💡First, I'll denote the time it takes for Danny to reach Steve's house as ( t ) minutes. Since Danny takes half the time compared to Steve, it takes Steve ( 2t ) minutes to reach Danny's house.Next, I'll consider the halfway point between their houses. Danny reaches this halfway point in ( frac{t}{2} ) minutes, and Steve reaches it in ( frac{2t}{2} = t ) minutes.According to the problem, Steve takes 16.5 minutes longer to reach the halfway point than Danny. This leads to the equation:[t - frac{t}{2} = 16.5]Simplifying the equation:[frac{2t}{2} - frac{t}{2} = 16.5][frac{t}{2} = 16.5]Multiplying both sides by 2 to solve for ( t ):[t = 16.5 times 2][t = 33]Therefore, it takes Danny 33 minutes to reach Steve's house.

💡First, I need to determine how much money Jennifer spent on each item. She spent 1/5 of her 150 on a sandwich, which is 30. She also spent 1/6 of her money on a museum ticket, amounting to 25. Additionally, she spent 1/2 of her money on a book, which totals 75.Next, I will add up the amounts she spent on each item to find the total expenditure: 30 (sandwich) + 25 (museum ticket) + 75 (book) = 130.Finally, to find out how much money Jennifer has left, I will subtract the total amount she spent from her original amount: 150 - 130 = 20.

💡First, I need to determine the number of days spent on jury selection. Let's denote this number as ( x ).The trial lasts 4 times as long as jury selection, so the trial duration is ( 4x ) days.The jury deliberation is equivalent to 6 full days, but since they spend 16 hours each day in deliberation, I need to convert these hours into days. A full day is typically 24 hours, so 6 full days amount to ( 6 times 24 = 144 ) hours. Dividing this by the 16 hours spent each day in deliberation gives ( frac{144}{16} = 9 ) days of deliberation.The total number of days John spends on jury duty is the sum of the days spent on jury selection, the trial, and the deliberation, which is given as 19 days. Therefore, the equation is:[x + 4x + 9 = 19]Combining like terms, we get:[5x + 9 = 19]Subtracting 9 from both sides:[5x = 10]Dividing both sides by 5:[x = 2]So, jury selection takes 2 days.

💡Connor is choosing a number between 50 and 100 to use as a password. He tells his friend that exactly three of the following four statements about it are true:1. It is a square number.2. It is even.3. It is divisible by 11.4. The tens digit is 6.To determine the last digit of Connor’s password number, we need to evaluate each statement and identify which one is false while the other three are true.### Step-by-Step Analysis:1. **Identify the range and constraints**: - The number is between 50 and 100. - It has three digits, so it must be between 100 and 999. However, since Connor is choosing a number between 50 and 100, we can narrow it down to 100 (since 50 is a two-digit number).2. **Evaluate the statements**: - **Statement (1): It is a square number**. - The square numbers between 50 and 100 are 64 (8²) and 81 (9²). 100 is also a square number but is at the upper limit. - **Statement (2): It is even**. - Among the square numbers identified, 64 is even, while 81 is odd. - **Statement (3): It is divisible by 11**. - Checking divisibility by 11 for the square numbers: - 64 ÷ 11 ≈ 5.818 (not divisible) - 81 ÷ 11 ≈ 7.364 (not divisible) - 100 ÷ 11 ≈ 9.091 (not divisible) - Therefore, none of the square numbers between 50 and 100 are divisible by 11. - **Statement (4): The tens digit is 6**. - For the square numbers: - 64: Tens digit is 6. - 81: Tens digit is 8. - 100: Tens digit is 0.3. **Determine which statement is false**: - If we assume that statement (3) is false (since none of the square numbers are divisible by 11), then statements (1), (2), and (4) must be true. - The only number that satisfies being a square number, even, and having a tens digit of 6 is 64.4. **Conclusion**: - The number 64 meets the criteria where three statements are true, and one (divisibility by 11) is false. - The last digit of 64 is 4.Thus, the last digit of Connor’s password number is **4**.

💡First, I recognize that Joe took four equally weighted tests and was allowed to drop his lowest score of 20. After dropping this score, his average improved to 40.To find the total sum of the three remaining test scores, I multiply the new average by the number of remaining tests:40 * 3 = 120.Next, I add the dropped score of 20 to this total to find the sum of all four test scores:120 + 20 = 140.Finally, to determine the original average before dropping the lowest score, I divide the total sum of all four test scores by the number of tests:140 / 4 = 35.Therefore, Joe's average test score before dropping the lowest score was 35.

💡First, I'll denote the monthly incomes of P, Q, and R as P, Q, and R respectively.From the given information, the average monthly income of Q and R is Rs. 6250. This can be expressed as:(Q + R) / 2 = 6250Multiplying both sides by 2 gives:Q + R = 12500Next, the average monthly income of P and R is Rs. 5200. This can be written as:(P + R) / 2 = 5200Multiplying both sides by 2 gives:P + R = 10400We are also given that the monthly income of P is Rs. 4000. Substituting this value into the second equation:4000 + R = 10400Solving for R:R = 10400 - 4000R = 6400Now, substitute the value of R back into the first equation to find Q:Q + 6400 = 12500Solving for Q:Q = 12500 - 6400Q = 6100Finally, to find the average monthly income of P and Q:(P + Q) / 2 = (4000 + 6100) / 2 = 10100 / 2 = 5050Therefore, the average monthly income of P and Q is Rs. 5050.