UND MATHEMATICS TRACK MEET Team Test 1

University of North Dakota Grades 7/8
January 12, 2026
School Team Name
Calculators are allowed. Solutions

• 1.

  What is the side length of a square with the same area as a triangle with base $10$ and height $6$? Enter your answer as a decimal rounded to three decimal places.

    (20 pts) 1. 5.477

  Solution: We have $s^2=\frac{1}{2}\cdot 10\cdot 6=30$, and so $s=\sqrt{30}\approx 5.477$.

• 2.

  What is the area of a circle with circumference $7\pi$? Enter your answer as a decimal rounded to three places.

    (20 pts) 2. $38.485$

  Solution: Since $2\pi r=7\pi$, we have $r=7/2$, and so $\pi r^2=\frac{49\pi }{4}\approx 38.485$

• 3.

  Find positive numbers $r$ and $s$ with $rs=31$ and $r+s=12$. Enter your answers as decimals rounded to three places.

    (20 pts) 3. $\{r,s\}\approx \{3.764,8.236\}$

  Solution: Since $(x-r)(x-s)=x^2-12x+31$, the quadratic formula gives $\{r,s\}\approx \{3.764,8.236\}$.

• 4.

  In a class your grade is determined by the average of five exam scores. If the average of your first three exams is 73%, what must your average on the last two exams be to get at least an 80% in the class?

    (20 pts) 4. $90.5\%$

  Solution: If $a$, $b$, $c$ are the first three exam scores and $d$, $e$ are the last two, we have $(a+b+c)/3=73$ and $(a+b+c+d+e)/5=80$. Solve to get $d+e=181$, and so $(c+d)/2=90.5$.

• 5.

  The value of a new car decreases by $10\%$ each year. If a new car costs $35,000, what will its value be after 5 years?

    (20 pts) 5. $20667.15

  Solution: $\mathrm{\$}35000\cdot {(.9)}^5=\mathrm{\$}20667.15$.

• 6.

  What is the greatest common divisor of $175$ and $280$?

    (20 pts) 6. 35

  Solution: We have $175=7\cdot 5^2$ and $280=2^3\cdot 5\cdot 7$ so the GCD is $5\cdot 7=35$.

• 7.

  Find the positive solution to $3^{x^2+1}=9^{2x+1}$. Enter your answer as a decimal rounded to three places.

    (20 pts) 7. 4.236

  Solution: We have $3^{x^2+1}=3^{4x+2}$, and so $x^2-4x-1=0$. The solutions are $2\pm \sqrt{5}$. The positive solution is $2+\sqrt{5}\approx 4.236$.

• 8.

  Suppose the price of an item is increased by $20$% and then later the new price is decreased by $30$%. Compared to the original price, what percentage decrease does the final price represent?

    (20 pts) 8. 16%

  Solution: If $P$ is the original price, the final price is $(.7)(1.2)P=(.84)P$. So the price has decreased by $16$%.

• 9.

  Solve the system of equations

    (20 pts) 9. $(x,y)=(1138,-2401)$

  Use Gaussian elimination or substitution to find $x=1138$, $y=-2401$.

• 10.

  Suppose $a$ and $b$ are the roots of $x^2+10x-3=0$. What is $a^{2}b+a{b}^2$ ?

    (20 pts) 10. 30

  Solution: We have $a^{2}b+a{b}^2=ab(a+b)=(-3)\cdot (-10)=30$. Alternatively, $a$ and $b$ could be calculated directly, etc.