💡Okay, so I've got this problem here about Caleb and his dad fishing. Let me try to understand what's going on. Caleb caught some trouts, and his dad caught three times as much as Caleb. But wait, it also says his dad caught 4 more trouts compared to Caleb. Hmm, that seems a bit confusing at first glance. Let me break it down step by step.First, let's define what we're looking for. We need to find out how many trouts Caleb caught. Let's call the number of trouts Caleb caught "C." So, C represents Caleb's catch.Now, according to the problem, Caleb's dad caught three times as much as Caleb. So, if Caleb caught C trouts, his dad caught 3 times that, which would be 3C trouts.But here's where it gets a bit tricky. The problem also says that his dad caught 4 more trouts compared to Caleb. So, if Caleb caught C trouts, then his dad caught C + 4 trouts.Wait a minute, so we have two expressions for the number of trouts Caleb's dad caught: one is 3C, and the other is C + 4. That means these two expressions should be equal because they both represent the same quantity—the number of trouts Caleb's dad caught.So, we can set up an equation:3C = C + 4Now, let's solve for C. To isolate C, we'll subtract C from both sides of the equation:3C - C = C + 4 - CThis simplifies to:2C = 4Now, to find the value of C, we'll divide both sides by 2:2C / 2 = 4 / 2Which simplifies to:C = 2So, Caleb caught 2 trouts.Let me double-check to make sure this makes sense. If Caleb caught 2 trouts, then his dad caught three times that, which would be 6 trouts. According to the problem, his dad also caught 4 more trouts than Caleb. So, if Caleb caught 2, adding 4 more would give us 6. That matches up perfectly.Wait, so both expressions give us the same number for his dad's catch, which is 6 trouts. That checks out. So, it seems like the solution is consistent and makes sense.I think I've got it. Caleb caught 2 trouts, and his dad caught 6 trouts, which is three times as much as Caleb and also 4 more trouts than Caleb. Everything adds up correctly.Just to recap:- Let C be the number of trouts Caleb caught.- Caleb's dad caught 3C trouts.- Caleb's dad also caught C + 4 trouts.- Setting these equal: 3C = C + 4.- Solving for C: 2C = 4 → C = 2.- Verification: 3 * 2 = 6 and 2 + 4 = 6, which matches.So, the answer is that Caleb caught 2 trouts.

💡First, I need to determine how many hours Doris works each week. She babysits for 3 hours each weekday, which totals 15 hours per week, and 5 hours on Saturday. This gives her a total of 20 hours per week.Next, I'll calculate her weekly earnings by multiplying the number of hours she works by her hourly rate of 20. This results in 400 earned each week.Finally, to find out how many weeks it will take her to earn at least 1200, I'll divide her monthly expense by her weekly earnings. This calculation shows that Doris needs 3 weeks to earn enough to cover her monthly expenses.

💡First, I need to determine the width of the rectangle. I know that the length of the rectangle is twice its width, so if I let ( w ) represent the width, the length will be ( 2w ).Next, I'll calculate the perimeter of the rectangle. The formula for the perimeter of a rectangle is ( P = 2(text{length} + text{width}) ). Substituting the known values, the perimeter becomes:[P = 2(2w + w) = 6w]Now, I'll find the circumference of the circle with a radius of 3 units. The formula for the circumference of a circle is ( C = 2pi times text{radius} ). Plugging in the radius:[C = 2pi times 3 = 6pi]Since the perimeter of the rectangle is equal to the circumference of the circle, I can set up the equation:[6w = 6pi]To solve for ( w ), I'll divide both sides of the equation by 6:[w = pi]Therefore, the width of the rectangle is ( pi ) units.

💡To convert (3589_{10}) to base 7, I'll start by determining the largest power of 7 that fits into 3589. Calculating the powers of 7, I find that (7^4 = 2401) is the highest power less than 3589.Next, I'll see how many times 2401 fits into 3589. It fits once, leaving a remainder of 1188. Moving to the next lower power of 7, which is (7^3 = 343), I find that it fits 3 times into 1188, leaving a remainder of 159.Then, I'll use (7^2 = 49) and determine that it fits 3 times into 159, leaving a remainder of 12. Finally, using (7^1 = 7), it fits once into 12, leaving a remainder of 5, which is (5 cdot 7^0).Putting it all together, (3589_{10}) is expressed as (1 cdot 7^4 + 3 cdot 7^3 + 3 cdot 7^2 + 1 cdot 7^1 + 5 cdot 7^0), which corresponds to the base 7 number (13315_7).

💡First, I need to calculate the total sum of test scores for each group of students.For the first group of 10 students, I will add up all their scores: 90 + 85 + 88 + 92 + 80 + 94 + 89 + 91 + 84 + 87. This gives a total of 880 marks.Next, for the second group of 15 students, I will add their scores: 85 + 80 + 83 + 87 + 75 + 89 + 84 + 86 + 79 + 82 + 77 + 74 + 81 + 78 + 70. This totals 1240 marks.Then, for the third group of 25 students, I will sum their scores: 40 + 62 + 58 + 70 + 72 + 68 + 64 + 66 + 74 + 76 + 60 + 78 + 80 + 82 + 84 + 86 + 88 + 61 + 63 + 65 + 67 + 69 + 71 + 73 + 75. This adds up to 1720 marks.After obtaining the total scores for each group, I will add them all together: 880 + 1240 + 1720, which equals 3840 marks.Finally, to find the average test score for the entire class of 50 students, I will divide the total sum of scores by the number of students: 3840 ÷ 50. This results in an average of 76.8 marks.

💡Let's denote the total inheritance amount as ( x ).1. **Federal Taxes**: Ms. Doe pays 25% of her inheritance in federal taxes. [ text{Federal Taxes} = 0.25x ]2. **Remaining Amount After Federal Taxes**: After paying federal taxes, the remaining amount is: [ x - 0.25x = 0.75x ]3. **State Taxes**: She pays 15% of the remaining amount in state taxes. [ text{State Taxes} = 0.15 times 0.75x = 0.1125x ]4. **Total Taxes Paid**: The sum of federal and state taxes is: [ 0.25x + 0.1125x = 0.3625x ]5. **Given Total Taxes**: The total taxes paid amount to 15,000. [ 0.3625x = 15000 ]6. **Solving for ( x )**: [ x = frac{15000}{0.3625} approx 41379 ]Therefore, the total amount of her inheritance is approximately 41,379.