answer：First, I need to understand the given information. The ratio of investments between partners P and Q is 7:5, and they invested for different durations—5 months and 9 months respectively. The goal is to find the ratio of their profits.I know that profit in a partnership is typically distributed based on both the amount invested and the time period for which the investment was made. Therefore, I should calculate the product of the investment amount and the time period for each partner.Let's denote the amount invested by P as 7x and by Q as 5x, where x is a common factor. Next, I'll calculate the investment multiplied by time for both partners:- For P: 7x * 5 = 35x- For Q: 5x * 9 = 45xNow, I'll find the ratio of their profits by comparing these products:- Ratio of P's profit to Q's profit = 35x : 45xTo simplify this ratio, I'll divide both terms by the common factor x:- 35x/x : 45x/x = 35 : 45Finally, I'll reduce this ratio to its simplest form by dividing both numbers by their greatest common divisor, which is 5:- 35/5 : 45/5 = 7 : 9Therefore, the ratio of their profits is 7:9.

answer：Alright, so I have this problem here where I need to find an integer ( a ) such that ( 0 leq a leq 20 ), and when I subtract ( a ) from the number ( 56831742_{85} ), the result is a multiple of 17. That means ( 56831742_{85} - a ) should be divisible by 17. First off, I need to understand what ( 56831742_{85} ) is. The subscript 85 tells me that this number is in base 85. So, each digit represents a power of 85, similar to how in base 10, each digit is a power of 10. Breaking it down, the number ( 56831742_{85} ) can be expanded as:[5 times 85^7 + 6 times 85^6 + 8 times 85^5 + 3 times 85^4 + 1 times 85^3 + 7 times 85^2 + 4 times 85^1 + 2 times 85^0]That's a pretty big number! Calculating this directly would be time-consuming and probably not necessary since I'm only interested in whether ( 56831742_{85} - a ) is divisible by 17. I remember that when dealing with divisibility by a number, especially primes like 17, modular arithmetic can be really helpful. Specifically, if I can find ( 56831742_{85} ) modulo 17, then I can figure out what ( a ) needs to be to make the whole expression divisible by 17.So, let's recall that ( 85 ) is a multiple of 17 because ( 85 = 5 times 17 ). That means ( 85 equiv 0 mod{17} ). This is super useful because any power of 85 will also be congruent to 0 modulo 17. Wait, let me think about that again. If ( 85 equiv 0 mod{17} ), then ( 85^k equiv 0^k mod{17} ) for any positive integer ( k ). But ( 0^k ) is 0 for ( k geq 1 ), right? So, that means every term in the expansion of ( 56831742_{85} ) except the last digit (which is multiplied by ( 85^0 = 1 )) will be congruent to 0 modulo 17. So, simplifying ( 56831742_{85} mod{17} ), we get:[56831742_{85} equiv 2 mod{17}]Because all the other terms are multiples of 85 and hence 0 modulo 17, leaving only the last digit, which is 2.Now, the problem states that ( 56831742_{85} - a ) is a multiple of 17. In modular arithmetic terms, this means:[56831742_{85} - a equiv 0 mod{17}]Substituting what we found earlier:[2 - a equiv 0 mod{17}]Which simplifies to:[a equiv 2 mod{17}]So, ( a ) must be congruent to 2 modulo 17. But we also know that ( a ) is between 0 and 20. The smallest non-negative integer that satisfies ( a equiv 2 mod{17} ) is 2 itself. The next one would be 19 (since 2 + 17 = 19), but 19 is still within our range of 0 to 20. Wait, hold on. If ( a equiv 2 mod{17} ), then possible values for ( a ) are 2, 19, 36, etc. But since ( a ) must be less than or equal to 20, the possible values are 2 and 19. But the problem says ( 0 leq a leq 20 ), so both 2 and 19 are within this range. Hmm, does that mean there are two possible solutions? But the problem asks for "the" integer ( a ), implying there's only one solution. Maybe I made a mistake somewhere.Let me double-check. The number ( 56831742_{85} ) modulo 17 is indeed 2, right? Because all the higher place values are multiples of 85, which is 0 modulo 17, so only the last digit remains. So, ( 56831742_{85} equiv 2 mod{17} ). Then, ( 56831742_{85} - a equiv 0 mod{17} ) implies ( 2 - a equiv 0 mod{17} ), so ( a equiv 2 mod{17} ). Given that ( a ) is between 0 and 20, the possible values are 2 and 19. But since 19 is also within the range, why is the answer 2 and not 19? Wait, maybe I need to consider the smallest non-negative integer in the range. Since 2 is smaller than 19, and both satisfy the condition, but the problem doesn't specify any other constraints, so both are valid. However, the problem asks for "the" integer ( a ), which suggests there's only one answer. Perhaps I need to re-examine the original number. Maybe I misread the digits or the base. Let me check again. The number is ( 56831742_{85} ). So, the digits are 5, 6, 8, 3, 1, 7, 4, 2. Wait, but in base 85, each digit must be less than 85. All these digits are single digits in base 85, so that's fine. Alternatively, maybe I need to consider that ( a ) is the remainder when ( 56831742_{85} ) is divided by 17, which would be 2. So, ( a = 2 ). But then why is 19 also a solution? Because 19 is congruent to 2 modulo 17, so both 2 and 19 would make ( 56831742_{85} - a ) divisible by 17. But the problem specifies ( 0 leq a leq 20 ), so both 2 and 19 are valid. However, usually, in such problems, the smallest non-negative solution is preferred unless specified otherwise. So, the answer should be 2. Wait, but to be thorough, let me test both values. If ( a = 2 ), then ( 56831742_{85} - 2 ) should be divisible by 17. Since ( 56831742_{85} equiv 2 mod{17} ), subtracting 2 gives 0 modulo 17, which is divisible by 17. If ( a = 19 ), then ( 56831742_{85} - 19 equiv 2 - 19 equiv -17 equiv 0 mod{17} ). So, that also works. Therefore, both 2 and 19 are solutions. But the problem asks for ( a ) such that ( 0 leq a leq 20 ). Since both are valid, but the problem might expect the smallest one, which is 2. Alternatively, maybe I'm overcomplicating. The key is that ( a equiv 2 mod{17} ), and within 0 to 20, the possible values are 2 and 19. But since the problem asks for "the" integer, perhaps it's expecting the smallest one, which is 2. So, after all this thinking, I believe the answer is 2.

answer：Alright, so I have this problem where I need to compute the sum of (x^2 + y^2) over all four ordered pairs ((x, y)) of real numbers that satisfy the system of equations:[begin{cases}x = y^2 - 20 y = x^2 + x - 21end{cases}]Hmm, okay. Let me try to figure this out step by step.First, since (x) is expressed in terms of (y) in the first equation, maybe I can substitute that into the second equation to eliminate (x). That way, I can have an equation solely in terms of (y), which I can then solve.So, substituting (x = y^2 - 20) into the second equation:[y = (y^2 - 20)^2 + (y^2 - 20) - 21]Let me expand this out. First, compute ((y^2 - 20)^2):[(y^2 - 20)^2 = y^4 - 40y^2 + 400]So, plugging that back into the equation:[y = (y^4 - 40y^2 + 400) + (y^2 - 20) - 21]Now, combine like terms:[y = y^4 - 40y^2 + 400 + y^2 - 20 - 21]Simplify the constants and the (y^2) terms:[y = y^4 - 39y^2 + 359]So, bringing all terms to one side to set the equation to zero:[y^4 - 39y^2 - y + 359 = 0]Alright, so now I have a quartic equation in terms of (y). Quartic equations can be tricky, but maybe I don't need to find the exact roots. The problem is asking for the sum of (x^2 + y^2) over all four solutions. Perhaps I can find this sum without explicitly finding each (x) and (y).Let me denote the four solutions as ((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)). I need to compute:[sum_{i=1}^4 (x_i^2 + y_i^2)]Since each (x_i) is related to (y_i) by (x_i = y_i^2 - 20), I can express (x_i^2) in terms of (y_i):[x_i^2 = (y_i^2 - 20)^2 = y_i^4 - 40y_i^2 + 400]Therefore, (x_i^2 + y_i^2) becomes:[x_i^2 + y_i^2 = y_i^4 - 40y_i^2 + 400 + y_i^2 = y_i^4 - 39y_i^2 + 400]So, the sum I need is:[sum_{i=1}^4 (x_i^2 + y_i^2) = sum_{i=1}^4 (y_i^4 - 39y_i^2 + 400)]This can be broken down into three separate sums:[sum_{i=1}^4 y_i^4 - 39 sum_{i=1}^4 y_i^2 + sum_{i=1}^4 400]Calculating each part:1. (sum_{i=1}^4 y_i^4): This is the sum of the fourth powers of the roots of the quartic equation.2. (sum_{i=1}^4 y_i^2): This is the sum of the squares of the roots.3. (sum_{i=1}^4 400): This is straightforward, it's just (4 times 400 = 1600).Now, I need to find (sum y_i^4) and (sum y_i^2). To do this, I can use Vieta's formulas, which relate the sums and products of the roots of a polynomial to its coefficients.Given the quartic equation:[y^4 - 39y^2 - y + 359 = 0]Let me write it in standard form:[y^4 + 0y^3 - 39y^2 - y + 359 = 0]So, the coefficients are:- (a_4 = 1)- (a_3 = 0)- (a_2 = -39)- (a_1 = -1)- (a_0 = 359)Vieta's formulas tell us that for a quartic equation (y^4 + a y^3 + b y^2 + c y + d = 0), the sums of the roots are:- (y_1 + y_2 + y_3 + y_4 = -a)- (y_1 y_2 + y_1 y_3 + y_1 y_4 + y_2 y_3 + y_2 y_4 + y_3 y_4 = b)- (y_1 y_2 y_3 + y_1 y_2 y_4 + y_1 y_3 y_4 + y_2 y_3 y_4 = -c)- (y_1 y_2 y_3 y_4 = d)In our case, (a = 0), so:- (y_1 + y_2 + y_3 + y_4 = 0)- (y_1 y_2 + y_1 y_3 + y_1 y_4 + y_2 y_3 + y_2 y_4 + y_3 y_4 = -39)- (y_1 y_2 y_3 + y_1 y_2 y_4 + y_1 y_3 y_4 + y_2 y_3 y_4 = 1)- (y_1 y_2 y_3 y_4 = 359)Okay, so I have these sums. Now, I need to find (sum y_i^2) and (sum y_i^4).First, let's compute (sum y_i^2). I know that:[left( sum y_i right)^2 = sum y_i^2 + 2 sum_{i < j} y_i y_j]We know that (sum y_i = 0) and (sum_{i < j} y_i y_j = -39). Plugging these in:[0^2 = sum y_i^2 + 2(-39)][0 = sum y_i^2 - 78][sum y_i^2 = 78]Great, so the sum of the squares of the roots is 78.Now, moving on to (sum y_i^4). To find this, I can use the identity:[sum y_i^4 = left( sum y_i^2 right)^2 - 2 sum_{i < j} y_i^2 y_j^2]But I don't have (sum_{i < j} y_i^2 y_j^2) directly. Alternatively, I can use Newton's identities, which relate power sums to elementary symmetric sums.Newton's identities for quartic equations can be a bit involved, but let me try to recall them.The general formula for the sum of the fourth powers is:[p_4 = (y_1^4 + y_2^4 + y_3^4 + y_4^4) = (y_1 + y_2 + y_3 + y_4) p_3 - (y_1 y_2 + y_1 y_3 + y_1 y_4 + y_2 y_3 + y_2 y_4 + y_3 y_4) p_2 + (y_1 y_2 y_3 + y_1 y_2 y_4 + y_1 y_3 y_4 + y_2 y_3 y_4) p_1 - y_1 y_2 y_3 y_4 (y_1 + y_2 + y_3 + y_4)]Where (p_k = y_1^k + y_2^k + y_3^k + y_4^k).But since (p_1 = sum y_i = 0), and (p_3) is the sum of the cubes, which we don't have yet, this might not be the most straightforward way.Alternatively, perhaps I can express (y_i^4) from the original equation.Given that each (y_i) satisfies:[y_i^4 - 39y_i^2 - y_i + 359 = 0]So, rearranged:[y_i^4 = 39y_i^2 + y_i - 359]Therefore, summing over all roots:[sum_{i=1}^4 y_i^4 = 39 sum_{i=1}^4 y_i^2 + sum_{i=1}^4 y_i - 4 times 359]We already know (sum y_i^2 = 78) and (sum y_i = 0). Plugging these in:[sum y_i^4 = 39 times 78 + 0 - 4 times 359]Calculating each term:- (39 times 78): Let's compute this. 40 x 78 = 3120, so subtract 1 x 78: 3120 - 78 = 3042- (4 times 359 = 1436)So,[sum y_i^4 = 3042 - 1436 = 1606]Wait, 3042 - 1436: Let me double-check that subtraction.3042 - 1436:- 3042 - 1000 = 2042- 2042 - 400 = 1642- 1642 - 36 = 1606Yes, that's correct.So, (sum y_i^4 = 1606).Now, going back to the expression for the sum we need:[sum_{i=1}^4 (x_i^2 + y_i^2) = sum y_i^4 - 39 sum y_i^2 + 1600]Wait, hold on. Earlier, I broke it down as:[sum (x_i^2 + y_i^2) = sum y_i^4 - 39 sum y_i^2 + sum 400]Which is:[sum y_i^4 - 39 sum y_i^2 + 4 times 400 = sum y_i^4 - 39 sum y_i^2 + 1600]So, plugging in the values we have:[1606 - 39 times 78 + 1600]Compute (39 times 78):As before, 40 x 78 = 3120, so 39 x 78 = 3120 - 78 = 3042So,[1606 - 3042 + 1600]Compute 1606 - 3042:1606 - 3042 = -1436Then, -1436 + 1600 = 164So, the total sum is 164.Wait, that seems low. Let me verify each step to make sure I didn't make a mistake.1. Substituted (x = y^2 - 20) into the second equation correctly, expanded, and got the quartic equation: (y^4 - 39y^2 - y + 359 = 0). That seems correct.2. Expressed (x_i^2 + y_i^2) in terms of (y_i): (y_i^4 - 39y_i^2 + 400). That also seems correct.3. Sum over all four pairs: (sum (y_i^4 - 39y_i^2 + 400)). Which breaks down into (sum y_i^4 - 39 sum y_i^2 + 1600). That makes sense.4. Calculated (sum y_i^2 = 78) using ((sum y_i)^2 = sum y_i^2 + 2 sum y_i y_j). Since (sum y_i = 0), we had (0 = sum y_i^2 + 2(-39)), so (sum y_i^2 = 78). Correct.5. For (sum y_i^4), used the equation (y_i^4 = 39y_i^2 + y_i - 359). Summed over all roots: (sum y_i^4 = 39 times 78 + sum y_i - 4 times 359). Since (sum y_i = 0), this becomes (39 times 78 - 4 times 359). Calculated 39 x 78 = 3042 and 4 x 359 = 1436, so 3042 - 1436 = 1606. Correct.6. Plugged into the sum: 1606 - 39 x 78 + 1600. Wait, hold on, 39 x 78 is 3042, so 1606 - 3042 + 1600 = (1606 + 1600) - 3042 = 3206 - 3042 = 164. Correct.So, all steps seem to check out. Therefore, the sum of (x^2 + y^2) over all four ordered pairs is 164.**Final Answer**boxed{164}

answer：First, I need to determine the cost of the first 100 channels, which is 100.Next, the cost of the next 100 channels is half of 100, which is 50.Adding these together, the total cost for 200 channels is 100 + 50 = 150.Since James splits the cost evenly with his roommate, I divide the total cost by 2 to find out how much James pays.Therefore, James pays 150 / 2 = 75.

answer：First, I need to understand the given ratio of Ram's and Shyam's weights, which is 4:5. This means I can represent their weights as 4k and 5k, respectively, where k is a constant.Next, the total weight of Ram and Shyam before any increase is 4k + 5k = 9k.The problem states that after a 15% increase, the total weight becomes 82.8 kg. I can set up the equation 9k * 1.15 = 82.8 to find the value of k.Solving for k, I get k = 8 kg.Now, I can determine the original weights of Ram and Shyam:- Ram's weight (R) = 4k = 32 kg- Shyam's weight (S) = 5k = 40 kgShyam's weight is increased by 19%, so his new weight is 40 kg * 1.19 = 47.6 kg.The total new weight is 82.8 kg, so Ram's new weight must be 82.8 kg - 47.6 kg = 35.2 kg.Finally, to find the percentage increase in Ram's weight, I calculate ((35.2 kg - 32 kg) / 32 kg) * 100% = 10%.

answer：Alright, so I've got this equation to solve: |3x - 2| + 5 = 23. Hmm, absolute value equations can sometimes be tricky, but I think I remember that you have to consider both the positive and negative scenarios. Let me try to work through this step by step.First, I need to isolate the absolute value part. The equation is |3x - 2| + 5 = 23. So, if I subtract 5 from both sides, that should help me get the absolute value by itself. Let's do that:|3x - 2| + 5 - 5 = 23 - 5Simplifying that, I get:|3x - 2| = 18Okay, so now I have |3x - 2| = 18. I remember that when you have an absolute value equation like |A| = B, it means that A can be equal to B or A can be equal to -B. So, applying that here, I can set up two separate equations:1. 3x - 2 = 182. 3x - 2 = -18Let me solve each of these separately.Starting with the first equation: 3x - 2 = 18I need to solve for x. So, first, I'll add 2 to both sides to get rid of the -2:3x - 2 + 2 = 18 + 2Simplifying that gives:3x = 20Now, to solve for x, I'll divide both sides by 3:3x / 3 = 20 / 3So, x = 20/3. That's one solution.Now, moving on to the second equation: 3x - 2 = -18Again, I need to solve for x. I'll start by adding 2 to both sides:3x - 2 + 2 = -18 + 2Simplifying that gives:3x = -16Now, divide both sides by 3:3x / 3 = -16 / 3So, x = -16/3. That's the second solution.Alright, so I have two solutions: x = 20/3 and x = -16/3. The problem is asking for the product of the values of x that satisfy the equation. So, I need to multiply these two solutions together.Let me write that out:(20/3) * (-16/3) = ?Multiplying the numerators together: 20 * (-16) = -320Multiplying the denominators together: 3 * 3 = 9So, putting it together, the product is -320/9.Wait, let me double-check that multiplication to make sure I didn't make a mistake. 20 times 16 is 320, and since one of them is negative, the result is negative. 3 times 3 is definitely 9. Yep, that seems right.Just to be thorough, I'll plug both values back into the original equation to ensure they work.First, let's check x = 20/3:|3*(20/3) - 2| + 5 = |20 - 2| + 5 = |18| + 5 = 18 + 5 = 23That works.Now, checking x = -16/3:|3*(-16/3) - 2| + 5 = |-16 - 2| + 5 = |-18| + 5 = 18 + 5 = 23That also works.So, both solutions are valid, and their product is indeed -320/9.I think that's it. I've isolated the absolute value, solved both cases, found the solutions, checked them, and calculated their product. Everything seems to add up correctly.