AMC 10 Problem Solutions¶
1. Problem: Simple Algebra - Solutions¶
What is the value of \( x \) if \( 3x + 7 = 19 \)?
Solution:
Thus, \( x = \boxed{4} \).
2. Problem: Geometry of a Triangle - Solutions¶
The perimeter of an equilateral triangle is \( 24 \). What is the length of one side?
Solution: The perimeter of an equilateral triangle is \( 3s \), where \( s \) is the side length. Given that the perimeter is \( 24 \):
Thus, the length of one side is \( \boxed{8} \).
3. Problem: Percentage Increase - Solutions¶
A store originally sells a product for \( 40 \) dollars. The price is increased by \( 20\% \). What is the new price?
Solution: The increase is \( 20\% \) of \( 40 \), which is:
Thus, the new price is: [ 40 + 8 = 48 ]
The new price is \( \boxed{48} \) dollars.
4. Problem: Factors - Solutions¶
How many positive divisors does \( 60 \) have?
Solution: The prime factorization of \( 60 \) is:
The number of divisors is given by \( (2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12 \). Thus, \( 60 \) has \( \boxed{12} \) divisors.
5. Problem: Time Problem - Solutions¶
A clock shows \( 4:30 \). What is the measure of the smaller angle between the hour and minute hands?
Solution: The minute hand at 30 minutes is \( 180^\circ \) from the 12 o'clock position. The hour hand at 4:30 is halfway between the 4 and 5 positions. Since each hour represents \( 30^\circ \), the hour hand is \( 4 \times 30 + 15 = 135^\circ \) from the 12 o'clock position.
The difference between the hands is:
Thus, the smaller angle is \( \boxed{45^\circ} \).
6. Problem: Consecutive Integers - Solutions¶
The sum of three consecutive integers is \( 33 \). What are the integers?
Solution: Let the integers be \( x-1 \), \( x \), and \( x+1 \). Their sum is:
Thus, the integers are \( 10, 11, 12 \), and the answer is \( \boxed{10, 11, 12} \).
7. Problem: Arithmetic Sequence - Solutions¶
In an arithmetic sequence, the first term is \( 5 \), and the common difference is \( 3 \). What is the sum of the first 20 terms?
Solution: The sum of the first \( n \) terms of an arithmetic sequence is:
The 20th term is \( a_{20} = a_1 + 19d = 5 + 19 \times 3 = 62 \). Thus, the sum is:
Thus, the sum of the first 20 terms is \( \boxed{670} \).
8. Problem: Probability - Solutions¶
Two dice are rolled. What is the probability that the sum of the numbers rolled is \( 8 \)?
Solution: There are \( 6 \times 6 = 36 \) possible outcomes. The combinations that sum to \( 8 \) are:
There are 5 favorable outcomes, so the probability is:
Thus, the probability is \( \boxed{\frac{5}{36}} \).
9. Problem: Quadratic Equation - Solutions¶
Solve the equation \( x^2 - 5x + 6 = 0 \).
Solution: Factor the quadratic:
Thus, \( x = 2 \) or \( x = 3 \). The solutions are \( \boxed{2} \) and \( \boxed{3} \).
10. Problem: Right Triangle - Solutions¶
In a right triangle, the legs are \( 6 \) and \( 8 \). What is the length of the hypotenuse?
Solution: Using the Pythagorean theorem:
Thus, the hypotenuse is \( \boxed{10} \).
11. Problem: Circles and Geometry - Solutions¶
Two concentric circles have radii of \( 3 \) and \( 5 \). What is the area of the ring-shaped region between the two circles?
Solution: The area of the ring is the difference between the areas of the larger and smaller circles:
[ \text{Area} = \pi \times 5^2 - \p i \times 3^2 = 25\pi - 9\pi = 16\pi ] Thus, the area of the ring is \( \boxed{16\pi} \).
12. Problem: Sum of Angles in a Polygon - Solutions¶
What is the sum of the interior angles of a pentagon?
Solution: The sum of the interior angles of a polygon with \( n \) sides is given by:
For a pentagon (\( n = 5 \)):
Thus, the sum of the interior angles is \( \boxed{540^\circ} \).