Practice Test 5 - DAT Quantitative Reasoning

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. In a dental survey, $30\%$ of patients have cavities, $20\%$ have gum disease, and $8\%$ have both conditions. If a patient is selected at random and has gum disease, what is the probability that this patient does not have cavities?
2. Bacteria in a dental waterline grows by factor $1.5$ hourly; if $N(t)=200(1.5)^t$, what is $N(3)$?
3. If $\frac{2^x + 2^{-x}}{2^x - 2^{-x}} = \frac{5}{3}$, what is the value of $2^x$?
4. An autoclave load cools 3 °C per minute on the counter and 8 °C per minute under a fan. After cooling on the counter for 4 minutes, the load is moved under the fan and reaches 40 °C exactly 2 minutes later. What was the starting temperature?
5. If $m$ and $n$ are integers with $0 < m < n$, compare the quantities: Quantity A: $\frac{m}{n} + \frac{n}{m}$ Quantity B: $\frac{m^2 + n^2}{mn}$
6. A dart is thrown at the target shown below and lands at a random location within the large square. Given that the dart lands inside the inscribed circle, what is the probability that it also lands inside the small central square (side length equal to the circle's radius)?
7. In a dental clinic, the probability that a patient needs a filling is $0.3$, needs a crown is $0.15$, and needs both a filling and a crown is $0.08$. Given that a randomly selected patient needs a crown, what is the probability that this patient also needs a filling?
8. Bacteria on a mouthguard increases $20\%$ per hour; if $N(t)=500(1.2)^t$, what is $N(2)$?
9. A car travels 60 miles at 30 mph and returns the same route at 20 mph. What is the average speed for the trip?
10. For $x > 0$, let $f(x) = \frac{x^3 + 8}{x^2 + 4}$ and $g(x) = x + \frac{4}{x}$. Compare the quantities: Quantity A: $f(2)$ Quantity B: $g(2)$
11. Given that $n$ is a positive integer, compare the quantities: Quantity A: $\frac{n!}{(n-2)!}$ Quantity B: $n^2 - n$
12. A pharmaceutical company produces two types of dental anesthetic. Type A contains 2% lidocaine and Type B contains 4% lidocaine. A dentist needs 50 mL of a 3.2% lidocaine solution. If she mixes Type A and Type B to achieve this concentration, and the cost per mL is $0.15 for Type A and $0.22 for Type B, what is the total cost of the mixture?
13. Market Analysis: Company A has profit $\$40{,}000$ and grows by $50\%$. Company B has profit $\$55{,}000$ and grows by $30\%$. Quantity A is Company A's projected profit after growth, and Quantity B is Company B's projected profit after growth. Determine whether Quantity A is greater, Quantity B is greater, or both are equal.
14. A dental supply company offers a bulk discount system. For orders up to 100 units, the price is $12 per unit. For orders from 101 to 300 units, the first 100 units cost $12 each and additional units cost $9.50 each. For orders over 300 units, the first 100 cost $12 each, the next 200 cost $9.50 each, and additional units cost $8.25 each. What is the total cost for an order of 450 units?
15. In a dental school, $70\%$ of students pass the written exam and $80\%$ pass the practical exam. If $60\%$ pass both exams, what is the probability that a randomly selected student passes at least one exam?
16. In a course, quizzes count 40\%, the midterm 25\%, and the final exam 35\% of the semester grade. A student's quiz average is 82 and the midterm score is 74. What score on the final exam will give a semester average of 80?
17. The table below compares two data sets. Based on the table shown, which statement must be true?
18. Distance Traveled: Route A is $72$ miles at $36$ mph. Route B is $90$ miles at $45$ mph. Quantity A is the travel time for Route A, and Quantity B is the travel time for Route B. Determine whether Quantity A is greater, Quantity B is greater, or both are equal.
19. Investment Returns: Option A invests $\$7{,}500$ at $6\%$ simple interest for 2 years. Option B invests $\$9{,}000$ at $5\%$ simple interest for 2 years. Quantity A is the interest earned in Option A, and Quantity B is the interest earned in Option B. Determine whether Quantity A is greater, Quantity B is greater, or both are equal.
20. Two fair dice are rolled. Given that at least one die shows 3, what is the probability the sum is at least 5?

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