Appearance
question:A certain item has an original price of 200 yuan. After two consecutive discounts of a%, the selling price becomes 148 yuan. Which of the following equations is correct? A: 200left(1+a%right)^{2}=148 B: 200left(1a%right)^{2}=148 C: 200left(12a%right)=148 D: 200left(1a^{2}%right)=148
answer:To solve this problem, let's break down the process of applying two consecutive discounts of a% on the original price of 200 yuan. 1. The first discount reduces the price by a%, so the price after the first discount is 200(1-a%). This is because a discount of a% means the item retains (100-a)% of its price, which can be written as 1-a%. 2. Applying the second discount of a% on the new price means we again multiply by (1-a%). Therefore, the price after the second discount is 200(1-a%) times (1-a%) = 200(1-a%)^2. 3. Given that the final selling price is 148 yuan, we can set up the equation as 200(1-a%)^2 = 148. Comparing this with the given options, we find that option B: 200(1-a%)^2=148 matches our equation. Therefore, the correct answer is boxed{text{B}}.
question:Find AX in the diagram where CX bisects angle ACB. [asy] import markers; real t=.64; pair A=(0,0); pair B=(4,3); pair C=(1,2); pair X=t*A+(1t)*B; draw(CABCX); label(A,A,SW); label(B,B,E); label(C,C,N); label(X,X,SE); label(34,.5*(B+X),SE); label(40,.5*(B+C),N); label(25,.5*(A+C),NW); [/asy]
answer:By the Angle Bisector Theorem, the ratio of the segments on one side of the angle bisector (CX) to the other side is equal to the ratio of the other two sides of the triangle: [ frac{AC}{AX} = frac{BC}{BX} ] Given (AC = 25), (BC = 40), and (BX = 34), we have: [ AX = frac{AC cdot BX}{BC} = frac{25 cdot 34}{40} ] Simplify the calculation: [ AX = frac{850}{40} = frac{85}{4} ] Thus, [ AX = boxed{frac{85}{4}} ]
question:A train that is 250 meters long is running at a certain speed. It takes 45 seconds to pass a station of 200 meters in length. What is the speed of the train in km/hour?
answer:To find the speed of the train, we first need to determine the total distance the train travels while passing the station. This distance is the length of the train plus the length of the station: Total distance = Length of train + Length of station Total distance = 250 meters + 200 meters Total distance = 450 meters The train takes 45 seconds to cover this distance. To find the speed in meters per second (m/s), we use the formula: Speed = Distance / Time Speed = 450 meters / 45 seconds Speed = 10 meters/second Now, we need to convert this speed from meters per second to kilometers per hour (km/h). There are 1000 meters in a kilometer and 3600 seconds in an hour. So, to convert m/s to km/h, we multiply by 3.6: Speed in km/h = Speed in m/s * 3.6 Speed in km/h = 10 m/s * 3.6 Speed in km/h = 36 km/h Therefore, the speed of the train is boxed{36} kilometers per hour.
question:Kiana, who is the youngest, has two older twin brothers. The product of their three ages is 180. What is the sum of their three ages? mathrm{(A)} 16qquad mathrm{(B)} 26qquad mathrm{(C)} 18qquad mathrm{(D)} 28qquad mathrm{(E)} 30
answer:1. **Identify factors of 180**: We start by noting that the product of the ages is 180 = 2^2 times 3^2 times 5. 2. **Assess twin ages and Kiana’s age**: The twins being older and of the same age suggests they could share common factors. The factors of 180 that are reasonable for ages and can be equal are 2, 3, and 9; since 9 being a square makes it suitable for twins. 3. **Explore possible combinations**: - If twins are each 9 years old, then the product 9 times 9 times K = 180 leads to K = 180 / 81 = 2.22, which is not possible as Kiana’s age must be an integer. - If twins are each 3 years old, then the product 3 times 3 times K = 180 leads to K = 180 / 9 = 20. This is plausible. 4. **Sum of their ages**: If the twins are each 3 years old and Kiana is 20 years old, the total sum is 3 + 3 + 20 = 26. 5. **Conclude with the final answer**: [ 26 ] The final answer is boxed{26}