💡Okay, so I have this problem where I need to figure out how many numbers from the set {1, 2, 3, 4, 5, 6} must be divisors of the expression 3b + 15 - c. The given condition is that 4b = 10 - 3a + c, and a, b, c are integers. Hmm, let me try to break this down step by step.First, I think I should express b in terms of a and c so that I can substitute it into the expression 3b + 15 - c. That way, I can see if I can simplify it or find some common factors.Starting with the equation:4b = 10 - 3a + cIf I solve for b, I divide both sides by 4:b = (10 - 3a + c) / 4Okay, so now I have b expressed in terms of a and c. Let me substitute this into 3b + 15 - c.So, substituting:3b + 15 - c = 3*( (10 - 3a + c)/4 ) + 15 - cLet me compute this step by step. First, multiply 3 by the fraction:3*(10 - 3a + c)/4 = (30 - 9a + 3c)/4Now, add 15 and subtract c:(30 - 9a + 3c)/4 + 15 - cTo combine these terms, I need a common denominator. Let me express 15 as 60/4 and c as 4c/4:= (30 - 9a + 3c)/4 + 60/4 - 4c/4Now, combine the numerators:= (30 - 9a + 3c + 60 - 4c)/4Simplify the numerator:30 + 60 = 90-9a remains as is3c - 4c = -cSo, the numerator becomes:90 - 9a - cTherefore, the entire expression simplifies to:(90 - 9a - c)/4Hmm, okay. So, 3b + 15 - c simplifies to (90 - 9a - c)/4. Now, I need to determine how many numbers from {1, 2, 3, 4, 5, 6} must divide this expression.Let me write it again for clarity:3b + 15 - c = (90 - 9a - c)/4I can factor out a 9 from the first two terms in the numerator:= (9*(10 - a) - c)/4Hmm, not sure if that helps immediately. Maybe I can think about the numerator: 90 - 9a - c. Let me factor 9 out of the first two terms:= 9*(10 - a) - cBut I'm not sure if that helps with divisibility. Maybe I should think about the entire expression (90 - 9a - c)/4 and see if it can be expressed in a way that reveals common factors.Alternatively, perhaps I can consider the expression modulo each of the numbers in the set {1, 2, 3, 4, 5, 6} to see if it's always divisible by them, regardless of the values of a and c, as long as they satisfy the original equation.Let me try that approach.First, let's consider modulo 1. Well, any integer is divisible by 1, so 1 is always a divisor. So, 1 is definitely in the set of divisors.Next, modulo 2. Let's see if (90 - 9a - c)/4 is an integer and if it's even or not. Wait, but since a, b, c are integers, (90 - 9a - c) must be divisible by 4 because b is an integer. So, (90 - 9a - c) is divisible by 4, which means that (90 - 9a - c)/4 is an integer.But does that integer have to be even? Let's see. Let me think about the numerator: 90 is even, 9a is 9 times an integer. 9 is odd, so 9a is odd if a is odd, even if a is even. c is an integer, so c can be either even or odd.So, 90 is even. 9a: if a is even, 9a is even; if a is odd, 9a is odd. c: same, even or odd.So, 90 - 9a - c: Let's see.Case 1: a is even, c is even.90 (even) - 9a (even) - c (even) = even - even - even = even.Case 2: a is even, c is odd.90 (even) - 9a (even) - c (odd) = even - even - odd = odd.Case 3: a is odd, c is even.90 (even) - 9a (odd) - c (even) = even - odd - even = odd.Case 4: a is odd, c is odd.90 (even) - 9a (odd) - c (odd) = even - odd - odd = even.So, depending on the parity of a and c, the numerator can be even or odd. But since the numerator must be divisible by 4, it must be even in any case because 4 divides it. Wait, but in cases 2 and 3, the numerator is odd, which would mean that (90 - 9a - c) is odd, which contradicts the fact that it must be divisible by 4, which requires it to be even.Wait, that can't be. So, perhaps my earlier assumption is wrong. Let me think again.If 4b = 10 - 3a + c, then 10 - 3a + c must be divisible by 4. So, 10 - 3a + c ≡ 0 mod 4.So, 10 mod 4 is 2, 3a mod 4 is (3 mod 4)*(a mod 4) = 3*(a mod 4). So, 2 - 3*(a mod 4) + (c mod 4) ≡ 0 mod 4.So, rearranged: -3*(a mod 4) + (c mod 4) ≡ -2 mod 4, which is equivalent to 2 mod 4.So, -3*(a mod 4) + (c mod 4) ≡ 2 mod 4.Let me write this as:(c mod 4) ≡ 3*(a mod 4) + 2 mod 4.So, depending on a mod 4, c mod 4 is determined.Let me tabulate possible values:If a ≡ 0 mod 4:c ≡ 3*0 + 2 ≡ 2 mod 4If a ≡ 1 mod 4:c ≡ 3*1 + 2 ≡ 5 ≡ 1 mod 4If a ≡ 2 mod 4:c ≡ 3*2 + 2 ≡ 6 + 2 ≡ 8 ≡ 0 mod 4If a ≡ 3 mod 4:c ≡ 3*3 + 2 ≡ 9 + 2 ≡ 11 ≡ 3 mod 4So, c mod 4 is determined by a mod 4 as above.Now, going back to the numerator: 90 - 9a - c.Let me compute this modulo 4.90 mod 4: 90 / 4 = 22*4 + 2, so 90 ≡ 2 mod 4.-9a mod 4: 9 ≡ 1 mod 4, so -9a ≡ -a mod 4.-c mod 4: -c mod 4.So, overall:90 - 9a - c ≡ 2 - a - c mod 4.But from earlier, we have that c ≡ 3a + 2 mod 4.So, substitute c ≡ 3a + 2 mod 4 into the expression:2 - a - (3a + 2) mod 4= 2 - a - 3a - 2 mod 4= (-4a) mod 4= 0 mod 4So, the numerator is indeed divisible by 4, which we already knew.But let's see if it's divisible by 8, which would make (90 - 9a - c)/4 divisible by 2.Wait, but we are interested in whether (90 - 9a - c)/4 is divisible by 2, 3, 4, 5, or 6.So, let's analyze each divisor.First, 1: as I said earlier, always a divisor.Now, 2: For (90 - 9a - c)/4 to be divisible by 2, the numerator must be divisible by 8.So, 90 - 9a - c ≡ 0 mod 8.Let me check if this is necessarily true.From earlier, we have that 90 - 9a - c ≡ 0 mod 4, but does it necessarily hold mod 8?Let me see.From 4b = 10 - 3a + c, we have that 10 - 3a + c ≡ 0 mod 4, which we used earlier.But to check mod 8, let's see.10 mod 8 is 2, 3a mod 8 is 3*(a mod 8), c mod 8 is c mod 8.So, 2 - 3a + c ≡ 0 mod 8.But we don't have information about a and c mod 8, only mod 4.So, it's possible that 10 - 3a + c is congruent to 0 mod 4 but not necessarily 0 mod 8.Therefore, 90 - 9a - c = 9*(10 - a) - c.But 10 - a ≡ (10 mod 8 - a mod 8) = 2 - a mod 8.So, 9*(10 - a) ≡ 9*(2 - a mod 8) ≡ (18 - 9a mod 8) ≡ (2 - 1a mod 8) since 18 mod 8 is 2 and 9 mod 8 is 1.So, 9*(10 - a) ≡ 2 - a mod 8.Then, subtract c: 2 - a - c mod 8.But from earlier, 10 - 3a + c ≡ 0 mod 4, which implies that 2 - 3a + c ≡ 0 mod 4.But mod 8, we don't have that information.So, 2 - a - c mod 8 can be various values depending on a and c.Therefore, (90 - 9a - c) mod 8 is not necessarily 0, so (90 - 9a - c)/4 is not necessarily even.Therefore, 2 is not necessarily a divisor.Wait, but let me think again. Since 90 - 9a - c is divisible by 4, but not necessarily by 8, so (90 - 9a - c)/4 is an integer, but it's not necessarily even. So, 2 is not necessarily a divisor.Okay, moving on to 3.We need to check if (90 - 9a - c)/4 is divisible by 3.So, 90 - 9a - c ≡ 0 mod 12, because if (90 - 9a - c)/4 is divisible by 3, then 90 - 9a - c must be divisible by 12.Let me check if 90 - 9a - c is divisible by 12.90 mod 12 is 6, 9a mod 12 is 9*(a mod 12), c mod 12 is c mod 12.So, 6 - 9a - c ≡ 0 mod 12.But from the original equation, 4b = 10 - 3a + c, which implies 10 - 3a + c ≡ 0 mod 4, but we don't have information mod 12.So, it's possible that 90 - 9a - c is not divisible by 12.For example, let me pick a=0, then from 4b = 10 - 0 + c, so c = 4b -10.Then, 90 - 9a - c = 90 - 0 - (4b -10) = 100 - 4b.So, (100 - 4b)/4 = 25 - b.Is 25 - b divisible by 3? Well, b can be any integer, so 25 - b can be any integer, so it's not necessarily divisible by 3.Therefore, 3 is not necessarily a divisor.Next, 4: Since (90 - 9a - c)/4 is an integer, it's automatically divisible by 1 and 4.Wait, no. Wait, (90 - 9a - c)/4 is an integer, but that doesn't mean it's divisible by 4. For example, if (90 - 9a - c)/4 = 5, then it's not divisible by 4.Wait, no, actually, if (90 - 9a - c)/4 is an integer, then 4 divides (90 - 9a - c). But that doesn't mean that (90 - 9a - c)/4 is divisible by 4. It just means that (90 - 9a - c) is divisible by 4.So, 4 is a divisor of the numerator, but not necessarily of the result after division.Wait, but the question is about the divisors of the expression 3b + 15 - c, which is equal to (90 - 9a - c)/4. So, we need to find the divisors of this integer.So, 4 is a divisor of the numerator, but not necessarily of the result. So, 4 is not necessarily a divisor of (90 - 9a - c)/4.Wait, but let me think again. If (90 - 9a - c) is divisible by 4, then (90 - 9a - c)/4 is an integer, but that integer could be any integer, so it's not necessarily divisible by 4.For example, if (90 - 9a - c) = 4, then (90 - 9a - c)/4 = 1, which is not divisible by 4.So, 4 is not necessarily a divisor.Wait, but actually, 4 is a divisor of the numerator, but not necessarily of the result. So, 4 is not necessarily a divisor of the expression.Wait, but the expression is (90 - 9a - c)/4, which is an integer, but 4 is not necessarily a divisor of that integer.So, 4 is not necessarily a divisor.Wait, but let me think again. If I have an integer k = (90 - 9a - c)/4, then 4k = 90 - 9a - c. So, 4k is divisible by 4, but k itself is not necessarily divisible by 4.So, 4 is not necessarily a divisor of k.Therefore, 4 is not necessarily a divisor.Next, 5: Let's check if (90 - 9a - c)/4 is divisible by 5.So, 90 - 9a - c ≡ 0 mod 20, because if (90 - 9a - c)/4 is divisible by 5, then 90 - 9a - c must be divisible by 20.Let me see if 90 - 9a - c is necessarily divisible by 20.90 mod 20 is 10, 9a mod 20 is 9*(a mod 20), c mod 20 is c mod 20.So, 10 - 9a - c ≡ 0 mod 20.But from the original equation, 4b = 10 - 3a + c, which implies 10 - 3a + c ≡ 0 mod 4, but we don't have information mod 20.So, it's possible that 90 - 9a - c is not divisible by 20.For example, let me pick a=0, then c = 4b -10.So, 90 - 9a - c = 90 - 0 - (4b -10) = 100 - 4b.So, (100 - 4b)/4 = 25 - b.Is 25 - b divisible by 5? Well, b can be any integer, so 25 - b can be any integer, so it's not necessarily divisible by 5.Therefore, 5 is not necessarily a divisor.Finally, 6: Let's check if (90 - 9a - c)/4 is divisible by 6.For that, it must be divisible by both 2 and 3.We already saw that it's not necessarily divisible by 2 or 3, so it's not necessarily divisible by 6.Therefore, the only number in the set {1, 2, 3, 4, 5, 6} that must be a divisor of 3b + 15 - c is 1.Wait, but let me double-check. Maybe I missed something.Wait, 3b + 15 - c = (90 - 9a - c)/4.Let me see if this expression is always an integer, which it is because 90 - 9a - c is divisible by 4.But does it have to be divisible by any other number besides 1?Wait, let me think about specific examples.Let me choose a=0. Then, from 4b = 10 - 0 + c, so c = 4b -10.Then, 3b + 15 - c = 3b +15 - (4b -10) = 3b +15 -4b +10 = -b +25.So, -b +25. Since b can be any integer, this can be any integer. So, for example, if b=0, then 25, which is not divisible by 2,3,4,5,6 except 5 and 25.Wait, but 25 is divisible by 5, but not necessarily always.Wait, but in this case, if a=0, c=4b -10, so 3b +15 -c =25 -b.If b=1, then 24, which is divisible by 2,3,4,6.If b=2, then 23, which is prime, only divisible by 1 and 23.So, in this case, 23 is not divisible by 2,3,4,5,6.Therefore, sometimes it's divisible by more, sometimes not.Similarly, if I choose a=1.From 4b =10 -3*1 +c => 4b=7 +c => c=4b -7.Then, 3b +15 -c =3b +15 - (4b -7)=3b +15 -4b +7= -b +22.So, -b +22. If b=0, 22, which is divisible by 2,11.If b=1, 21, which is divisible by 3,7.If b=2, 20, divisible by 2,4,5,10.If b=3, 19, prime.So, again, sometimes divisible by more, sometimes not.Therefore, the only divisor that is always present is 1.Wait, but let me think again. Maybe I can factor the expression differently.We have 3b +15 -c = (90 -9a -c)/4.Let me factor 9 from the first two terms:= (9*(10 -a) -c)/4.Hmm, not sure if that helps.Alternatively, maybe I can write it as:= (90 - c -9a)/4.But I don't see a clear factorization that would reveal a common divisor.Alternatively, perhaps I can think about the expression modulo each number.But I think I already did that.So, in conclusion, the only number in the set {1,2,3,4,5,6} that must divide 3b +15 -c is 1.Therefore, the answer is 1.

💡Okay, so I have this problem about a triangle with sides 5, 12, and 13. It says that the triangle has both an inscribed circle and a circumscribed circle, and I need to find the distance between the centers of those two circles. Hmm, okay, let me think about how to approach this.First, I remember that in any triangle, the center of the inscribed circle is called the incenter, and the center of the circumscribed circle is called the circumcenter. So, I need to find the distance between the incenter and the circumcenter of this triangle.Now, the triangle has sides 5, 12, and 13. Wait a minute, 5, 12, 13 is a Pythagorean triple, right? So, this is a right-angled triangle. That should make things a bit easier because some properties of right-angled triangles might help here.In a right-angled triangle, the circumcenter is located at the midpoint of the hypotenuse. That's a useful fact. So, if I consider the triangle with sides 5, 12, and 13, the hypotenuse is 13. Therefore, the circumradius, which is the distance from the circumcenter to any vertex, should be half of the hypotenuse. So, the circumradius R is 13/2, which is 6.5.Now, what about the inradius? I recall that the inradius r of a triangle can be found using the formula r = A/s, where A is the area of the triangle and s is the semi-perimeter. Let me calculate that.First, the semi-perimeter s is (5 + 12 + 13)/2 = (30)/2 = 15.Next, the area A of the triangle. Since it's a right-angled triangle, the area is (base * height)/2. Here, the legs are 5 and 12, so the area is (5*12)/2 = 60/2 = 30.So, the inradius r is A/s = 30/15 = 2. Got it, the inradius is 2.Now, I need to find the distance between the incenter and the circumcenter. I remember there's a formula that relates the distance between the incenter and circumcenter with other parameters of the triangle. Let me try to recall it.I think the formula is IO² = R(R - 2r), where IO is the distance between the incenter and circumcenter, R is the circumradius, and r is the inradius. Let me check if that makes sense.Wait, if I plug in the values I have, R is 6.5 and r is 2. So, IO² = 6.5*(6.5 - 2*2) = 6.5*(6.5 - 4) = 6.5*2.5. Let me calculate that: 6.5*2.5 is 16.25. So, IO is the square root of 16.25, which is sqrt(16.25). Simplifying that, sqrt(16.25) is sqrt(65/4), which is (sqrt(65))/2. So, the distance is sqrt(65)/2.Wait, but I should make sure that this formula applies to all triangles or if it's specific to certain types. Hmm, I think it's a general formula for any triangle, but let me think if there's a different approach just to verify.Alternatively, I could try to find the coordinates of the incenter and circumcenter and then compute the distance between them. Maybe that would be a good way to verify my answer.Let's place the right-angled triangle on a coordinate system. Let me assume that the right angle is at the origin (0,0), one leg along the x-axis, and the other along the y-axis. So, the vertices are at (0,0), (5,0), and (0,12). Wait, but the hypotenuse is 13, so actually, the legs should be 5 and 12, making the hypotenuse 13. So, the coordinates would be (0,0), (5,0), and (0,12).First, let's find the circumcenter. Since it's the midpoint of the hypotenuse, which is the side opposite the right angle. The hypotenuse is between (5,0) and (0,12). The midpoint of this hypotenuse would be ((5+0)/2, (0+12)/2) = (2.5, 6). So, the circumcenter O is at (2.5, 6).Now, let's find the incenter. The incenter is located at the intersection of the angle bisectors. For a right-angled triangle, there's a formula for the inradius, which we already found to be 2. The coordinates of the incenter can be found using the formula:If the triangle has vertices at (0,0), (a,0), and (0,b), then the inradius r is given by r = (a + b - c)/2, where c is the hypotenuse. Wait, we already calculated r as 2, which matches because (5 + 12 - 13)/2 = (4)/2 = 2.The coordinates of the incenter are (r, r) in this coordinate system because it's equidistant from both legs. So, since r = 2, the incenter I is at (2, 2).Now, to find the distance between the incenter I(2,2) and the circumcenter O(2.5,6), I can use the distance formula:Distance IO = sqrt[(x2 - x1)^2 + (y2 - y1)^2] = sqrt[(2.5 - 2)^2 + (6 - 2)^2] = sqrt[(0.5)^2 + (4)^2] = sqrt[0.25 + 16] = sqrt[16.25] = sqrt(65/4) = (sqrt(65))/2.Okay, so that matches the result I got earlier using the formula IO² = R(R - 2r). So, both methods give me the same answer, which is reassuring.Just to make sure I didn't make any calculation errors, let me go through the steps again.First, confirming the circumradius R: hypotenuse is 13, so R = 13/2 = 6.5. Correct.Inradius r: area A = 30, semi-perimeter s = 15, so r = A/s = 30/15 = 2. Correct.Using the formula IO² = R(R - 2r): 6.5*(6.5 - 4) = 6.5*2.5 = 16.25. So, IO = sqrt(16.25) = sqrt(65/4) = sqrt(65)/2. Correct.Using coordinates: circumcenter at (2.5,6), incenter at (2,2). Distance squared is (0.5)^2 + (4)^2 = 0.25 + 16 = 16.25. So, distance is sqrt(16.25) = sqrt(65)/2. Correct.Therefore, I'm confident that the distance between the incenter and circumcenter is sqrt(65)/2.I think I've covered all the steps and verified the result using two different methods. So, I don't see any mistakes in my reasoning.

💡Alright, so I have this problem here: Given that ( x ) is a real number, the statement ( frac{1}{x} < 1 ) is the ( ) of ( x > 1 ). The options are A, B, C, D, which correspond to different types of conditions. I need to figure out whether ( frac{1}{x} < 1 ) is a necessary condition, a sufficient condition, both, or neither for ( x > 1 ).First, let me recall what necessary and sufficient conditions mean. A necessary condition is something that must be true for another statement to be true, but it doesn't guarantee it. On the other hand, a sufficient condition is something that, if it's true, guarantees the truth of another statement, but it's not necessarily the only way for that statement to be true.So, in this case, I need to see if ( frac{1}{x} < 1 ) is necessary for ( x > 1 ), or if it's sufficient, or both, or neither.Let me start by solving the inequality ( frac{1}{x} < 1 ). To do this, I'll consider different cases for ( x ) because the inequality can behave differently depending on whether ( x ) is positive or negative.Case 1: ( x > 0 )If ( x ) is positive, I can multiply both sides of the inequality by ( x ) without changing the direction of the inequality. So:( frac{1}{x} < 1 )Multiply both sides by ( x ):( 1 < x )Which simplifies to:( x > 1 )So, for positive ( x ), ( frac{1}{x} < 1 ) implies ( x > 1 ).Case 2: ( x < 0 )If ( x ) is negative, multiplying both sides of the inequality by ( x ) will reverse the inequality sign. So:( frac{1}{x} < 1 )Multiply both sides by ( x ) (negative, so inequality flips):( 1 > x )Which simplifies to:( x < 1 )But since ( x ) is already negative in this case, the inequality ( x < 1 ) is automatically true because all negative numbers are less than 1. So, for negative ( x ), ( frac{1}{x} < 1 ) is always true.Putting both cases together, the solution to ( frac{1}{x} < 1 ) is ( x > 1 ) or ( x < 0 ).Now, let's analyze the relationship between ( frac{1}{x} < 1 ) and ( x > 1 ).First, is ( frac{1}{x} < 1 ) a necessary condition for ( x > 1 )?A necessary condition must be true whenever the statement it's conditioning is true. So, if ( x > 1 ), does ( frac{1}{x} < 1 ) necessarily hold?From Case 1, when ( x > 1 ), ( frac{1}{x} < 1 ) is indeed true. So, ( frac{1}{x} < 1 ) is a necessary condition for ( x > 1 ).Next, is ( frac{1}{x} < 1 ) a sufficient condition for ( x > 1 )?A sufficient condition means that if it's true, then the statement it's conditioning must also be true. So, if ( frac{1}{x} < 1 ), does that necessarily mean ( x > 1 )?From the solution above, ( frac{1}{x} < 1 ) can be true in two cases: when ( x > 1 ) or when ( x < 0 ). So, ( frac{1}{x} < 1 ) doesn't guarantee that ( x > 1 ); it could also be that ( x < 0 ). Therefore, ( frac{1}{x} < 1 ) is not a sufficient condition for ( x > 1 ).Putting it all together, ( frac{1}{x} < 1 ) is a necessary but not sufficient condition for ( x > 1 ).I think that makes sense. Let me double-check. If ( x > 1 ), then ( frac{1}{x} < 1 ) must be true, so it's necessary. But ( frac{1}{x} < 1 ) can also be true when ( x ) is negative, so it doesn't guarantee ( x > 1 ). Therefore, it's not sufficient.Yeah, that seems right. So the answer should be option A: Necessary but not sufficient condition.Wait, hold on. The original problem says "the statement ( frac{1}{x} < 1 ) is the ( ) of ( x > 1 )." So, is ( frac{1}{x} < 1 ) the condition for ( x > 1 ), or is ( x > 1 ) the condition for ( frac{1}{x} < 1 )?I think I interpreted it as ( frac{1}{x} < 1 ) being the condition for ( x > 1 ), but maybe it's the other way around.Let me read it again: "the statement ( frac{1}{x} < 1 ) is the ( ) of ( x > 1 )."Hmm, that phrasing is a bit unclear. It could mean that ( frac{1}{x} < 1 ) is a condition related to ( x > 1 ). So, perhaps it's asking whether ( frac{1}{x} < 1 ) is a necessary or sufficient condition for ( x > 1 ).In that case, my previous reasoning still applies. ( frac{1}{x} < 1 ) is necessary for ( x > 1 ) because whenever ( x > 1 ), ( frac{1}{x} < 1 ) holds. But it's not sufficient because ( frac{1}{x} < 1 ) can also be true when ( x < 0 ).So, yes, the answer should be A: Necessary but not sufficient condition.Alternatively, if the question was reversed, asking whether ( x > 1 ) is a condition for ( frac{1}{x} < 1 ), then ( x > 1 ) would be a sufficient condition because if ( x > 1 ), then ( frac{1}{x} < 1 ). But it's not necessary because, as we saw, ( frac{1}{x} < 1 ) can also be true for ( x < 0 ).But the question specifically says ( frac{1}{x} < 1 ) is the ( ) of ( x > 1 ). So, it's about the relationship of ( frac{1}{x} < 1 ) to ( x > 1 ).Therefore, ( frac{1}{x} < 1 ) is a necessary condition for ( x > 1 ) because ( x > 1 ) implies ( frac{1}{x} < 1 ). But it's not sufficient because ( frac{1}{x} < 1 ) doesn't imply ( x > 1 ); it could also be ( x < 0 ).So, I think I'm confident now that the answer is A: Necessary but not sufficient condition.**Final Answer**boxed{A}

💡Okay, so I have this problem where Susie buys 5 muffins and 4 bananas, and Calvin buys 10 muffins and 18 bananas. Calvin spends three times as much as Susie. I need to figure out how many times more expensive a muffin is compared to a banana.Alright, let's start by assigning variables to the unknowns. Let me denote the cost of one muffin as ( m ) and the cost of one banana as ( b ). So, Susie's total cost would be ( 5m + 4b ), and Calvin's total cost would be ( 10m + 18b ).According to the problem, Calvin spends three times what Susie spends. So, I can write the equation:[ 10m + 18b = 3 times (5m + 4b) ]Let me simplify the right side of the equation first. Multiplying 3 by both terms inside the parentheses:[ 3 times 5m = 15m ][ 3 times 4b = 12b ]So, the equation becomes:[ 10m + 18b = 15m + 12b ]Now, I need to solve for ( m ) in terms of ( b ) to find out how many times more expensive a muffin is compared to a banana. Let's rearrange the equation to get all the ( m ) terms on one side and the ( b ) terms on the other side.Subtract ( 10m ) from both sides:[ 10m + 18b - 10m = 15m + 12b - 10m ][ 18b = 5m + 12b ]Now, subtract ( 12b ) from both sides:[ 18b - 12b = 5m + 12b - 12b ][ 6b = 5m ]So, ( 6b = 5m ). To find ( m ) in terms of ( b ), I can divide both sides by 5:[ m = frac{6b}{5} ]Simplifying that, ( m = 1.2b ). This means that a muffin is 1.2 times more expensive than a banana.Let me double-check my steps to make sure I didn't make any mistakes. I set up the equations correctly based on the problem statement. When I expanded the equation, I correctly multiplied 3 by both 5m and 4b. Then, I moved the terms correctly to isolate ( m ) and ( b ). The algebra seems straightforward, and I don't see any errors in the calculations.So, yes, a muffin is 1.2 times more expensive than a banana. That corresponds to option A.

💡Okay, so I have this problem where there's a ten-digit number that uses all ten digits from 0 to 9, and zero isn't in the first position. Then, the digits are reversed, and this reversed sequence is added to the right of the original number, making a twenty-digit number. I need to prove that this twenty-digit number is divisible by 99. Hmm, okay.First, I remember that 99 is 9 times 11, so if a number is divisible by both 9 and 11, it's divisible by 99. Maybe I can check both conditions separately.Starting with divisibility by 9: I know that a number is divisible by 9 if the sum of its digits is divisible by 9. The original ten-digit number uses all digits from 0 to 9, so the sum of its digits is 0+1+2+3+4+5+6+7+8+9. Let me calculate that: 0+1 is 1, plus 2 is 3, plus 3 is 6, plus 4 is 10, plus 5 is 15, plus 6 is 21, plus 7 is 28, plus 8 is 36, plus 9 is 45. So, the sum is 45.Since we're appending the reversed digits, the total sum of the twenty-digit number will be 45 (from the original) plus 45 (from the reversed), which is 90. 90 divided by 9 is 10, so 90 is divisible by 9. That means the twenty-digit number is divisible by 9. Okay, that part seems done.Now, for divisibility by 11: I remember that a number is divisible by 11 if the alternating sum of its digits is a multiple of 11. That is, if you take the sum of the digits in the odd positions and subtract the sum of the digits in the even positions, the result should be 0 or a multiple of 11.Let me think about how the twenty-digit number is structured. It's the original ten-digit number followed by its reverse. So, if the original number is ABCDEFGHIJ, the reversed number is JIHGFEDCBA, and the twenty-digit number is ABCDEFGHIJJ IHGFEDCBA.Wait, actually, it's ABCDEFGHIJ followed by JIHGFEDCBA, so the full number is ABCDEFGHIJJ IHGFEDCBA. Hmm, but that might not be the exact structure. Let me clarify: the original number is 10 digits, and the reversed number is also 10 digits, so together, it's 20 digits.Let me denote the original number as N = a1a2a3a4a5a6a7a8a9a10, where a1 is not zero. Then, the reversed number is a10a9a8a7a6a5a4a3a2a1. So, the twenty-digit number is N concatenated with reversed N, which is a1a2a3a4a5a6a7a8a9a10a10a9a8a7a6a5a4a3a2a1.Now, to check the divisibility by 11, I need to compute the alternating sum of the digits. Let's index the positions from 1 to 20. So, the first digit is a1 (position 1), second is a2 (position 2), ..., tenth is a10 (position 10), eleventh is a10 (position 11), twelfth is a9 (position 12), ..., twentieth is a1 (position 20).So, the sum of digits in odd positions (1,3,5,...,19) would be a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2.Similarly, the sum of digits in even positions (2,4,6,...,20) would be a2 + a4 + a6 + a8 + a10 + a9 + a7 + a5 + a3 + a1.Wait a minute, let me write that out:Sum of odd positions: a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2Sum of even positions: a2 + a4 + a6 + a8 + a10 + a9 + a7 + a5 + a3 + a1Hmm, interesting. If I look closely, the sum of odd positions is a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2, which is the same as (a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10) + (a10 + a9 + a8 + a7 + a6 + a5 + a4 + a3 + a2 + a1) minus the overlapping terms? Wait, no, that might not be the right way.Wait, actually, the sum of odd positions is the sum of the first digit, third digit, fifth digit, etc., up to the nineteenth digit. Similarly, the sum of even positions is the sum of the second digit, fourth digit, etc., up to the twentieth digit.But in the twenty-digit number, the first ten digits are the original number, and the last ten digits are the reversed number. So, the odd positions in the first ten digits are positions 1,3,5,7,9, and the odd positions in the last ten digits are positions 11,13,15,17,19.Similarly, the even positions in the first ten digits are positions 2,4,6,8,10, and the even positions in the last ten digits are positions 12,14,16,18,20.So, let me break it down:Sum of odd positions:- From the first ten digits: a1, a3, a5, a7, a9- From the last ten digits: a10 (position 11), a8 (position 13), a6 (position 15), a4 (position 17), a2 (position 19)So, total sum of odd positions: a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2Similarly, sum of even positions:- From the first ten digits: a2, a4, a6, a8, a10- From the last ten digits: a9 (position 12), a7 (position 14), a5 (position 16), a3 (position 18), a1 (position 20)So, total sum of even positions: a2 + a4 + a6 + a8 + a10 + a9 + a7 + a5 + a3 + a1Wait, so the sum of odd positions is a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2, and the sum of even positions is a2 + a4 + a6 + a8 + a10 + a9 + a7 + a5 + a3 + a1.If I rearrange the terms, both sums are actually the same: a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10.Because in the sum of odd positions, it's a1 + a3 + a5 + a7 + a9 + a10 + a8 + a6 + a4 + a2, which is just a permutation of all the digits. Similarly, the sum of even positions is a2 + a4 + a6 + a8 + a10 + a9 + a7 + a5 + a3 + a1, which is also a permutation of all the digits.Therefore, both sums are equal to 45, since the sum of all digits from 0 to 9 is 45. So, the sum of odd positions is 45, and the sum of even positions is also 45.Therefore, the alternating sum is 45 - 45 = 0, which is a multiple of 11 (since 0 is a multiple of every integer). Therefore, the twenty-digit number is divisible by 11.Since the number is divisible by both 9 and 11, it must be divisible by their least common multiple, which is 99. Therefore, the twenty-digit number is divisible by 99.Wait, let me double-check. I concluded that the sum of odd positions and the sum of even positions are both 45, so their difference is 0, which is divisible by 11. That seems correct. And earlier, the total sum of digits is 90, which is divisible by 9. So, yes, it should be divisible by 99.I think that covers both conditions. So, the twenty-digit number is indeed divisible by 99.

💡First, I need to determine the new price of the suit after a 25% increase from the original price of 200.To calculate the increase amount, I multiply the original price by 25%:25% of 200 is 50.Adding this increase to the original price gives:200 + 50 = 250.Next, I need to apply the 25% discount coupon to the increased price of 250.Calculating the discount amount:25% of 250 is 62.50.Subtracting the discount from the increased price:250 - 62.50 = 187.50.Therefore, the final price that consumers paid for the suit after using the coupon is 187.50.