💡First, I need to determine the individual work rates of the man and his son.The man can complete the work in 6 days, so his work rate is ( frac{1}{6} ) of the work per day.The son can also complete the work in 6 days, so his work rate is ( frac{1}{6} ) of the work per day.Next, I'll add their work rates to find the combined work rate when they work together.( frac{1}{6} + frac{1}{6} = frac{2}{6} = frac{1}{3} ) of the work per day.Finally, to find out how many days it takes for them to complete the work together, I'll take the reciprocal of the combined work rate.( frac{1}{frac{1}{3}} = 3 ) days.

💡First, I need to determine the initial amount of money before any spending occurred. I know that after spending 30% of the initial amount, 350 remained. This means that 350 represents 70% of the initial amount because 100% minus the spent 30% equals 70%.To find the initial amount, I can set up the equation where 70% of the initial amount equals 350. Converting 70% to a decimal gives 0.70. So, the equation becomes:0.70 × Initial Amount = 350To solve for the Initial Amount, I divide both sides of the equation by 0.70:Initial Amount = 350 / 0.70Calculating this gives:Initial Amount = 500Therefore, the initial amount of money was 500.

💡Okay, so I have this quadratic equation: kx² - 2kx + 4 = 0. The question is asking for the value of k such that the equation has two equal real roots. Hmm, I remember that for a quadratic equation ax² + bx + c = 0, the discriminant determines the nature of the roots. The discriminant is b² - 4ac. If the discriminant is zero, the equation has exactly one real root, which means two equal real roots.Alright, so let's identify a, b, and c in the given equation. Here, a is k, b is -2k, and c is 4. Plugging these into the discriminant formula, we get:Discriminant = (-2k)² - 4 * k * 4Let me compute that step by step. First, (-2k) squared is 4k². Then, 4 times k times 4 is 16k. So, the discriminant becomes:4k² - 16kAnd since we want two equal real roots, the discriminant must be zero. So:4k² - 16k = 0Hmm, okay, I can factor out a common term here. Both terms have a 4k, so factoring that out:4k(k - 4) = 0So, setting each factor equal to zero gives us:4k = 0 or k - 4 = 0Which simplifies to:k = 0 or k = 4Wait a second, but if k is zero, then the original equation becomes 0x² - 0x + 4 = 0, which simplifies to 4 = 0. That doesn't make sense because 4 isn't equal to zero. So, k can't be zero because it would make the equation invalid. Therefore, the only valid solution is k = 4.Let me double-check that. If k is 4, then the equation becomes 4x² - 8x + 4 = 0. Let's compute the discriminant for this specific case:Discriminant = (-8)² - 4 * 4 * 4 = 64 - 64 = 0Yes, that's correct. The discriminant is zero, so there are two equal real roots. Therefore, k must be 4.I think that's it. So, the answer should be D: 4.

💡First, I need to understand the data presented in the chart. The x-axis represents the number of home runs, and the y-axis represents the number of top hitters who achieved that number of home runs.Looking at the chart, I can identify the following data points:- 5 home runs: 5 players- 9 home runs: 3 players- 7 home runs: 4 players- 11 home runs: 2 playersNext, I calculate the total number of home runs by multiplying the number of home runs by the number of players who achieved that number and then summing these products:[5 times 5 + 9 times 3 + 7 times 4 + 11 times 2 = 25 + 27 + 28 + 22 = 102 text{ total home runs}]Then, I determine the total number of players by adding the number of players for each category:[5 + 3 + 4 + 2 = 14 text{ players}]Finally, I calculate the mean (average) number of home runs by dividing the total number of home runs by the total number of players:[frac{102}{14} = boxed{7.29}]

💡Okay, so Josie got 50 as a gift. She wants to buy two cassette tapes and a headphone set. Let's see, each cassette tape costs 9, and the headphone set is 25. I need to figure out how much money she'll have left after buying all these things.First, I should calculate the total cost of the cassette tapes. Since she wants two of them and each is 9, I can multiply the number of tapes by the cost per tape. So, 2 times 9 equals... let's see, 2 times 9 is 18. Okay, so the cassette tapes will cost her 18 in total.Next, there's the headphone set, which is 25. I need to add that to the cost of the tapes to find out how much she's spending altogether. So, 18 for the tapes plus 25 for the headphones. Let me add those together: 18 plus 25 is 43. So, the total amount she's going to spend is 43.Now, Josie started with 50. To find out how much she'll have left after her purchases, I subtract the total amount she spent from the amount she received as a gift. That means 50 minus 43. Let me do that subtraction: 50 minus 43 equals 7. So, Josie will have 7 left after buying the cassette tapes and the headphone set.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Two cassette tapes at 9 each: 2 times 9 is indeed 18. The headphone set is 25, so adding that to the tapes: 18 plus 25 is 43. Subtracting that from her 50 gift: 50 minus 43 is 7. Yep, that seems correct.I think I've got it right. Josie will have 7 left after her purchases.

💡First, I need to expand the expression ((3x + 2)(2x - 3)) to find the values of (a), (k), and (n).I'll start by applying the distributive property (FOIL method) to expand the expression:[(3x + 2)(2x - 3) = 3x cdot 2x + 3x cdot (-3) + 2 cdot 2x + 2 cdot (-3)]Calculating each term:[3x cdot 2x = 6x^2][3x cdot (-3) = -9x][2 cdot 2x = 4x][2 cdot (-3) = -6]Combining all the terms:[6x^2 - 9x + 4x - 6 = 6x^2 - 5x - 6]Now, comparing this with the general form (ax^2 + kx + n), I can identify:[a = 6,quad k = -5,quad n = -6]Finally, I'll calculate (a - n + k):[a - n + k = 6 - (-6) + (-5) = 6 + 6 - 5 = 7]