling at 100 km/h, Melanie travels 200 km. When Melanie travels 200 km at 80 km/h, it takes ${\displaystyle \frac{200km}{80km/h}}=2.5h$. Melanie travels a total of $200km+200km=400km$ for $2h+2.5h=4.5h$. Therefore, Melanie’s average speed is ${\displaystyle \frac{400km}{4.5h}}\approx 88.89km/h$.

In the diagram, $\mathrm{△}ABC$ has $AB=BC=3x+4$ and $AC=2x$ and rectangle $DEFG$ has $EF=2x-2$ and $FG=3x-1$. The perimeter of $\mathrm{△}ABC$ is equal to the perimeter of rectangle $DEFG$. What is the area of $\mathrm{△}ABC$?

  (3 pts) 7. $168$

Solution: The perimeter of $\mathrm{△}ABC$ is $(3x+4)+(3x+4)+2x=8x+8$. The perimeter of rectangle $DEFG$ is $2(2x-2)+2(3x-1)=4x-4+6x-2=10x-6$. Since these perimeters are equal, we have $10x-6=8x+8$ which gives $2x=14$ and so $x=7$. Thus $\mathrm{△}ABC$ has $AC=2\cdot 7=14$ and $AB=BC=3\cdot 7+4=25$. Suppose $T$ is the midpoint of $AC$, so that BT is the altitude of $\mathrm{△}ABC$. By the Pythagorean Theorem, $BT=\sqrt{B{C}^2+T{C}^2}=\sqrt{{25}^2-7^2}=\sqrt{625-49}=\sqrt{576}=24$. Therefore, the area of $\mathrm{△}ABC$ is equal to $\frac{1}{2}\cdot AC\cdot BT=\frac{1}{2}\cdot 14\cdot 24=168$.

What is $x+{\displaystyle \frac{1}{x}}$ if $x^2=2-{\displaystyle \frac{1}{x^2}}$?

  (2 pts) 1.

If $-2\le x\le 1,-1\le y\le 3$, then $a\le x^{2}y\le 12$. What is $a$?

  (3 pts) 2.

$x=1$ is a root of $x^3-x^2+ax+4=0$. What are the other roots?

  (3 pts) 3.

If $(a,b)$ is the minimum point on the graph of $y=x^2-6x+10$, what is the distance of $(a,b)$ from the $y$-axis?

  (3 pts) 4.

If ${\log }_{10}(x^2-3)>0$, then or $x>b$. What is $a$?

  (3 pts) 5.

  (3 pts) 6.

In a survey of 100 people, 50 liked baseball, 70 liked basketball and 20 did not like either sport. How many people liked both baseball and basketball?

  (3 pts) 7.

What is $x+{\displaystyle \frac{1}{x}}$ if $x^2=2-{\displaystyle \frac{1}{x^2}}$?

  (2 pts) 1. $\pm 2$

If $-2\le x\le 1,-1\le y\le 3$, then $a\le x^{2}y\le 12$. What is $a$?

  (3 pts) 2. $-4$

$x=1$ is a root of $x^3-x^2+ax+4=0$. What are the other roots?

  (3 pts) 3. $\pm 2$

If $(a,b)$ is the minimum point on the graph of $y=x^2-6x+10$, what is the distance of $(a,b)$ from the $y$-axis?

  (3 pts) 4. $3$

If ${\log }_{10}(x^2-3)>0$, then or $x>b$. What is $a$?

  (3 pts) 5. $-2$

  (3 pts) 6. $4$

In a survey of 100 people, 50 liked baseball, 70 liked basketball and 20 did not like either sport. How many people liked both baseball and basketball?

  (3 pts) 7. $40$

What is $x+{\displaystyle \frac{1}{x}}$ if $x^2=2-{\displaystyle \frac{1}{x^2}}$?

  (2 pts) 1. $\pm 2$

Solution: $x^2+2+1/x^2=4\Rightarrow {(x+1/x)}^2=4\Rightarrow x+\frac{1}{x}=\pm 2$.

If $-2\le x\le 1,-1\le y\le 3$, then $a\le x^{2}y\le 12$. What is $a$?

  (3 pts) 2. $-4$

Solution: $0\le x^2\le 4$ and $-1\le y\le 3\Rightarrow -4\le x^{2}y\le 12$.

$x=1$ is a root of $x^3-x^2+ax+4=0$. What are the other roots?

  (3 pts) 3. $\pm 2$

Solution: $a=-4\Rightarrow x^3-x^2-4x+4=(x-1)(x^2-4)\Rightarrow x=1,\pm 2$.

If $(a,b)$ is the minimum point on the graph of $y=x^2-6x+10$, what is the distance of $(a,b)$ from the $y$-axis?

  (3 pts) 4. $3$

Solution: $y={(x-3)}^2+1\Rightarrow (a,b)=(3,1)$.

If ${\log }_{10}(x^2-3)>0$, then or $x>b$. What is $a$?

  (3 pts) 5. $-2$

Solution: .

  (3 pts) 6. $4$

Solution: $4\pi r^2=2\pi r^2+8\pi r\Rightarrow r^2-4r=r(r-4)=0$.

In a survey of 100 people, 50 liked baseball, 70 liked basketball and 20 did not like either sport. How many people liked both baseball and basketball?

  (3 pts) 7. $40$

Since 80 liked some sport, 10 liked baseball but not basketball. Therefore, 40 liked both.

Find the two numbers that add to 20 and multiply to 91.

  (2 pts) 1.

How many $(x,y)$ points have integer coordinates and are exactly 5 units from the origin?

  (3 pts) 2.

How many positive integers evenly divide 2025, including 1 and 2025?

  (3 pts) 3.

A standard deck of 52 cards has 13 values in each of 4 different suits. How many ways are there to get only one suit in a hand of 5 cards?

  (3 pts) 4.

Find the smallest natural number $n$ so that the sum of the integers from 1 to $n$ (inclusive) is at least 2025.

  (3 pts) 5.

A quadratic function of the form $f(x)=a{x}^2+c$ has $f(3)=5$ and $f(5)=3$. What is $f(-3)$?

  (3 pts) 6.

Write down 2025 copies of the letter A. Beginning from the left, replace every pair of A’s with a single B, so that there is one A left over. Repeat the process, replacing pairs of B’s with C’s, pairs of C’s with D’s, and so on. When you cannot replace any more, what is left?

  (3 pts) 7.

Find the two numbers that add to 20 and multiply to 91.

  (2 pts) 1. 7 and 13

How many $(x,y)$ points have integer coordinates and are exactly 5 units from the origin?

  (3 pts) 2. 12

How many positive integers evenly divide 2025, including 1 and 2025?

  (3 pts) 3. 15

A standard deck of 52 cards has 13 values in each of 4 different suits. How many ways are there to get only one suit in a hand of 5 cards?

  (3 pts) 4. $4\left(\genfrac{}{}{0pt}{}{13}{5}\right)=5148$

Find the smallest natural number $n$ so that the sum of the integers from 1 to $n$ (inclusive) is at least 2025.

  (3 pts) 5. 64

A quadratic function of the form $f(x)=a{x}^2+c$ has $f(3)=5$ and $f(5)=3$. What is $f(-3)$?

  (3 pts) 6. 5

Write down 2025 copies of the letter A. Beginning from the left, replace every pair of A’s with a single B, so that there is one A left over. Repeat the process, replacing pairs of B’s with C’s, pairs of C’s with D’s, and so on. When you cannot replace any more, what is left?

  (3 pts) 7. KJIHGFDA

Find the two numbers that add to 20 and multiply to 91.

Solution: $x+y=20$ and $xy=91$ means that $x(20-x)=91$, or $0=x^2-20x+91=(x-7)(x-13)$.

  (2 pts) 1. 7 and 13

How many $(x,y)$ points have integer coordinates and are exactly 5 units from the origin?

Solution: The four points $(\pm 3,\pm 4)$, the four points $(\pm 4,\pm 3)$, the two points $(0,\pm 5)$, and the two points $(\pm 5,0)$.

  (3 pts) 2. 12

How many positive integers evenly divide 2025, including 1 and 2025?

Solution: Since $2025=3^4\cdot 5^2$, there are $(4+1)(2+1)$ factors.

  (3 pts) 3. 15

A standard deck of 52 cards has 13 values in each of 4 different suits. How many ways are there to get only one suit in a hand of 5 cards?

Solution: There are four options for the suit and $\left(\genfrac{}{}{0pt}{}{13}{5}\right)$ options for the values.

  (3 pts) 4. $4\left(\genfrac{}{}{0pt}{}{13}{5}\right)=5148$

Find the smallest natural number $n$ so that the sum of the integers from 1 to $n$ (inclusive) is at least 2025.

Solution: We need $n(n+1)/2\ge 2025$, so $n(n+1)=4050$. Since this is about $2^{12}$, we can start by checking $2^6$, seeing that $64\cdot 65/2=2080\ge 2025$, and .

  (3 pts) 5. 64

A quadratic function of the form $f(x)=a{x}^2+c$ has $f(3)=5$ and $f(5)=3$. What is $f(-3)$?

Solution: The function is even. Alternatively, $9a+c=5$ and $25a+c=3$ means $16a=-2$ so that $a=-1/8$ and $c=49/8$.

  (3 pts) 6. 5

Write down 2025 copies of the letter A. Beginning from the left, replace every pair of A’s with a single B, so that there is one A left over. Repeat the process, replacing pairs of B’s with C’s, pairs of C’s with D’s, and so on. When you cannot replace any more, what is left?

Solution: This is equivalent to finding the binary representation of 2025. We have

  (3 pts) 7. KJIHGFDA

For the figure below, how many rotations and reflections carry it onto itself? Do not count the reflection or rotation that leaves all points in the same place.

  (2 pts) 1.

You are driving to your cabin that is $x$ miles away. On the first weekend, you could travel $75mph$. The next weekend there was a snowstorm, and you could only travel $50mph$ to your cabin. The second weekend took you $42min$ longer to travel than the first. How far away is your cabin?

  (3 pts) 2.

A moving truck rental company charges $\$50$ to rent a truck for $10$ miles and $\$80$ for $25$ miles. Assume the truck rental charges are linear with miles. How much will the company charge for you rent a truck for $85$ miles?

  (3 pts) 3.

Determine the length, $l$ of a rectangle, with width $w$, if its perimeter is $44$, its area is $105$, and $l>w$.

  (3 pts) 4.

A number is divided by 3 and then 15 is added to the result to give 46. What is the original number?

  (3 pts) 5.

Each of the numbers $1,\mathrm{\hspace{0.33em}2},\mathrm{\hspace{0.33em}3},$ and $4$ is substituted, in some order, for $p,q,r,$ and $s$. What is the highest possible value of $p^q+r^s$?

  (3 pts) 6.

Find the side length $x$.

  (3 pts) 7.

For the figure below, how many rotations and reflections carry it onto itself? Do not count the reflection or rotation that leaves all points in the same place.

  (3 pts) 1. 9

You are driving to your cabin that is $x$ miles away. On the first weekend, you could travel $75mph$. The next weekend there was a snowstorm, and you could only travel $50mph$ to your cabin. The second weekend took you $42min$ longer to travel than the first. How far away is your cabin?

  (3 pts) 2. 105 miles

A moving truck rental company charges $\$50$ to rent a truck for $10$ miles and $\$80$ for $25$ miles. Assume the truck rental charges are linear with miles. How much will the company charge for you rent a truck for $85$ miles?

  (3 pts) 3. $200

Determine the length, $l$ of a rectangle, with width $w$, if its perimeter is $44$, its area is $105$, and $l>w$.

  (3 pts) 4. 15

A number is divided by 3 and then 15 is added to the result to give 46. What is the original number?

  (3 pts) 5. 93

Each of the numbers $1,\mathrm{\hspace{0.33em}2},\mathrm{\hspace{0.33em}3},$ and $4$ is substituted, in some order, for $p,q,r,$ and $s$. What is the highest possible value of $p^q+r^s$?

  (3 pts) 6. 83

Find the side length $x$.

  (3 pts) 7. 4

Determine exactly the solution to

  (20 pts) 1.

At a school dance class attended by $200$ juniors and seniors, the seniors are asked to teach the juniors how to dance. Anna danced with $7$ juniors, Joey danced with $8$ juniors, Hailey danced with $9$ juniors, and so on through the last senior, who danced with all of the juniors. How many juniors attended the dance?

  (20 pts) 2.

$a$, $b$, and $c$ are positive integers such that the sum of $a$ and $b$ is $3$ more than the value of $c$, the value of $a$ is double that of $b$, and the value of $b$ is five less than $c$. Determine exactly the value of $b$.

  (20 pts) 3.

The base of a triangular flower bed is $4$ meters longer than its height. If the area of the flower bed is $30$ m², find the height of the triangle.

  (20 pts) 4.

Find the sum of the two solutions to this absolute value equation: $\left|2x+1\right|=9$

  (20 pts) 5.

When the height of a triangle is quadrupled, its area increased by $2025$ cm². What is the area of the original triangle?

  (20 pts) 6.

Five years ago, Maria was three times as old as Alex. Five years from now, Maria will be twice as old as Alex. What is Alex’s current age?

  (20 pts) 7.

In $2023$, the number of total enrolled students at the University of North Dakota was $14,172$. This student population represented approximiately $24\%$ of the population of Grand Forks. About how many people lived in Grand Forks in $2023$? Round your answer to the nearest integer.

  (20 pts) 8.

$(x,y)$ is the intersection of ${\displaystyle \frac{3x}{y}}-4=11$ and $x-y=44$. Determine the value of $x+y$.

  (20 pts) 9.

Summing the powers of $3$ produces an interesting pattern that allows you to find the sum without adding. Discover the pattern and use it to find the sum of:
$1+3+9+27+\mathrm{\cdots }+531441$.

  (20 pts) 10.

Determine exactly the solution to

  (20 pts) 1. $x=-{\displaystyle \frac{15}{2}}$

At a school dance class attended by $200$ juniors and seniors, the seniors are asked to teach the juniors how to dance. Anna danced with $7$ juniors, Joey danced with $8$ juniors, Hailey danced with $9$ juniors, and so on through the last senior, who danced with all of the juniors. How many juniors attended the dance?

  (20 pts) 2. $103$

$a$, $b$, and $c$ are positive integers such that the sum of $a$ and $b$ is $3$ more than the value of $c$, the value of $a$ is double that of $b$, and the value of $b$ is five less than $c$. Determine exactly the value of $b$.

  (20 pts) 3. $4$

The base of a triangular flower bed is $4$ meters longer than its height. If the area of the flower bed is $30$ m², find the height of the triangle.

  (20 pts) 4. $6$

Find the sum of the two solutions to this absolute value equation: $\left|2x+1\right|=9$

  (20 pts) 5. $-1$

When the height of a triangle is quadrupled, its area increased by $2025$ cm². What is the area of the original triangle?

  (20 pts) 6. $675$

Five years ago, Maria was three times as old as Alex. Five years from now, Maria will be twice as old as Alex. What is Alex’s current age?

  (20 pts) 7. $15$

In $2023$, the number of total enrolled students at the University of North Dakota was $14,172$. This student population represented approximiately $24\%$ of the population of Grand Forks. About how many people lived in Grand Forks in $2023$? Round your answer to the nearest integer.

  (20 pts) 8. $59,050$

$(x,y)$ is the intersection of ${\displaystyle \frac{3x}{y}}-4=11$ and $x-y=44$. Determine the value of $x+y$.

  (20 pts) 9. $66$

Summing the powers of $3$ produces an interesting pattern that allows you to find the sum without adding. Discover the pattern and use it to find the sum of:
$1+3+9+27+\mathrm{\cdots }+531441$.

  (20 pts) 10. 797161

Determine exactly the solution to

Solution:

At a school dance class attended by $200$ juniors and seniors, the seniors are asked to teach the juniors how to dance. Anna danced with $7$ juniors, Joey danced with $8$ juniors, Hailey danced with $9$ juniors, and so on through the last senior, who danced with all of the juniors. How many juniors attended the dance?

Solution: The first senior danced with $6+1$ juniors, the second senior danced with $6+2$ juniors, etc., so we can say that the $n^{\text{th}}$ senior danced with $6+n$ juniors. Since there were $n$ seniors and $6+n$ juniors at the dance,
$n+\left(6+n\right)=200\Rightarrow 2n+6=200\Rightarrow n=97$, so there were $97$ seniors and $103$ juniors.

$a$, $b$, and $c$ are positive integers such that the sum of $a$ and $b$ is $3$ more than the value of $c$, the value of $a$ is double that of $b$, and the value of $b$ is five less than $c$. Determine exactly the value of $b$.

Solution: If the sum of $a$ and $b$ is $3$ more than $c$, then $a+b=c+3$. Since $a$ is double $b$ and $b$ is five less than $c$, then $a=2b$ and $b=c-5$ respectively. Substituting the second and third equations into the first gives us:

Therefore, $b=9-5=4$

The base of a triangular flower bed is $4$ meters longer than its height. If the area of the flower bed is $30$ m², find the height of the triangle.

Solution: Let the height of the triangle be $h$. Then, the base is $h+4$. Substitute into the formula for the area of a triangle:

Solving for $h$, we get $h=-10$ and $h=6$, but height here cannot be negative, so our height must be $6$ meters.

Find the sum of the two solutions to this absolute value equation: $\left|2x+1\right|=9$

Solution: We get two equations to solve, either $2x+1=9$ or $2x+1=-9$. In the first case, $x=4$. In the second case, $x=-5$. Their sum is $-1$.

When the height of a triangle is quadrupled, its area increased by $2025$ cm². What is the area of the original triangle?

Solution:

Five years ago, Maria was three times as old as Alex. Five years from now, Maria will be twice as old as Alex. What is Alex’s current age?

Solution: Let Alex’s current age be $x$ and Maria’s current age be $y$.

1. 1.

   Five years ago:
   Maria’s age was $y-5$, and Alex’s age was $x-5$. From the problem:

   $$y-5=3\left(x-5\right).$$

2. 2.

   Five years from now:
   Maria’s age will be $y+5$, and Alex’s age will be $x+5$. From the problem:

   $$y+5=2\left(x+5\right).$$

3. 3.

   Solve the system of equations:
   From the first equation:

   $$y-5=3x-15\Rightarrow y=3x-10.$$

   Substitute $y=3x-10$ into the second equation:

   $$\left(3x-10\right)+5=2\left(x+5\right).$$

   Simplify:

   $$3x-5=2x+10.$$

   Solve for $x$:

   $$x=15.$$

Hence, Alex is currently $15$ years old.

In $2023$, the number of total enrolled students at the University of North Dakota was $14,172$. This student population represented approximiately $24\%$ of the population of Grand Forks. About how many people lived in Grand Forks in $2023$? Round your answer to the nearest integer.

Solution: Let the total population of Grand Forks in 2023 be $x$. We know that $24\%$ of this total is the number of enrolled students at the University of North Dakota, which is $14,172$. This relationship can be written as:

To solve for $x$, divide both sides of the equation by $0.24$:

Simplify the division:

Hence, the population of Grand Forks in $2023$ was approximately $59,050$ people.

$(x,y)$ is the intersection of ${\displaystyle \frac{3x}{y}}-4=11$ and $x-y=44$. Determine the value of $x+y$.

Solution: $3x=15y\Rightarrow x=5y.$ Therefore, $5y-y=44\Rightarrow y=11$. So, $x=55$ and $x+y=66$

Summing the powers of $3$ produces an interesting pattern that allows you to find the sum without adding. Discover the pattern and use it to find the sum of:
$1+3+9+27+\mathrm{\cdots }+531441$.

Solution: Take the last power of three, muliply by $3$, subract $1$, and divide by $2$.

Step 1:

Step 2:

Step 3:

A movie theater in a small town usually open its doors 3 days in a row and then closes the next day for maintenance. Another movie theater is open 4 days in a row and then closes the next day for the same reason. Suppose both movie theaters are closed today and that today is Wednesday. What day is it the next time they are both closed again at the same time?

  (20 pts) 1.

Let ${\mathrm{\ell }}_1$ be the line segment whose endpoints lie at the top left and bottom right corner of a square and let ${\mathrm{\ell }}_2$ be the line segment whose endpoints are the bottom left corner of the square and the center of the top side of the square. What fraction of the area of the square is not occupied by the triangle formed from the point at the intersection of ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$ and the bottom two corners of the square?

  (20 pts) 2.

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many more children than adults attended?

  (20 pts) 3.

The sum of the squares of two consecutive positive odd integers is 74. What is the value of the larger integer?

  (20 pts) 4.

How many ways are there to pick two nonconsecutive numbers from the first 50 positive integers?

  (20 pts) 5.

Suppose the rational function $f(x)$ is given by $f(x)={\displaystyle \frac{5x+7}{2x+4}}$. Find $f^{-1}(x)$.

  (20 pts) 6.

A parallelogram with sides of length $x$ and ${\displaystyle \frac{3x+2}{2}}$ has perimeter 32. What is the length of the longer side?

  (20 pts) 7.

If the cost of a bat and a baseball combined is $1.10 and the ball costs $1.00 less than the bat, how much is the ball?

  (20 pts) 8.

The square root of the value obtained by adding twelve to some positive number is the same as the number itself. What is it?

  (20 pts) 9.

What is the third smallest two digit positive integer that is equal to seven times the sum of its digits?

  (20 pts) 10.

A movie theater in a small town usually open its doors 3 days in a row and then closes the next day for maintenance. Another movie theater is open 4 days in a row and then closes the next day for the same reason. Suppose both movie theaters are closed today and that today is Wednesday. What day is it the next time they are both closed again at the same time?

  (20 pts) 1. Tuesday

Let ${\mathrm{\ell }}_1$ be the line segment whose endpoints lie at the top left and bottom right corner of a square and let ${\mathrm{\ell }}_2$ be the line segment whose endpoints are the bottom left corner of the square and the center of the top side of the square. What fraction of the area of the square is not occupied by the triangle formed from the point at the intersection of ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$ and the bottom two corners of the square?

  (20 pts) 2. ${\displaystyle \frac{2}{3}}$

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many more children than adults attended?

  (20 pts) 3. $800$

The sum of the squares of two consecutive positive odd integers is 74. What is the value of the larger integer?

  (20 pts) 4. $7$

How many ways are there to pick two nonconsecutive numbers from the first 50 positive integers?

  (20 pts) 5. $1176$

Suppose the rational function $f(x)$ is given by $f(x)={\displaystyle \frac{5x+7}{2x+4}}$. Find $f^{-1}(x)$.

  (20 pts) 6. $f^{-1}(x)={\displaystyle \frac{7-4x}{2x-5}}$

A parallelogram with sides of length $x$ and ${\displaystyle \frac{3x+2}{2}}$ has perimeter 32. What is the length of the longer side?

  (20 pts) 7. $10$

If the cost of a bat and a baseball combined is $1.10 and the ball costs $1.00 less than the bat, how much is the ball?

  (20 pts) 8. $0.05

The square root of the value obtained by adding twelve to some positive number is the same as the number itself. What is it?

  (20 pts) 9. $4$

What is the third smallest two digit positive integer that is equal to seven times the sum of its digits?

  (20 pts) 10. $63$

A movie theater in a small town usually open its doors 3 days in a row and then closes the next day for maintenance. Another movie theater is open 4 days in a row and then closes the next day for the same reason. Suppose both movie theaters are closed today and that today is Wednesday. What day is it the next time they are both closed again at the same time?

  (20 pts) 1. Tuesday

Solution: The first theater has an open-closed cycle that repeats every 4 days, while the second has an open-closed cycle that repeats every 5 days. The least common multiple of 4 and 5 is 20, so we need only note that 20 days after Wednesday is Tuesday. That is, the next time both theaters will be closed at once will occur on a Tuesday.

Let ${\mathrm{\ell }}_1$ be the line segment whose endpoints lie at the top left and bottom right corner of a square and let ${\mathrm{\ell }}_2$ be the line segment whose endpoints are the bottom left corner of the square and the center of the top side of the square. What fraction of the area of the square is not occupied by the triangle formed from the point at the intersection of ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$ and the bottom two corners of the square?

  (20 pts) 2. ${\displaystyle \frac{2}{3}}$

Solution: Consider a square of side length 1 for simplicity, so that the area of the triangle is the same as the proportion of the square’s area that it occupies. The triangle formed is shaded in blue below:

Notice that the height of the triangle and the triangle formed above it sum to 1, and that these two triangles are similar. Let $h$ be the height of the larger triangle, so that $1-h$ is the height of the smaller. As the ratio of the height to the base of similar triangles is the same, we then have

So, the area of the triangle is $\frac{1}{2}(1)(h)=\frac{1}{3}$, which is also the proportion of the area of any square a triangle formed in such a fashion will occupy. Hence, the proportion of the square not occupied by the triangle is $\frac{2}{3}$.

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many more children than adults attended?

  (20 pts) 3. $800$

Solution: Let $x$ denote the number of adults and let $y$ denote the number of children. Then, we have the system

Substituting $x=2200-y$ into the second equation shows

and $x=2200-1500=700$ so that $1500-700=800$ more children than adults attended.

The sum of the squares of two consecutive positive odd integers is 74. What is the value of the larger integer?

  (20 pts) 4. $7$

Solution: Every odd integer is of the form $2k+1$ for some integer $k\ge 0$. So, the sum of the squares of any two consecutive odd integers is of the form ${(2k+1)}^2+{(2k+3)}^2=8k^2+16k+10$. If the value of this sum is to be 74, then we need only solve $8k^2+16k+10=74$, which amounts to finding the roots of the quadratic $8k^2+16k-64=0$, or (factoring out an 8) $k^2+2k-8=0$. Via the quadratic formula, this produces $k=-4$ or $k=2$. Since we are looking for a positive odd integer, we may conclude that $k=2$. Thus, the smaller integer is $2(2)+1=5$ and the larger is $2(2)+3=7$.

How many ways are there to pick two nonconsecutive numbers from the first 50 positive integers?

  (20 pts) 5. $1176$

Solution: There are $\left(\genfrac{}{}{0pt}{}{50}{2}\right)=\frac{50!}{2!(50-2)!}=\frac{(50)(49)(48!)}{(2)(48!)}=(25)(49)=1225$ ways to choose two numbers from the first 50 positive integers. As a set of two consecutive integers $\{x,x+1\}$ is determined uniquely by the smaller of the two values (i.e. by x), notice that there are $50-1=49$ possible choices for this smallest value (as $50+1=51$ is not in the first 50 positive integers). So, there are 49 ways to choose two consecutive numbers from the first 50 positive integers, and any of the other 1225 possible ways to choose two numbers from the first 50 positive integers other than these 49 ways results in a pair of nonconsecutive values. Thus, we reach our answer: There are $1225-49=1176$ ways to pick two nonconsecutive numbers from the first 50 positive integers.

Suppose the rational function $f(x)$ is given by $f(x)={\displaystyle \frac{5x+7}{2x+4}}$. Find $f^{-1}(x)$.

  (20 pts) 6. $f^{-1}(x)={\displaystyle \frac{7-4x}{2x-5}}$

Solution: Writing $y=f(x)$, we have $y=\frac{5x+7}{2x+4}$. To find $f^{-1}(x)$, we interchange $x$ and $y$ and solve for $y$:

So, replacing $y$, we have found $f^{-1}(x)={\displaystyle \frac{7-4x}{2x-5}}$.

A parallelogram with sides of length $x$ and ${\displaystyle \frac{3x+2}{2}}$ has perimeter 32. What is the length of the longer side?

  (20 pts) 7. $10$

Solution: The perimeter of a parallelogram with sides of length $x$ and ${\displaystyle \frac{3x+2}{2}}$ is $2x+2\cdot {\displaystyle \frac{3x+2}{2}}=5x+2$. Therefore, we may conclude that $5x+2=32$ and solve for $x$ to produce $x=6$. Correspondingly, the sides of the parallelogram are of length $x=6$ and ${\displaystyle \frac{3x+2}{2}}={\displaystyle \frac{3(6)+2}{2}}=10$. Thus, the longer side is of length 10.