AMC 8 Problem Solutions¶
1. Problem: Geometry of a Square - Solution¶
What is the area of a square with a perimeter of \( 24 \) units?
Solution: The perimeter of a square is given by \( 4s \), where \( s \) is the length of one side. If the perimeter is \( 24 \), we have:
The area of the square is \( s^2 \), so:
Thus, the area of the square is \( \boxed{36} \) square units.
2. Problem: Coin Flip - Solution¶
A fair coin is flipped three times. What is the probability of getting exactly two heads?
Solution:
There are \( 2^3 = 8 \) total outcomes when flipping the coin three times. To get exactly two heads, we can have the following outcomes: \( HHT, HTH, THH \). There are \( 3 \) favorable outcomes.
Thus, the probability is:
So, the probability of getting exactly two heads is \( \boxed{\frac{3}{8}} \).
3. Problem: Average of a Set - Solution¶
The average of five numbers is \( 20 \). If one of the numbers is removed, the average of the remaining four numbers is \( 15 \). What is the number that was removed?
Solution: Let the sum of the five numbers be \( S \). Since the average of five numbers is \( 20 \), we have:
Let the number removed be \( x \). The sum of the remaining four numbers is \( 100 - x \), and their average is \( 15 \):
Thus, the number that was removed is \( \boxed{40} \).
4. Problem: Rectangle Area - Solution¶
A rectangle has a length of \( 8 \) units and a width of \( 5 \) units. What is the length of the diagonal of the rectangle?
Solution: The length of the diagonal \( d \) of a rectangle can be found using the Pythagorean theorem:
Thus, the length of the diagonal is \( \boxed{\sqrt{89}} \) units.
5. Problem: Digit Sum* - Solution¶
What is the sum of the digits of \( 2023 \)?
Solution: To find the sum of the digits of \( 2023 \), we add:
Thus, the sum of the digits is \( \boxed{7} \).
6. Problem: Time Problem - Solution¶
A clock shows 3:15. What is the angle between the hour hand and the minute hand?
Solution: At 3:00, the hour hand is at \( 90^\circ \) (3 o'clock). The minute hand moves \( 6^\circ \) per minute, so at 3:15, the minute hand is at:
The hour hand moves \( 0.5^\circ \) per minute. By 3:15, the hour hand has moved:
Thus, the angle between the hour and minute hands is:
Therefore, the angle between the hands is \( \boxed{82.5^\circ} \).
7. Problem: Factors - Solution¶
How many positive divisors does \( 36 \) have?
Solution: The prime factorization of \( 36 \) is:
[ 36 = 2^2 \times 3^2
] To find the number of divisors, use the formula \( (e_1+1)(e_2+1) \dots (e_n+1) \), where \( e_1, e_2, \dots, e_n \) are the exponents in the prime factorization. For \( 36 = 2^2 \times 3^2 \), we have:
Thus, \( 36 \) has \( \boxed{9} \) divisors.
8. Problem: Percentages - Solution¶
A shirt originally costs \( 40 \) dollars. After a \( 20\% \) discount, what is the price of the shirt?
Solution: A \( 20\% \) discount means the price is reduced by \( 20\% \) of \( 40 \):
Thus, the discounted price is:
Therefore, the price of the shirt is \( \boxed{32} \) dollars.
9. Problem: Volume of a Box - Solution¶
A rectangular box has dimensions \( 4 \), \( 5 \), and \( 6 \). What is the volume of the box?
Solution: The volume of a rectangular box is the product of its dimensions:
Thus, the volume of the box is \( \boxed{120} \) cubic units.
10. Problem: Arithmetic Sequence - Solution¶
In an arithmetic sequence, the first term is \( 2 \) and the common difference is \( 5 \). What is the 10th term of the sequence?
Solution: The general formula for the \( n \)-th term of an arithmetic sequence is:
Substitute \( a_1 = 2 \), \( d = 5 \), and \( n = 10 \):
Thus, the 10th term is \( \boxed{47} \).