answer：Okay, so I have this problem here about a function f(x) that's odd, defined over all real numbers, and it has a period of 3. Also, they tell me that f(2) equals zero. I need to figure out the minimum number of solutions to f(x) = 0 within the interval (0, 6). The options are 2, 3, 7, or 5. Hmm, let me think about how to approach this.First, let's recall what it means for a function to be odd. An odd function satisfies the property f(-x) = -f(x) for all x in its domain. That means if I know the value of the function at some point x, I automatically know it at -x. Also, for odd functions, f(0) must be zero because f(-0) = -f(0) implies f(0) = 0.Next, the function has a period of 3. That means f(x + 3) = f(x) for all x. So, the function repeats its values every 3 units. This periodicity will help me find zeros of the function beyond the initial interval.Given that f(2) = 0, I can use the periodicity to find other zeros. Since the period is 3, adding or subtracting multiples of 3 to the argument will give me other zeros. So, f(2 + 3k) = 0 for any integer k. Let's see, within the interval (0, 6), the relevant multiples would be k = 0, 1, and maybe 2, but 2 + 3*2 = 8, which is outside of 6, so we can stop at k=1.So, f(2) = 0 and f(5) = 0 because 5 = 2 + 3*1. That gives me two zeros at x=2 and x=5.But wait, since the function is odd, f(-2) = -f(2) = -0 = 0. So, f(-2) is also zero. But -2 is outside of our interval (0,6). However, because of the periodicity, f(-2) = f(-2 + 3) = f(1). So, f(1) = 0 as well. That's another zero at x=1.Similarly, f(-5) = f(-5 + 6) = f(1) = 0, but that's the same as before. Hmm, maybe I can find more zeros by considering the periodicity and oddness together.Let me think about other points. Since f(3) is a multiple of the period, f(3) = f(0) = 0 because f is periodic and f(0) = 0. So, f(3) = 0. That's another zero at x=3.Now, let's consider the midpoint between 0 and 3, which is 1.5. Since f is odd and periodic, what can I say about f(1.5)? Let me see, f(1.5) = f(1.5 - 3) = f(-1.5). But f(-1.5) = -f(1.5) because it's odd. So, f(1.5) = -f(1.5), which implies that 2f(1.5) = 0, so f(1.5) = 0. Therefore, x=1.5 is another zero.Similarly, because of periodicity, f(1.5 + 3) = f(4.5) = 0. So, x=4.5 is another zero.Let me list out all the zeros I have so far within (0,6):- x=1- x=1.5- x=2- x=3- x=4.5- x=5Wait, that's six zeros. But the options don't have 6. The closest is 7. Did I miss one?Let me check again. Starting from x=0, which is f(0)=0, but 0 is not in (0,6). Then, x=1, x=1.5, x=2, x=3, x=4.5, x=5. Hmm, that's six points. Maybe I missed another one.Wait, let's think about x=4. Since f(4) = f(4 - 3) = f(1) = 0. So, f(4)=0. That's another zero at x=4. So, now I have:- x=1- x=1.5- x=2- x=3- x=4- x=4.5- x=5That's seven zeros. So, within (0,6), there are at least seven solutions where f(x)=0.Let me double-check if all these points are indeed zeros:- x=1: Yes, because f(1) = f(-2 + 3) = f(-2) = -f(2) = 0.- x=1.5: Yes, because f(1.5) = -f(1.5) implies f(1.5)=0.- x=2: Given.- x=3: f(3)=f(0)=0.- x=4: f(4)=f(1)=0.- x=4.5: f(4.5)=f(1.5)=0.- x=5: f(5)=f(2)=0.Yes, all of these are valid zeros. So, the minimum number of solutions is 7.I think I might have initially missed x=4 and x=4.5, but upon considering the periodicity and oddness, they also become zeros. So, the correct answer should be 7, which is option (C).

answer：Alright, I've got this problem about an ellipse and a hyperbola, and then a geometry problem involving points and areas. Let me try to work through it step by step.**Part 1: Finding the Equation of the Ellipse**First, the ellipse is given by the equation ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ) where ( a > b > 0 ). The minor axis length is ( 2sqrt{3} ). Since the minor axis is ( 2b ), that means ( 2b = 2sqrt{3} ), so ( b = sqrt{3} ).Next, there's a hyperbola given by ( frac{x^2}{m} - frac{y^2}{n} = 1 ) with asymptotes ( y = pm sqrt{3}x ). The asymptotes of a hyperbola ( frac{x^2}{m} - frac{y^2}{n} = 1 ) are ( y = pm frac{sqrt{n}}{sqrt{m}}x ). So, ( frac{sqrt{n}}{sqrt{m}} = sqrt{3} ), which implies ( frac{n}{m} = 3 ).The eccentricity of the hyperbola ( e_2 ) is calculated as ( e_2 = sqrt{1 + frac{n}{m}} = sqrt{1 + 3} = 2 ).Given that ( e_1 e_2 = 1 ), and since ( e_2 = 2 ), we have ( e_1 = frac{1}{2} ).For the ellipse, the eccentricity ( e_1 ) is given by ( e_1 = frac{c}{a} ), where ( c = sqrt{a^2 - b^2} ). So, ( frac{1}{2} = frac{sqrt{a^2 - 3}}{a} ). Squaring both sides, we get ( frac{1}{4} = frac{a^2 - 3}{a^2} ). Simplifying, ( a^2 - 3 = frac{a^2}{4} ), so ( frac{3a^2}{4} = 3 ), which gives ( a^2 = 4 ). Therefore, ( a = 2 ).So, the equation of the ellipse is ( frac{x^2}{4} + frac{y^2}{3} = 1 ).**Part 2: Finding the Equation of the Line MN**Given point ( A ) is the right vertex of the ellipse, so ( A = (2, 0) ). Point ( P ) is ( (-1, frac{3}{2}) ). The line ( AP ) intersects the y-axis at ( H ).First, let's find the equation of line ( AP ). The slope ( m ) of ( AP ) is ( frac{frac{3}{2} - 0}{-1 - 2} = frac{frac{3}{2}}{-3} = -frac{1}{2} ). So, the equation is ( y = -frac{1}{2}(x - 2) ), which simplifies to ( y = -frac{1}{2}x + 1 ). Therefore, point ( H ) is at ( (0, 1) ).Now, another line through ( H ) intersects the ellipse at points ( M ) and ( N ). Let's denote this line as ( y = kx + 1 ). Substituting this into the ellipse equation:( frac{x^2}{4} + frac{(kx + 1)^2}{3} = 1 )Multiplying through by 12 to eliminate denominators:( 3x^2 + 4(kx + 1)^2 = 12 )Expanding:( 3x^2 + 4(k^2x^2 + 2kx + 1) = 12 )Simplify:( 3x^2 + 4k^2x^2 + 8kx + 4 = 12 )Combine like terms:( (4k^2 + 3)x^2 + 8kx - 8 = 0 )Let ( x_1 ) and ( x_2 ) be the roots of this quadratic equation. Then:( x_1 + x_2 = frac{-8k}{4k^2 + 3} )( x_1 x_2 = frac{-8}{4k^2 + 3} )Given that the area of ( triangle HMA ) is ( S_1 ) and the area of ( triangle PHN ) is ( S_2 ), with ( S_1 = 6 S_2 ). Since areas are proportional to the base times height, and considering the coordinates, it's likely that the distances from ( H ) and ( P ) to points ( M ) and ( N ) are in a ratio related to 6:1. Assuming the line MN is such that ( HM = 3 HN ) (since ( S_1 = 6 S_2 ) and heights might be similar), we can set ( x_1 = -3 x_2 ). Substituting ( x_1 = -3 x_2 ) into the sum and product of roots:Sum: ( -3x_2 + x_2 = -2x_2 = frac{-8k}{4k^2 + 3} ) => ( x_2 = frac{4k}{4k^2 + 3} )Product: ( (-3x_2) x_2 = -3x_2^2 = frac{-8}{4k^2 + 3} )Substituting ( x_2 ):( -3 left( frac{4k}{4k^2 + 3} right)^2 = frac{-8}{4k^2 + 3} )Simplify:( -3 cdot frac{16k^2}{(4k^2 + 3)^2} = frac{-8}{4k^2 + 3} )Multiply both sides by ( (4k^2 + 3)^2 ):( -48k^2 = -8(4k^2 + 3) )Simplify:( -48k^2 = -32k^2 - 24 )Bring all terms to one side:( -16k^2 + 24 = 0 )Solve for ( k^2 ):( 16k^2 = 24 ) => ( k^2 = frac{24}{16} = frac{3}{2} ) => ( k = pm frac{sqrt{6}}{2} )Thus, the equations of line MN are ( y = frac{sqrt{6}}{2}x + 1 ) and ( y = -frac{sqrt{6}}{2}x + 1 ).**Final Answer**(1) The equation of the ellipse is ( boxed{dfrac{x^2}{4} + dfrac{y^2}{3} = 1} ).(2) The equations of the line ( MN ) are ( boxed{y = pm dfrac{sqrt{6}}{2} x + 1} ).

answer：First, I need to determine the amount of alcohol in the initial 15-liter mixture. Since the mixture contains 20% alcohol, I calculate 20% of 15 liters.Next, I add 5 liters of water to the mixture, which increases the total volume to 20 liters. The amount of alcohol remains unchanged at 3 liters.Finally, to find the new percentage of alcohol in the mixture, I divide the volume of alcohol by the total volume of the new mixture and multiply by 100.

answer：First, denote the total profit as ( P ).1. **Equal Division for Efforts:** - One-third of the profit is divided equally between Mary and Mike. - Each receives ( frac{P}{6} ).2. **Division Based on Investments:** - The remaining two-thirds of the profit is divided in the ratio of their investments. - Mary's investment ratio: ( frac{550}{1000} = frac{11}{20} ). - Mike's investment ratio: ( frac{450}{1000} = frac{9}{20} ). - Mary's share from investments: ( frac{11}{20} times frac{2P}{3} = frac{22P}{60} ). - Mike's share from investments: ( frac{9}{20} times frac{2P}{3} = frac{18P}{60} ).3. **Total Shares:** - Mary's total share: ( frac{P}{6} + frac{22P}{60} ). - Mike's total share: ( frac{P}{6} + frac{18P}{60} ).4. **Difference in Shares:** - The difference between Mary's and Mike's shares is 1000. - Set up the equation: ( left( frac{P}{6} + frac{22P}{60} right) - left( frac{P}{6} + frac{18P}{60} right) = 1000 ). - Simplify: ( frac{4P}{60} = 1000 ). - Solve for ( P ): ( P = 15000 ).Therefore, the total profit made by the business is 15,000.

answer：Alright, let's try to tackle this problem step by step. It's about geometric sequences, which I remember involve a common ratio between consecutive terms. So, let's break down what's given and what we need to find.First, we have a geometric sequence {a_n} with the first term 'a' and common ratio 'q'. That means each term is a multiplied by q raised to the power of (n-1). So, a_1 = a, a_2 = aq, a_3 = aq^2, and so on.Next, there's another sequence {b_n} defined as b_n = 1 + a_1 + a_2 + ... + a_n for n = 1, 2, 3, etc. So, each term b_n is the sum of the first n terms of the geometric sequence {a_n} plus 1. I think I can use the formula for the sum of a geometric series here. The sum of the first n terms of a geometric sequence is S_n = a(1 - q^n)/(1 - q), right? So, b_n should be 1 + a(1 - q^n)/(1 - q).Then, we have the sequence {c_n} defined as c_n = 2 + b_1 + b_2 + ... + b_n. So, each term c_n is the sum of the first n terms of the sequence {b_n} plus 2. Since {c_n} is given to be a geometric sequence, that means the ratio between consecutive terms c_{n+1}/c_n should be constant for all n.Okay, so my goal is to find the values of 'a' and 'q' such that {c_n} is a geometric sequence, and then find a + q.Let me write down the expressions step by step.First, express b_n:b_n = 1 + a_1 + a_2 + ... + a_n = 1 + a(1 - q^n)/(1 - q).Simplify that:b_n = 1 + (a/(1 - q)) - (a q^n)/(1 - q).So, b_n = [1 + a/(1 - q)] - (a q^n)/(1 - q).Now, let's find c_n, which is 2 + sum_{k=1}^n b_k.So, c_n = 2 + sum_{k=1}^n [1 + a/(1 - q) - (a q^k)/(1 - q)].Let's break this sum into three parts:sum_{k=1}^n 1 = n.sum_{k=1}^n [a/(1 - q)] = n * [a/(1 - q)].sum_{k=1}^n [ - (a q^k)/(1 - q) ] = - (a/(1 - q)) * sum_{k=1}^n q^k.The sum sum_{k=1}^n q^k is another geometric series, which is q(1 - q^n)/(1 - q).So, putting it all together:c_n = 2 + n + [n * a/(1 - q)] - [a/(1 - q)] * [q(1 - q^n)/(1 - q)].Simplify term by term:First term: 2.Second term: n.Third term: (a n)/(1 - q).Fourth term: - [a q (1 - q^n)] / (1 - q)^2.So, combining all these:c_n = 2 + n + (a n)/(1 - q) - [a q (1 - q^n)] / (1 - q)^2.Hmm, that seems a bit complicated. Let's see if we can simplify it further.Let me factor out some terms:c_n = 2 + n [1 + a/(1 - q)] - [a q (1 - q^n)] / (1 - q)^2.Since {c_n} is a geometric sequence, it must satisfy c_{n+1} / c_n = constant for all n. That ratio should be the same regardless of n. So, let's denote the common ratio as r. Then, c_{n+1} = r * c_n.Given that, let's compute c_{n+1} and c_n and set up the ratio.But before that, maybe it's better to express c_n in a form that resembles a geometric sequence. A geometric sequence has the form c_n = c_1 * r^{n-1}.But our expression for c_n has terms involving n and q^n. For c_n to be geometric, the coefficients of n must be zero because otherwise, the sequence would have a linear term, which isn't possible for a geometric sequence (unless it's trivial, but in this case, it's non-trivial since c_n starts with 2 + ...).So, to eliminate the linear term in n, the coefficient of n must be zero.Looking back at c_n:c_n = 2 + n [1 + a/(1 - q)] - [a q (1 - q^n)] / (1 - q)^2.So, the coefficient of n is [1 + a/(1 - q)]. For this to be zero:1 + a/(1 - q) = 0.Let's solve this equation:1 + a/(1 - q) = 0.Multiply both sides by (1 - q):(1)(1 - q) + a = 0.1 - q + a = 0.So, a = q - 1.Okay, that's one equation relating a and q.Now, let's look at the remaining terms in c_n:c_n = 2 - [a q (1 - q^n)] / (1 - q)^2.But we also have the term 2, which is a constant. For c_n to be a geometric sequence, the constant term must also satisfy the condition that when we take the ratio c_{n+1}/c_n, it's constant.But let's substitute a = q - 1 into the expression.First, a = q - 1.So, let's substitute into the expression:c_n = 2 - [(q - 1) q (1 - q^n)] / (1 - q)^2.Let me simplify this:First, note that (1 - q)^2 is in the denominator, and (q - 1) is in the numerator. Notice that (q - 1) = -(1 - q). So, let's rewrite:(q - 1) = -(1 - q).So, substituting:c_n = 2 - [ - (1 - q) * q (1 - q^n) ] / (1 - q)^2.Simplify the negatives:= 2 + [ (1 - q) q (1 - q^n) ] / (1 - q)^2.Cancel out one (1 - q) from numerator and denominator:= 2 + [ q (1 - q^n) ] / (1 - q).So, c_n = 2 + q(1 - q^n)/(1 - q).Hmm, that's interesting. Let's write that as:c_n = 2 + [q(1 - q^n)] / (1 - q).Now, let's see if we can write this as a geometric sequence.First, let's compute c_1, c_2, c_3, etc., to see if we can find a pattern.Compute c_1:c_1 = 2 + [q(1 - q^1)] / (1 - q) = 2 + [q(1 - q)] / (1 - q) = 2 + q.Similarly, c_2:c_2 = 2 + [q(1 - q^2)] / (1 - q) = 2 + [q(1 - q)(1 + q)] / (1 - q) = 2 + q(1 + q).Simplify: 2 + q + q^2.Similarly, c_3:c_3 = 2 + [q(1 - q^3)] / (1 - q) = 2 + [q(1 - q)(1 + q + q^2)] / (1 - q) = 2 + q(1 + q + q^2).Simplify: 2 + q + q^2 + q^3.Wait a minute, so c_n seems to be 2 + q + q^2 + ... + q^n.But wait, that's the sum from k=0 to n of q^k minus 1, because 2 is 1 + 1, but actually, let's see:Wait, 2 + q + q^2 + ... + q^n is equal to 1 + (1 + q + q^2 + ... + q^n). Since the sum from k=0 to n of q^k is (1 - q^{n+1})/(1 - q). So, 2 + q + q^2 + ... + q^n = 1 + [1 + q + q^2 + ... + q^n] = 1 + (1 - q^{n+1})/(1 - q).But wait, in our case, c_n = 2 + [q(1 - q^n)] / (1 - q). Let's see:2 + [q(1 - q^n)] / (1 - q) = 2 + [q - q^{n+1}]/(1 - q).Let me write 2 as 2*(1 - q)/(1 - q) to combine the terms:= [2(1 - q) + q - q^{n+1}]/(1 - q).Simplify numerator:2(1 - q) + q = 2 - 2q + q = 2 - q.So, numerator is (2 - q) - q^{n+1}.Thus, c_n = (2 - q - q^{n+1}) / (1 - q).Hmm, so c_n = (2 - q - q^{n+1}) / (1 - q).But for c_n to be a geometric sequence, the expression must be of the form c_n = C * r^{n}, where C is a constant and r is the common ratio.Looking at our expression, c_n = (2 - q - q^{n+1}) / (1 - q).Let me factor out q^{n+1}:c_n = [ (2 - q) - q^{n+1} ] / (1 - q ) = (2 - q)/(1 - q) - q^{n+1}/(1 - q).So, c_n = A - B q^{n+1}, where A = (2 - q)/(1 - q) and B = 1/(1 - q).But for c_n to be a geometric sequence, the term involving q^{n+1} must dominate, and the constant term A must be zero because otherwise, the sequence would have a constant term and a geometric term, which wouldn't be a pure geometric sequence.Wait, but if A is not zero, then c_n would be a combination of a constant and a geometric term, which isn't a geometric sequence unless the constant term is zero.So, to have c_n purely geometric, the constant term A must be zero.So, set A = 0:(2 - q)/(1 - q) = 0.This implies that 2 - q = 0, so q = 2.But wait, if q = 2, let's check if that works.Wait, but earlier we had a = q - 1, so if q = 2, then a = 2 - 1 = 1.So, a = 1 and q = 2.Let me check if this makes {c_n} a geometric sequence.Compute c_n with a = 1 and q = 2.From earlier, c_n = 2 + [q(1 - q^n)] / (1 - q).Substitute q = 2:c_n = 2 + [2(1 - 2^n)] / (1 - 2) = 2 + [2(1 - 2^n)] / (-1) = 2 - 2(1 - 2^n) = 2 - 2 + 2^{n+1} = 2^{n+1}.So, c_n = 2^{n+1}, which is indeed a geometric sequence with first term 4 (when n=1: 2^{2}=4) and common ratio 2.Wait, but let's check c_1:c_1 = 2 + [2(1 - 2^1)] / (1 - 2) = 2 + [2(1 - 2)] / (-1) = 2 + [2*(-1)] / (-1) = 2 + 2 = 4.c_2 = 2 + [2(1 - 2^2)] / (1 - 2) = 2 + [2(1 - 4)] / (-1) = 2 + [2*(-3)] / (-1) = 2 + 6 = 8.c_3 = 2 + [2(1 - 2^3)] / (1 - 2) = 2 + [2(1 - 8)] / (-1) = 2 + [2*(-7)] / (-1) = 2 + 14 = 16.So, c_n = 4, 8, 16, 32,... which is 2^{n+1}, a geometric sequence with ratio 2. Perfect.But wait, earlier when we set A = 0, we got q = 2, and a = q - 1 = 1. So, a + q = 1 + 2 = 3, which is option B.But let me double-check if there are other possibilities. Because sometimes, when solving equations, especially with quadratics, there might be multiple solutions.Wait, when we set A = 0, we got q = 2. But let's go back to the earlier step where we had:c_n = (2 - q - q^{n+1}) / (1 - q).We set A = (2 - q)/(1 - q) = 0, leading to q = 2.But what if instead of setting A = 0, we consider another approach? Maybe the term involving q^{n+1} can be expressed in a way that the entire expression becomes a geometric sequence.Alternatively, perhaps I made a mistake in assuming that the coefficient of n must be zero. Let me revisit that.We had c_n = 2 + n [1 + a/(1 - q)] - [a q (1 - q^n)] / (1 - q)^2.For c_n to be a geometric sequence, the expression must not have any terms with n, except in the exponent. So, the coefficient of n must be zero, which gave us 1 + a/(1 - q) = 0, leading to a = q - 1.Then, after substituting a = q - 1, we arrived at c_n = (2 - q - q^{n+1}) / (1 - q).To make this a geometric sequence, the constant term (2 - q)/(1 - q) must be zero, leading to q = 2.So, that seems consistent.Alternatively, maybe there's another way to approach this problem.Let me consider the fact that {c_n} is geometric. So, the ratio c_{n+1}/c_n must be constant.Let me compute c_{n+1}/c_n and set it equal to some constant r.From earlier, c_n = (2 - q - q^{n+1}) / (1 - q).So, c_{n+1} = (2 - q - q^{n+2}) / (1 - q).Thus, the ratio:c_{n+1}/c_n = [ (2 - q - q^{n+2}) / (1 - q) ] / [ (2 - q - q^{n+1}) / (1 - q) ] = (2 - q - q^{n+2}) / (2 - q - q^{n+1}).For this ratio to be constant for all n, the dependence on n must cancel out. Let's denote D = 2 - q, so:c_{n+1}/c_n = (D - q^{n+2}) / (D - q^{n+1}).We need this ratio to be independent of n. Let's denote r = c_{n+1}/c_n.So,(D - q^{n+2}) = r (D - q^{n+1}).Let's rearrange:D - q^{n+2} = r D - r q^{n+1}.Bring all terms to one side:D - r D - q^{n+2} + r q^{n+1} = 0.Factor:D(1 - r) + q^{n+1}(-q + r) = 0.For this equation to hold for all n, the coefficients of q^{n+1} and the constant term must both be zero.So,1. Coefficient of q^{n+1}: (-q + r) = 0 => r = q.2. Constant term: D(1 - r) = 0 => D(1 - q) = 0.Since D = 2 - q, we have:(2 - q)(1 - q) = 0.So, either 2 - q = 0 => q = 2, or 1 - q = 0 => q = 1.But q = 1 would make the original geometric sequence {a_n} have all terms equal to a, since q = 1. Let's check if q = 1 is possible.If q = 1, then a_n = a for all n. Then, b_n = 1 + a + a + ... + a (n times) = 1 + a n.Then, c_n = 2 + sum_{k=1}^n b_k = 2 + sum_{k=1}^n (1 + a k) = 2 + n + a sum_{k=1}^n k.Sum_{k=1}^n k = n(n+1)/2.So, c_n = 2 + n + a [n(n+1)/2].For c_n to be a geometric sequence, the expression 2 + n + (a/2) n(n+1) must be geometric. But this is a quadratic in n, which can't be a geometric sequence unless the coefficients of n and n^2 are zero.So, set coefficients to zero:Coefficient of n^2: a/2 = 0 => a = 0.Coefficient of n: 1 + a/2 = 1 + 0 = 1 ≠ 0.But if a = 0, then c_n = 2 + n, which isn't a geometric sequence. So, q = 1 is not a valid solution.Therefore, the only valid solution is q = 2.Then, from earlier, a = q - 1 = 2 - 1 = 1.Thus, a + q = 1 + 2 = 3, which is option B.I think that's solid. I considered both possibilities from the equation (2 - q)(1 - q) = 0, checked q = 1 and found it invalid, so q = 2 is the only solution. Then, a = 1, so a + q = 3.

answer：Okay, so I need to figure out how many miles I can ride in a taxi in Rivertown for 15, including a 3 tip. Let me try to break this down step by step.First, the total amount I have is 15. Out of this, I plan to give a 3 tip to the driver. So, the amount left for the actual fare is 15 minus 3, which is 12. That means I have 12 to spend on the taxi ride itself.Now, looking at the fare structure: The first 3/4 mile costs 3.00. After that, the fare is 0.25 for each additional 0.1 mile. Hmm, okay, so beyond the initial 3/4 mile, every 0.1 mile adds 0.25 to the fare.Let me write this down to make it clearer.Total money available: 15Tip: 3Money left for fare: 15 - 3 = 12Fare structure:- First 3/4 mile: 3.00- Additional mileage: 0.25 per 0.1 mileSo, after the first 3/4 mile, I have 12 - 3 = 9 left to spend on additional mileage.Now, I need to figure out how many additional miles I can cover with 9, given that each 0.1 mile costs 0.25.Let me denote the additional miles beyond the first 3/4 mile as x. Since the fare is 0.25 for each 0.1 mile, the cost for x miles would be:Cost = (x / 0.1) * 0.25Because for every 0.1 mile, it's 0.25, so I divide x by 0.1 to find out how many 0.1 miles are in x miles, and then multiply by 0.25.So, the equation becomes:(x / 0.1) * 0.25 = 9Let me solve for x.First, divide both sides by 0.25:(x / 0.1) = 9 / 0.25(x / 0.1) = 36Now, multiply both sides by 0.1:x = 36 * 0.1x = 3.6So, I can travel an additional 3.6 miles beyond the initial 3/4 mile.Now, let's add the initial 3/4 mile to this additional distance to get the total miles I can ride.Total miles = 3/4 + 3.6First, convert 3/4 to decimal to make it easier to add:3/4 = 0.75So,Total miles = 0.75 + 3.6 = 4.35Therefore, I can ride for a total of 4.35 miles with 15, including the 3 tip.Let me double-check my calculations to make sure I didn't make any mistakes.1. Total money: 152. Tip: 3, so fare money: 123. First 0.75 miles: 3, remaining fare: 94. Additional cost: 0.25 per 0.1 mile5. Number of 0.1 miles with 9: 9 / 0.25 = 366. Total additional miles: 36 * 0.1 = 3.67. Total miles: 0.75 + 3.6 = 4.35Everything seems to add up correctly. So, the answer should be 4.35 miles.