IAR - Illinois Assessment of Readiness Math Concepts and Reasoning Questions and Answers 1 — Questions and Answers
Question 1: A firefighter leans a 25-foot ladder against a building. The base of the ladder is 7 feet from the building's wall. How high up the wall does the ladder reach?
- 18 feet
- 24 feet (Correct answer)
- 26 feet
- 32 feet
Correct answer: 24 feet
This scenario forms a right triangle with the ladder as the hypotenuse (c=25 ft), the distance from the wall as one leg (a=7 ft), and the height on the wall as the other leg (b). Using the Pythagorean theorem, a² + b² = c², we can solve for b: 7² + b² = 25². This simplifies to 49 + b² = 625. Subtracting 49 from both sides gives b² = 576. Taking the square root of 576 gives b = 24 feet.
Question 2: The following dataset represents the number of minutes 7 students spent on homework: {45, 50, 52, 55, 60, 65, 150}. Which of the following statements is true about the measures of central tendency for this dataset?
- The median is greater than the mean.
- The mean is greater than the median. (Correct answer)
- The mean and median are equal.
- The mode is the most accurate measure of center.
Correct answer: The mean is greater than the median.
The value 150 is an outlier, which significantly affects the mean but not the median. The median is the middle value of the ordered set, which is 55. The mean is the sum of the values (477) divided by the count (7), which is approximately 68.14. Because the outlier pulls the average up, the mean is greater than the median.
Question 3: A movie theater sold 200 tickets for a total of $2,050. Adult tickets cost $12 each and child tickets cost $8 each. If 'a' represents the number of adult tickets and 'c' represents the number of child tickets, which system of equations correctly models this situation?
- a + c = 2050; 12a + 8c = 200
- a - c = 200; 12a + 8c = 2050
- a + c = 200; 12a + 8c = 2050 (Correct answer)
- 12a + 8c = 200; a + c = 2050
Correct answer: a + c = 200; 12a + 8c = 2050
The problem can be modeled with two separate equations. The first equation represents the total number of tickets sold: the number of adult tickets (a) plus the number of child tickets (c) equals 200. The second equation represents the total revenue: the price per adult ticket times the number of adult tickets (12a) plus the price per child ticket times the number of child tickets (8c) equals the total revenue of $2,050.
Question 4: A map has a scale where 1 inch represents 25 miles. If the distance between two cities on the map is 5.2 inches, what is the actual distance between them?
- 4.81 miles
- 30.2 miles
- 125.5 miles
- 130 miles (Correct answer)
Correct answer: 130 miles
This problem can be solved by setting up a proportion: (1 inch / 25 miles) = (5.2 inches / x miles). To find the unknown distance 'x', you can cross-multiply: 1 * x = 25 * 5.2. The calculation 25 * 5.2 equals 130. Therefore, the actual distance between the cities is 130 miles.
Question 5: The expression 5000(1 + 0.03/4)^(4t) is used to calculate the value of an investment after 't' years. Which of the following best describes the meaning of the value 0.03/4 in this context?
- The number of times the interest is compounded per year.
- The total interest earned after t years.
- The interest rate applied during each compounding period. (Correct answer)
- The annual interest rate.
Correct answer: The interest rate applied during each compounding period.
In the standard compound interest formula, A = P(1 + r/n)^(nt), 'r' is the annual interest rate and 'n' is the number of times the interest is compounded per year. Here, 0.03 is the annual rate (r) and 4 is the number of compounding periods (n). The term r/n, or 0.03/4, represents the interest rate that is applied for each compounding period (quarterly, in this case).
Question 6: A cylindrical water tank has a radius of 5 feet and a height of 12 feet. What is the approximate volume of water the tank can hold? (Use π ≈ 3.14)
- 942 cubic feet (Correct answer)
- 188.4 cubic feet
- 376.8 cubic feet
- 3,768 cubic feet
Correct answer: 942 cubic feet
The formula for the volume of a cylinder is V = πr²h. Substitute the given values into the formula: V ≈ 3.14 * (5 ft)² * (12 ft). First, square the radius: 5² = 25. Then, multiply the values: V ≈ 3.14 * 25 * 12, which equals 942. The volume is approximately 942 cubic feet.
