answer：First, I need to find the equation of the linear function ( y = kx + b ) that passes through the points ( (1, 5) ) and ( (-1, 1) ).To determine the slope ( k ), I'll use the formula for the slope between two points:[k = frac{y_2 - y_1}{x_2 - x_1} = frac{1 - 5}{-1 - 1} = frac{-4}{-2} = 2]Now that I have the slope ( k = 2 ), I'll substitute one of the points into the equation to find ( b ). Using the point ( (1, 5) ):[5 = 2(1) + b implies 5 = 2 + b implies b = 3]So, the equation of the linear function is ( y = 2x + 3 ).Next, to find the value of ( y ) when ( x = -4 ), I'll substitute ( x = -4 ) into the equation:[y = 2(-4) + 3 = -8 + 3 = -5]

answer：Okay, so I have this problem about three pirates dividing a sack of coins. Let me try to figure it out step by step. First, the problem says the first pirate took 3/7 of all the coins. Hmm, so if the total number of coins is, let's say, N, then the first pirate took (3/7)N coins. That leaves the remainder for the other two pirates. So the remainder would be N - (3/7)N, right? Let me calculate that: N - (3/7)N is the same as (4/7)N. So the first pirate took 3/7, leaving 4/7 of the coins for the other two pirates.Next, the second pirate took 51% of the remainder. The remainder after the first pirate took his share is (4/7)N. So 51% of that would be 0.51 * (4/7)N. Let me compute that: 0.51 * (4/7) is equal to (0.51 * 4)/7, which is 2.04/7. Hmm, that's approximately 0.2914. So the second pirate took about 0.2914N coins. But maybe I should keep it as fractions to be precise. Let me see: 51% is 51/100, so 51/100 * 4/7 = (51*4)/(100*7) = 204/700. Simplifying that, both numerator and denominator can be divided by 4: 204 ÷ 4 is 51, and 700 ÷ 4 is 175. So 51/175. Therefore, the second pirate took 51/175 of the total coins. So, the second pirate took 51/175N coins. That means the remainder after the second pirate took his share is (4/7)N - (51/175)N. Let me compute that. First, I need a common denominator. 4/7 is equal to 100/175, right? Because 4*25=100 and 7*25=175. So, 100/175 - 51/175 = 49/175. So the remainder is 49/175N, which simplifies to 7/25N because 49 divided by 7 is 7 and 175 divided by 7 is 25. So the third pirate gets 7/25N coins.But wait, the problem says that after the first two pirates took their shares, the third pirate had 8 coins less than the second one. So, the third pirate's share is 7/25N, and the second pirate's share is 51/175N. So, according to the problem, 7/25N = 51/175N - 8. Let me write that equation down:7/25N = 51/175N - 8.Hmm, okay, let's solve for N. First, let me subtract 51/175N from both sides to get all the terms involving N on one side. So:7/25N - 51/175N = -8.To subtract these fractions, I need a common denominator. The denominators are 25 and 175. 25 goes into 175 seven times, so 175 is the common denominator. Let me convert 7/25 to 49/175 because 7*7=49 and 25*7=175. So now, the equation becomes:49/175N - 51/175N = -8.Subtracting these gives:(49 - 51)/175N = -8,which simplifies to:(-2)/175N = -8.So, (-2/175)N = -8. Let me multiply both sides by -1 to eliminate the negative signs:(2/175)N = 8.Now, to solve for N, I can multiply both sides by 175/2:N = 8 * (175/2).Calculating that: 8 divided by 2 is 4, so 4 * 175 = 700. So N is 700.Wait, let me verify that. If N is 700, then the first pirate took 3/7 of 700, which is 300 coins. That leaves 400 coins. The second pirate took 51% of 400, which is 0.51 * 400 = 204 coins. Then, the third pirate took the remainder, which is 400 - 204 = 196 coins. According to the problem, the third pirate should have 8 coins less than the second one. The second pirate had 204, so 204 - 8 = 196. That's exactly what the third pirate took. So, yes, that checks out.Therefore, the total number of coins in the sack was 700.**Final Answer**The total number of coins in the sack was boxed{700}.

answer：First, I'll define the number of trips Bill makes as ( x ).Since Jean makes 6 more trips than Bill, the number of trips Jean makes is ( x + 6 ).Together, the total number of trips they make is 40. So, I can set up the equation:( x + (x + 6) = 40 ).Combining like terms, the equation becomes:( 2x + 6 = 40 ).Subtracting 6 from both sides gives:( 2x = 34 ).Dividing both sides by 2, I find:( x = 17 ).Therefore, Jean makes ( 17 + 6 = 23 ) trips.

answer：Alright, so I've got this problem about a water tank with five pipes. It's a bit tricky, but let me try to figure it out step by step. First, let me parse the information given:1. The first four pipes together take 6 hours to fill the tank.2. The last four pipes together take 8 hours to fill the tank.3. If only the first and fifth pipes are opened, it takes 12 hours to fill the tank.And the question is: How many hours are needed to fill the tank if only the fifth pipe is used?Hmm, okay. So, I think I need to model the rates at which each pipe fills the tank. Since all these pipes are working together, their rates should add up. Let me denote the rates of the pipes as follows:- Let pipe 1 have a rate of ( r_1 ) (tank per hour)- Pipe 2: ( r_2 )- Pipe 3: ( r_3 )- Pipe 4: ( r_4 )- Pipe 5: ( r_5 )So, the first piece of information says that the first four pipes together take 6 hours. That means their combined rate is ( frac{1}{6} ) tank per hour. So, mathematically, that's:( r_1 + r_2 + r_3 + r_4 = frac{1}{6} ) ...(1)Similarly, the last four pipes take 8 hours together. So their combined rate is ( frac{1}{8} ) tank per hour. That gives:( r_2 + r_3 + r_4 + r_5 = frac{1}{8} ) ...(2)And the third piece of information is that only the first and fifth pipes take 12 hours together. So their combined rate is ( frac{1}{12} ) tank per hour:( r_1 + r_5 = frac{1}{12} ) ...(3)Okay, so now I have three equations with five variables. Hmm, that seems underdetermined, but maybe I can find a relationship between the variables.Let me see. If I subtract equation (1) from equation (2), I can eliminate ( r_2 + r_3 + r_4 ). Let's try that:Equation (2) - Equation (1):( (r_2 + r_3 + r_4 + r_5) - (r_1 + r_2 + r_3 + r_4) = frac{1}{8} - frac{1}{6} )Simplify:( r_5 - r_1 = frac{1}{8} - frac{1}{6} )Compute the right-hand side:( frac{1}{8} - frac{1}{6} = frac{3}{24} - frac{4}{24} = -frac{1}{24} )So, ( r_5 - r_1 = -frac{1}{24} )Which can be rewritten as:( r_1 = r_5 + frac{1}{24} ) ...(4)Okay, so now I have an expression for ( r_1 ) in terms of ( r_5 ). Let's plug this into equation (3):Equation (3): ( r_1 + r_5 = frac{1}{12} )Substitute ( r_1 ):( (r_5 + frac{1}{24}) + r_5 = frac{1}{12} )Simplify:( 2r_5 + frac{1}{24} = frac{1}{12} )Subtract ( frac{1}{24} ) from both sides:( 2r_5 = frac{1}{12} - frac{1}{24} )Compute the right-hand side:( frac{2}{24} - frac{1}{24} = frac{1}{24} )So, ( 2r_5 = frac{1}{24} )Divide both sides by 2:( r_5 = frac{1}{48} )So, the rate of the fifth pipe is ( frac{1}{48} ) tank per hour. Therefore, the time it takes to fill the tank alone is the reciprocal of this rate, which is 48 hours.Wait, let me just verify that. If ( r_5 = frac{1}{48} ), then time ( t = frac{1}{r_5} = 48 ) hours. That seems right.But just to make sure I didn't make any mistakes, let me check my steps again.1. Equations (1) and (2) subtracted to get ( r_5 - r_1 = -1/24 ). That seems correct.2. Then equation (3) substituted ( r_1 = r_5 + 1/24 ) into it, leading to ( 2r_5 + 1/24 = 1/12 ). Then solving for ( r_5 ) gives ( 1/48 ). That seems correct.Alternatively, let me compute the rates of the other pipes to see if everything adds up.From equation (4): ( r_1 = r_5 + 1/24 = 1/48 + 1/24 = 1/48 + 2/48 = 3/48 = 1/16 ). So, ( r_1 = 1/16 ).Then, from equation (1): ( r_1 + r_2 + r_3 + r_4 = 1/6 ). So, ( 1/16 + (r_2 + r_3 + r_4) = 1/6 ). Therefore, ( r_2 + r_3 + r_4 = 1/6 - 1/16 ).Compute ( 1/6 - 1/16 ):Convert to common denominator, which is 48:( 8/48 - 3/48 = 5/48 ). So, ( r_2 + r_3 + r_4 = 5/48 ).From equation (2): ( r_2 + r_3 + r_4 + r_5 = 1/8 ). We know ( r_2 + r_3 + r_4 = 5/48 ) and ( r_5 = 1/48 ). So, adding them together: ( 5/48 + 1/48 = 6/48 = 1/8 ). Which matches equation (2). So that's consistent.Also, equation (3): ( r_1 + r_5 = 1/16 + 1/48 = 3/48 + 1/48 = 4/48 = 1/12 ). Which is correct.So, all the equations are consistent with ( r_5 = 1/48 ). Therefore, the time needed for the fifth pipe alone is 48 hours.I think that makes sense. So, the answer is 48 hours.**Final Answer**The fifth pipe alone takes boxed{48} hours to fill the tank.

answer：Alright, so I have this geometry problem here. Let me try to visualize it first. There's a right-angled triangle, BAD, right-angled at B. So, point B is the right angle, and the triangle is BAD. That means sides BA and BD are the legs, and AD is the hypotenuse.Now, on AD, there's a point C such that AC equals CD. So, point C is the midpoint of AD. That makes sense because if AC = CD, then C divides AD into two equal parts. So, C is the midpoint.Also, it's given that AB equals BC. Hmm, so AB is one of the legs of the triangle BAD, and BC is a segment from B to C, which is the midpoint of AD. So, AB = BC.We need to find the magnitude of angle DAB. So, angle at point A between sides AB and AD. Let's denote angle DAB as θ. So, θ is the angle we need to find.Let me try to draw this triangle to get a better idea. So, triangle BAD is right-angled at B. Let me place point B at the origin (0,0) for simplicity. Let me assume point A is somewhere on the x-axis, say at (a, 0), and point D is somewhere on the y-axis, say at (0, d). So, then AD would be the hypotenuse from (a, 0) to (0, d).Point C is the midpoint of AD, so its coordinates would be the average of the coordinates of A and D. So, coordinates of C would be ((a + 0)/2, (0 + d)/2) = (a/2, d/2).Now, AB is the distance from A to B, which is just the length along the x-axis from (a, 0) to (0, 0), so AB = a.BC is the distance from B to C. Since B is at (0,0) and C is at (a/2, d/2), the distance BC can be calculated using the distance formula: sqrt[(a/2 - 0)^2 + (d/2 - 0)^2] = sqrt[(a^2)/4 + (d^2)/4] = (sqrt(a^2 + d^2))/2.It's given that AB = BC, so AB = a, and BC = (sqrt(a^2 + d^2))/2. Therefore, setting them equal:a = (sqrt(a^2 + d^2))/2Let me solve this equation for a and d. Multiply both sides by 2:2a = sqrt(a^2 + d^2)Now, square both sides to eliminate the square root:(2a)^2 = (sqrt(a^2 + d^2))^24a^2 = a^2 + d^2Subtract a^2 from both sides:3a^2 = d^2So, d^2 = 3a^2, which means d = a*sqrt(3)So, the length of BD is d = a*sqrt(3), and AB is a.Now, we need to find angle DAB, which is angle at point A between AB and AD.In triangle ABD, which is right-angled at B, we can find angle DAB.In triangle ABD, angle at A is θ. The sides adjacent to θ are AB and AD.Wait, no. Let me think again. In triangle ABD, angle at A is θ, and the sides are AB, AD, and BD.Wait, triangle ABD is right-angled at B, so sides AB and BD are the legs, and AD is the hypotenuse.So, to find angle DAB, which is angle at A, we can use trigonometric ratios.In triangle ABD, tan(theta) = opposite/adjacent = BD / AB.We have BD = d = a*sqrt(3), and AB = a.So, tan(theta) = (a*sqrt(3)) / a = sqrt(3)So, tan(theta) = sqrt(3). Therefore, theta is arctangent of sqrt(3). We know that tan(60 degrees) = sqrt(3). So, theta is 60 degrees.Wait, but let me double-check because sometimes I might confuse the sides.Wait, in triangle ABD, angle at A is theta. The sides are AB (adjacent), BD (opposite), and AD (hypotenuse). So, yes, tan(theta) = opposite / adjacent = BD / AB = (a*sqrt(3))/a = sqrt(3). So, theta is 60 degrees.But wait, let me make sure that this is consistent with point C.We have point C as the midpoint of AD, and AB = BC.We found that d = a*sqrt(3), so BD = a*sqrt(3). So, in triangle ABD, sides AB = a, BD = a*sqrt(3), and AD = sqrt(a^2 + (a*sqrt(3))^2) = sqrt(a^2 + 3a^2) = sqrt(4a^2) = 2a. So, AD = 2a, which makes sense because AC = CD = a each.So, point C is at (a/2, d/2) = (a/2, (a*sqrt(3))/2). So, coordinates of C are (a/2, (a*sqrt(3))/2).Now, distance from B to C is sqrt[(a/2)^2 + ((a*sqrt(3))/2)^2] = sqrt[(a^2)/4 + (3a^2)/4] = sqrt[(4a^2)/4] = sqrt(a^2) = a. So, BC = a, which is equal to AB = a. So, that checks out.Therefore, angle DAB is 60 degrees, which is option B.Wait, but let me think again. Is there another way to approach this problem?Alternatively, maybe using coordinate geometry or vectors.But I think the approach I took is correct. Let me just recap:1. Assign coordinates to the triangle with B at (0,0), A at (a,0), D at (0,d).2. Midpoint C is at (a/2, d/2).3. Distance BC is sqrt[(a/2)^2 + (d/2)^2] = (sqrt(a^2 + d^2))/2.4. Given AB = BC, so a = (sqrt(a^2 + d^2))/2.5. Solving gives d = a*sqrt(3).6. Therefore, in triangle ABD, tan(theta) = BD / AB = sqrt(3), so theta = 60 degrees.Yes, that seems consistent.Alternatively, maybe using triangle properties. Since C is the midpoint of AD and AB = BC, triangle ABC is also an isosceles triangle with AB = BC.Wait, but point C is on AD, so triangle ABC is not necessarily a triangle unless we connect points A, B, and C.Wait, let me see. So, points A, B, and C form triangle ABC, where AB = BC, so it's an isosceles triangle with AB = BC. So, angles opposite those sides are equal.But in triangle ABC, sides AB = BC, so angles at A and C are equal.Wait, but point C is on AD, so maybe we can find some relationships there.Alternatively, maybe using the median properties. Since C is the midpoint of AD, BC is the median from B to AD.In a right-angled triangle, the median to the hypotenuse is equal to half the hypotenuse. Wait, but in triangle ABD, AD is the hypotenuse, so the median from B to AD is BC, which should be equal to half of AD.But in our case, BC is equal to AB, which is a. And AD is 2a, so half of AD is a, which is equal to BC. So, that's consistent.So, BC = a = AB, which is given.Therefore, everything checks out.So, angle DAB is 60 degrees, which is option B.I think that's solid. I don't see any mistakes in the reasoning. So, I feel confident that the answer is 60 degrees.**Final Answer**boxed{B}

answer：First, I need to find a three-digit number that, when multiplied by 7, results in a product where the last three digits are 638. Let's denote this three-digit number as ( x ).The equation based on the problem statement is:[ 7x = 1000k + 638 ]where ( k ) is an integer representing the thousands place digit.Rearranging the equation to solve for ( x ):[ x = frac{1000k + 638}{7} ]To ensure ( x ) is an integer, ( 1000k + 638 ) must be divisible by 7. I'll check the divisibility by calculating the remainder when ( 1000k + 638 ) is divided by 7.Breaking it down:- ( 1000 div 7 ) leaves a remainder of 6.- ( 638 div 7 ) leaves a remainder of 5.So, the total remainder is:[ (6k + 5) mod 7 = 0 ]Solving for ( k ):[ 6k + 5 equiv 0 pmod{7} ][ 6k equiv -5 pmod{7} ][ 6k equiv 2 pmod{7} ][ k equiv 6 pmod{7} ]The smallest positive integer value for ( k ) is 6. Substituting ( k = 6 ) back into the equation for ( x ):[ x = frac{1000 times 6 + 638}{7} = frac{6638}{7} = 948 ]Finally, verifying the result:[ 948 times 7 = 6636 ]The last three digits of 6636 are indeed 636, which matches the problem's condition.