Q. A bakery sells cakes and cookies. If the total number of items is 150 and the number of cakes is twice the number of cookies, how many cakes are there?
A.
100
B.
50
C.
75
D.
25
Solution
Let the number of cookies be x. Then the number of cakes is 2x. The equation is x + 2x = 150. Solving gives 3x = 150, so x = 50. Therefore, cakes = 2x = 100.
Q. A bookstore sells novels and magazines. If the total number of books is 200 and the number of novels is 3 times the number of magazines, how many novels are there?
A.
150
B.
120
C.
75
D.
100
Solution
Let the number of magazines be x. Then the number of novels is 3x. The equation is x + 3x = 200. Solving gives 4x = 200, so x = 50. Therefore, novels = 3x = 150.
Q. A building is 20 meters tall. If the angle of elevation from a point on the ground 10 meters away from the base of the building is θ, what is tan(θ)?
Q. A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?
A.
4 cm
B.
3 cm
C.
2 cm
D.
5 cm
Solution
The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron's formula. The radius r = A/s. The area is 24 cm², so r = 24/12 = 2 cm.
Q. A circle is inscribed in a triangle. If the sides of the triangle are 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?
A.
3 cm
B.
4 cm
C.
5 cm
D.
6 cm
Solution
The radius r of the inscribed circle can be found using the formula r = A/s, where A is the area and s is the semi-perimeter. The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron's formula: A = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12*5*4*3] = √720 = 12√5. Thus, r = A/s = (12√5)/12 = √5 cm, which is approximately 2.24 cm.
Q. A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?
A.
3 cm
B.
4 cm
C.
5 cm
D.
6 cm
Solution
The area A of the triangle can be calculated using Heron's formula. The semi-perimeter s = (7 + 8 + 9) / 2 = 12. The area A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)) = √(12*5*4*3) = 12√5. The radius r = A/s = (12√5)/12 = √5 cm, which is approximately 4 cm.
Q. A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?
A.
3 square units
B.
4 square units
C.
5 square units
D.
6 square units
Solution
The area of the triangle is 24 square units (using Heron's formula). The semi-perimeter is 12 units. The radius r = Area/semi-perimeter = 24/12 = 2 units.
Q. A circle is inscribed in a triangle. What is the radius of the incircle if the triangle has sides of lengths 7, 8, and 9 units?
A.
4 square units
B.
3 square units
C.
5 square units
D.
2 square units
Solution
The area of the triangle can be calculated using Heron's formula. The semi-perimeter s = (7+8+9)/2 = 12. The area A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)) = √(12*5*4*3) = 12√5. The radius r = A/s = 12√5/12 = √5. The radius is approximately 3 square units.
Q. A company produces pens and pencils. If the total production is 500 items and the number of pens is 4 times the number of pencils, how many pens are produced?
A.
400
B.
100
C.
200
D.
300
Solution
Let the number of pencils be x. Then the number of pens is 4x. The equation is x + 4x = 500. Solving gives 5x = 500, so x = 100. Therefore, pens = 4x = 400.
Q. A concert hall has 300 seats. If the number of reserved seats is 50 more than the number of general admission seats, how many reserved seats are there?
A.
125
B.
175
C.
100
D.
150
Solution
Let the number of general admission seats be x. Then the number of reserved seats is x + 50. The equation is x + (x + 50) = 300. Solving gives 2x + 50 = 300, so 2x = 250, x = 125. Therefore, reserved seats = x + 50 = 175.
