💡First, I need to determine the amount donated to Treetown National Park and The Forest Reserve combined. Let's denote the amount donated to Animal Preservation Park as ( x ).According to the problem, the donation to Treetown National Park and The Forest Reserve together is 140 more than the donation to Animal Preservation Park. Therefore, the combined donation to these two parks is ( x + 140 ).The total donation to all three parks is 1000. This means that the sum of the donations to Animal Preservation Park and the combined donations to Treetown National Park and The Forest Reserve equals 1000. This can be expressed as:[x + (x + 140) = 1000]Simplifying the equation:[2x + 140 = 1000]Subtracting 140 from both sides:[2x = 860]Dividing both sides by 2:[x = 430]So, the amount donated to Animal Preservation Park is 430. To find the combined donation to Treetown National Park and The Forest Reserve, add 140 to the donation to Animal Preservation Park:[430 + 140 = 570]Therefore, the total donation to Treetown National Park and The Forest Reserve is 570.

💡Okay, so Jerry is suing his convenience store because he slipped and fell. He's asking for a bunch of damages: loss of salary, medical bills, and punitive damages. I need to figure out what percentage of his total claim he actually received if he got 5,440,000.First, let's break down what he's asking for. He wants 50,000 annually for 30 years. That sounds like his lost salary. So, I guess I need to multiply 50,000 by 30 to get the total salary damages. Let me do that: 50,000 times 30 is 1,500,000. Okay, that's straightforward.Next, he's asking for 200,000 in medical bills. That seems clear too. So, medical damages are 200,000.Now, the tricky part is the punitive damages. It says punitive damages are triple the medical and salary damages. So, I think that means I need to add the salary and medical damages first and then triple that amount. Let me add 1,500,000 and 200,000. That gives me 1,700,000. Now, tripling that would be 1,700,000 times 3, which is 5,100,000. So, punitive damages are 5,100,000.Now, to find the total claim Jerry is asking for, I need to add up all these amounts: salary damages, medical bills, and punitive damages. So, 1,500,000 plus 200,000 is 1,700,000, and then plus 5,100,000 is 6,800,000. So, his total claim is 6,800,000.He actually received 5,440,000. To find out what percentage this is of his total claim, I need to divide the amount he received by the total claim and then multiply by 100 to get the percentage. So, 5,440,000 divided by 6,800,000. Let me do that division: 5,440,000 divided by 6,800,000 equals 0.8. Multiplying by 100 gives me 80%.So, Jerry received 80% of his total claim. That seems pretty high, but I guess that's what the calculation shows.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting with the salary: 50,000 times 30 is indeed 1,500,000. Medical bills are 200,000, so adding those gives 1,700,000. Tripling that for punitive damages gives 5,100,000. Adding all together: 1,500,000 plus 200,000 is 1,700,000, plus 5,100,000 is 6,800,000. That looks correct.Then, 5,440,000 divided by 6,800,000 equals 0.8, which is 80%. Yeah, that seems right. So, Jerry got 80% of what he was asking for.

💡First, I need to find the principal amount (P) using the simple interest formula. The simple interest is 400, and the rate is 20% per annum. By rearranging the simple interest formula, I can solve for P.Next, I'll use the compound interest formula to set up an equation involving the principal amount, the rate, and the time period. I'll substitute the value of P obtained from the simple interest calculation into this equation.This will give me an equation that relates the time period (t) to the compound interest. Since this equation cannot be solved algebraically, I'll use trial and error with different values of t to find the one that satisfies the equation.After testing t = 1 and t = 2, I find that t = 2 satisfies the equation, indicating that the interest is calculated for 2 years.

💡Okay, so I have this problem here: I need to find the coefficient of the ( x^2 ) term in the expansion of ( left( a sqrt{x} - frac{1}{sqrt{x}} right)^6 ), where ( a ) is given as the integral from 0 to ( pi ) of ( sin x + cos x ) dx. Hmm, let me break this down step by step.First, I think I need to calculate the value of ( a ). The integral is from 0 to ( pi ) of ( sin x + cos x ) dx. I remember that integrating ( sin x ) gives ( -cos x ) and integrating ( cos x ) gives ( sin x ). So, let me compute that.So, ( a = int_{0}^{pi} (sin x + cos x) dx ). Let me split this into two separate integrals: ( int_{0}^{pi} sin x , dx + int_{0}^{pi} cos x , dx ).Calculating the first integral: ( int sin x , dx = -cos x ). Evaluated from 0 to ( pi ), that's ( -cos pi - (-cos 0) ). I know that ( cos pi = -1 ) and ( cos 0 = 1 ). So, plugging those in: ( -(-1) - (-1) = 1 + 1 = 2 ). Wait, no, hold on. Let me recast that. It's ( [-cos pi] - [-cos 0] ) which is ( -(-1) - (-1) ) which is ( 1 + 1 = 2 ). Okay, so the first integral is 2.Now, the second integral: ( int cos x , dx = sin x ). Evaluated from 0 to ( pi ), that's ( sin pi - sin 0 ). I know that ( sin pi = 0 ) and ( sin 0 = 0 ). So, that integral is ( 0 - 0 = 0 ).Therefore, ( a = 2 + 0 = 2 ). Got it, so ( a = 2 ).Now, moving on to the expansion part. I need to expand ( left( 2 sqrt{x} - frac{1}{sqrt{x}} right)^6 ) and find the coefficient of the ( x^2 ) term.I remember that when expanding a binomial like ( (A + B)^n ), the general term is given by ( binom{n}{k} A^{n - k} B^k ), where ( k ) ranges from 0 to ( n ). In this case, ( A = 2 sqrt{x} ) and ( B = -frac{1}{sqrt{x}} ), and ( n = 6 ).So, the general term ( T_{k+1} ) is ( binom{6}{k} (2 sqrt{x})^{6 - k} left( -frac{1}{sqrt{x}} right)^k ).Let me write that out:( T_{k+1} = binom{6}{k} times (2 sqrt{x})^{6 - k} times left( -frac{1}{sqrt{x}} right)^k ).Simplify this expression step by step.First, let's handle the constants and the coefficients:( (2)^{6 - k} times (-1)^k times binom{6}{k} ).Then, the variable part:( (sqrt{x})^{6 - k} times left( frac{1}{sqrt{x}} right)^k ).Let me simplify the variable part:( (sqrt{x})^{6 - k} = x^{frac{6 - k}{2}} ).Similarly, ( left( frac{1}{sqrt{x}} right)^k = x^{-frac{k}{2}} ).Multiplying these together:( x^{frac{6 - k}{2}} times x^{-frac{k}{2}} = x^{frac{6 - k - k}{2}} = x^{frac{6 - 2k}{2}} = x^{3 - k} ).So, the exponent of ( x ) in each term is ( 3 - k ).We need the term where the exponent is 2, so set ( 3 - k = 2 ). Solving for ( k ):( 3 - k = 2 ) => ( k = 1 ).So, the term corresponding to ( k = 1 ) will have ( x^2 ).Now, let's compute the coefficient for ( k = 1 ):First, the binomial coefficient ( binom{6}{1} ). I remember that ( binom{6}{1} = 6 ).Next, the constants:( (2)^{6 - 1} = 2^5 = 32 ).And ( (-1)^1 = -1 ).Multiplying all these together:Coefficient = ( binom{6}{1} times 2^{5} times (-1)^1 = 6 times 32 times (-1) ).Calculating that:6 times 32 is 192, and 192 times (-1) is -192.So, the coefficient of the ( x^2 ) term is -192.Wait, let me just double-check my steps to make sure I didn't make any mistakes.1. Calculated ( a ) correctly: integral of ( sin x + cos x ) from 0 to ( pi ) is 2. That seems right.2. Set up the binomial expansion correctly: ( (2 sqrt{x} - 1/sqrt{x})^6 ). Yes.3. General term: ( binom{6}{k} (2 sqrt{x})^{6 - k} (-1/sqrt{x})^k ). That looks correct.4. Simplified the variable part to ( x^{3 - k} ). Let me verify:( (sqrt{x})^{6 - k} = x^{(6 - k)/2} ).( (1/sqrt{x})^k = x^{-k/2} ).Adding exponents: ( (6 - k)/2 - k/2 = (6 - 2k)/2 = 3 - k ). Correct.5. Set exponent equal to 2: ( 3 - k = 2 ) => ( k = 1 ). Correct.6. Calculated the coefficient for ( k = 1 ):( binom{6}{1} = 6 ).( 2^{5} = 32 ).( (-1)^1 = -1 ).Multiply together: 6 * 32 = 192; 192 * (-1) = -192. Correct.So, all steps seem to check out. Therefore, I'm confident that the coefficient of ( x^2 ) is indeed -192.**Final Answer**The coefficient of the ( x^2 ) term is boxed{-192}.

💡Okay, so I have this problem here: Given positive integers x and y such that x is not equal to y, and 1/x plus 1/y equals 1/12. I need to find the smallest possible value for x + y. Hmm, let me think about how to approach this.First, I remember that when dealing with equations involving reciprocals, it's often helpful to combine them into a single fraction. So, let me rewrite the equation:1/x + 1/y = 1/12To combine the left side, I need a common denominator, which would be xy. So, adding the fractions:(y + x) / xy = 1/12That simplifies to:(x + y) / xy = 1/12Now, cross-multiplying both sides to eliminate the fractions:12(x + y) = xyHmm, okay, so now I have 12x + 12y = xy. I think I can rearrange this equation to make it easier to work with. Let me move all terms to one side:xy - 12x - 12y = 0Hmm, this looks a bit like a quadratic equation, but it's in two variables. I remember something called Simon's Favorite Factoring Trick, which might help here. The idea is to add a constant term to both sides so that the left side can be factored into a product of two binomials.Looking at the equation:xy - 12x - 12y = 0I notice that the coefficients of x and y are both -12. So, if I add (12)^2, which is 144, to both sides, I can factor the left side:xy - 12x - 12y + 144 = 144Now, let's factor the left side. It should factor into (x - 12)(y - 12). Let me check:(x - 12)(y - 12) = xy - 12x - 12y + 144Yes, that's correct. So, the equation becomes:(x - 12)(y - 12) = 144Okay, so now I have the product of (x - 12) and (y - 12) equal to 144. Since x and y are positive integers and x ≠ y, both (x - 12) and (y - 12) must be positive integers as well. So, I need to find pairs of positive integers that multiply to 144 and then add 12 back to each to get x and y.Let me list the factor pairs of 144:1 and 1442 and 723 and 484 and 366 and 248 and 189 and 1612 and 12Wait, but x and y must be different, so the pair (12, 12) would result in x = 24 and y = 24, which are equal. Since x ≠ y, I should exclude this pair.So, the valid factor pairs are:(1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16)Now, for each of these pairs, I can find the corresponding x and y by adding 12 to each factor:For (1, 144):x = 1 + 12 = 13y = 144 + 12 = 156Sum x + y = 13 + 156 = 169For (2, 72):x = 2 + 12 = 14y = 72 + 12 = 84Sum x + y = 14 + 84 = 98For (3, 48):x = 3 + 12 = 15y = 48 + 12 = 60Sum x + y = 15 + 60 = 75For (4, 36):x = 4 + 12 = 16y = 36 + 12 = 48Sum x + y = 16 + 48 = 64For (6, 24):x = 6 + 12 = 18y = 24 + 12 = 36Sum x + y = 18 + 36 = 54For (8, 18):x = 8 + 12 = 20y = 18 + 12 = 30Sum x + y = 20 + 30 = 50For (9, 16):x = 9 + 12 = 21y = 16 + 12 = 28Sum x + y = 21 + 28 = 49Wait, hold on, 49 is smaller than 50. Did I make a mistake earlier?Let me double-check the calculations for (9, 16):x = 9 + 12 = 21y = 16 + 12 = 28Sum = 21 + 28 = 49Yes, that's correct. So, 49 is smaller than 50. Hmm, so maybe 49 is the smallest sum.But wait, let me check if I missed any factor pairs. I listed all the factor pairs where the first number is less than the second to avoid repetition. So, I think I covered all possible pairs.But just to be thorough, let me list all possible factor pairs, including those where the first number is greater than the second:(144, 1), (72, 2), (48, 3), (36, 4), (24, 6), (18, 8), (16, 9), (12, 12)But since x and y are just positive integers, the order doesn't matter, so these are just the same pairs as before, just reversed. So, the sums would be the same.So, the sums I calculated are:169, 98, 75, 64, 54, 50, 49So, the smallest sum is 49.Wait, but earlier I thought 50 was the smallest. So, why did I get confused?Because in my initial calculation, I thought (8, 18) gives 50, but (9, 16) gives 49, which is smaller.So, 49 is actually the smallest sum.But wait, let me double-check if x and y are positive integers and x ≠ y.For (9, 16):x = 21, y = 28. Both are positive integers, and 21 ≠ 28. So, that's valid.Similarly, for (8, 18):x = 20, y = 30. Also valid.So, 49 is smaller than 50, so 49 should be the answer.But wait, let me check if I made a mistake in the factor pairs.Wait, 144 can also be factored as:(12, 12), but that gives x = 24, y = 24, which are equal, so we exclude that.Are there any other factor pairs?Wait, 144 is 12 squared, so its factors are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.So, the factor pairs I listed earlier are correct.Therefore, the smallest sum is 49.Wait, but in the initial problem, the user wrote:"Given positive integers x and y such that x ≠ y and 1/x + 1/y = 1/12, what is the smallest possible value for x + y?"So, according to my calculations, the smallest possible value is 49.But wait, in the initial thought process, I thought 50 was the answer, but upon re-evaluating, 49 is smaller.So, perhaps I made a mistake in the initial thought process.Wait, let me check again:For (9, 16):x = 9 + 12 = 21y = 16 + 12 = 28Sum = 21 + 28 = 49Yes, that's correct.So, why did I initially think 50 was the answer? Maybe because I stopped at (8, 18) which gives 50, but (9, 16) gives a smaller sum.Therefore, the correct smallest sum is 49.Wait, but let me check if 21 and 28 satisfy the original equation:1/21 + 1/28 = ?Let me compute:1/21 is approximately 0.04761/28 is approximately 0.0357Adding them together: 0.0476 + 0.0357 ≈ 0.08331/12 is approximately 0.0833, so yes, it checks out.Similarly, 1/20 + 1/30 = 0.05 + 0.0333 ≈ 0.0833, which also equals 1/12.So, both pairs (20, 30) and (21, 28) satisfy the equation, but (21, 28) gives a smaller sum.Therefore, the smallest possible value for x + y is 49.Wait, but in the initial problem, the user wrote:"Given positive integers x and y such that x ≠ y and 1/x + 1/y = 1/12, what is the smallest possible value for x + y?"So, according to my calculations, the answer should be 49.But in the initial thought process, I thought 50 was the answer, but upon re-evaluating, 49 is smaller.Therefore, the correct answer is 49.But wait, let me make sure I didn't miss any factor pairs that could give an even smaller sum.Looking back at the factor pairs:(1, 144) gives sum 169(2, 72) gives 98(3, 48) gives 75(4, 36) gives 64(6, 24) gives 54(8, 18) gives 50(9, 16) gives 49So, 49 is indeed the smallest.Therefore, the smallest possible value for x + y is 49.

💡To determine how many lemons Cristine initially bought, let's follow these steps:1. **Let ( L ) represent the total number of lemons Cristine bought initially.**2. **Cristine gave away ( frac{1}{4} ) of the lemons to her neighbor. This means she gave away ( frac{1}{4}L ) lemons.**3. **The number of lemons Cristine has left after giving some to her neighbor is ( L - frac{1}{4}L = frac{3}{4}L ).**4. **According to the problem, Cristine has 9 lemons left. Therefore, we can set up the equation:** [ frac{3}{4}L = 9 ]5. **To find ( L ), solve the equation by multiplying both sides by ( frac{4}{3} ):** [ L = 9 times frac{4}{3} = 12 ]6. **Thus, Cristine initially bought 12 lemons.**