💡First, I need to understand the costs involved. Grisha has 5000 rubles. Each chocolate bunny costs 45 rubles, and each bag costs 30 rubles. Each bag can hold up to 30 bunnies.Next, I'll calculate the cost of a full bag, which includes both the bunnies and the bag itself. - The cost of 30 bunnies is 30 multiplied by 45, which equals 1350 rubles.- Adding the cost of the bag, the total cost for a full bag is 1350 plus 30, totaling 1380 rubles.Now, I'll determine how many full bags Grisha can buy with his 5000 rubles.- Dividing 5000 by 1380 gives approximately 3.623, so Grisha can buy 3 full bags.Calculating the money spent on these 3 full bags:- 3 bags multiplied by 1380 rubles per bag equals 4140 rubles.Subtracting this from Grisha's total money:- 5000 minus 4140 leaves him with 860 rubles.With the remaining 860 rubles, Grisha can buy additional bunnies and possibly another bag. - The cost of one more empty bag is 30 rubles, leaving him with 830 rubles.- With 830 rubles, he can buy 18 additional bunnies (since 18 multiplied by 45 is 810 rubles).Finally, subtracting the cost of the additional bunnies:- 830 minus 810 leaves Grisha with 20 rubles remaining.

💡Alright, so I have this math problem here: "The reciprocal of -frac{1}{2} is" with options A: -2, B: 2, C: -frac{1}{2}, and D: frac{1}{2}. Hmm, okay, I need to find the reciprocal of -frac{1}{2}. First off, what does reciprocal mean? I remember that the reciprocal of a number is like flipping it, right? So if I have a fraction, I just switch the numerator and the denominator. For example, the reciprocal of frac{2}{3} is frac{3}{2}. But wait, what if the number is negative? Does that affect anything?Let me think. If I have a negative fraction like -frac{1}{2}, the reciprocal should still be flipping the numerator and the denominator, but keeping the negative sign, right? So flipping -frac{1}{2} would give me -2/1, which simplifies to -2. So that would be option A.But just to make sure I'm not making a mistake, let me double-check. The reciprocal of a number x is defined as frac{1}{x}. So if x = -frac{1}{2}, then the reciprocal should be frac{1}{-frac{1}{2}}. Calculating that, frac{1}{-frac{1}{2}} is the same as multiplying by the reciprocal, which is 1 times -2 = -2. Yep, that confirms it. So the reciprocal is indeed -2, which is option A.Wait, but what if I approached it differently? Let's say I think about it in terms of multiplication. If I multiply -frac{1}{2} by its reciprocal, the result should be 1. So let's test that. If I take -frac{1}{2} and multiply it by -2, what do I get?-frac{1}{2} times -2 = frac{2}{2} = 1. Perfect, that works out. So multiplying -frac{1}{2} by -2 gives me 1, which is exactly what the reciprocal should do. That further solidifies that option A is correct.Is there any chance I could have messed up the signs? Let me see. If I had thought the reciprocal was positive 2, that would be option B. But if I multiply -frac{1}{2} by 2, I get -frac{2}{2} = -1, which is not 1. So that wouldn't work. Similarly, if I had picked option C, which is -frac{1}{2}, multiplying that by -frac{1}{2} would give me frac{1}{4}, which is also not 1. And option D, frac{1}{2}, multiplied by -frac{1}{2} would give me -frac{1}{4}, still not 1. So none of the other options work, only option A gives me the product of 1 when multiplied by -frac{1}{2}.I guess another way to look at it is that the reciprocal of a number is essentially the number that, when multiplied together, equals 1. So for -frac{1}{2}, I need a number that when multiplied by it gives 1. Since -frac{1}{2} is a negative fraction, its reciprocal should also be negative to ensure the product is positive 1. And since the original number is a fraction less than 1, its reciprocal should be a number greater than 1 in absolute value. So -2 fits both criteria: it's negative and its absolute value is greater than 1.Just to recap, the steps I took were:1. Understand what a reciprocal is.2. Apply the reciprocal definition to -frac{1}{2}.3. Check the result by multiplying to ensure it equals 1.4. Verify that the sign and magnitude make sense.All these steps led me to conclude that the reciprocal of -frac{1}{2} is indeed -2, which is option A.

💡Okay, so I have this problem about an ellipse given by the equation (frac{x^2}{25} + frac{y^2}{9} = 1). I need to find the length of (|ON|), where (O) is the origin, (M) is a point on the ellipse that's 2 units away from one of the foci, (F_1), and (N) is the midpoint of the segment (MF_1).First, I remember that the standard form of an ellipse is (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a) is the semi-major axis and (b) is the semi-minor axis. In this case, (a^2 = 25) and (b^2 = 9), so (a = 5) and (b = 3). That means the major axis is along the x-axis because the larger denominator is under (x^2).Next, I recall that the distance from the center to each focus of an ellipse is given by (c), where (c^2 = a^2 - b^2). Plugging in the values, (c^2 = 25 - 9 = 16), so (c = 4). Therefore, the foci (F_1) and (F_2) are located at ((-4, 0)) and ((4, 0)) respectively since the major axis is along the x-axis.Now, the problem states that point (M) is on the ellipse and is 2 units away from focus (F_1). I remember that one of the defining properties of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant and equal to the major axis length, which is (2a). In this case, that sum is (2 times 5 = 10).So, if (|MF_1| = 2), then (|MF_2|) must be (10 - 2 = 8). That makes sense because the total distance from (M) to both foci is always 10, so if it's 2 units away from one, it must be 8 units away from the other.Now, point (N) is the midpoint of segment (MF_1). To find the coordinates of (N), I can use the midpoint formula. If (M) has coordinates ((x, y)) and (F_1) is at ((-4, 0)), then the midpoint (N) would have coordinates (left(frac{x + (-4)}{2}, frac{y + 0}{2}right)) which simplifies to (left(frac{x - 4}{2}, frac{y}{2}right)).But I need to find the length (|ON|), which is the distance from the origin (O(0, 0)) to the point (N). Using the distance formula, (|ON| = sqrt{left(frac{x - 4}{2}right)^2 + left(frac{y}{2}right)^2}).Hmm, this seems a bit complicated. Maybe there's a smarter way to approach this without having to find the exact coordinates of (M). Let me think about the properties of ellipses and midpoints.I remember that in an ellipse, the midpoint of the segment joining a point on the ellipse and one of the foci has a special relationship with the other focus. Specifically, the distance from the origin to this midpoint is half the distance from the point on the ellipse to the other focus. So, (|ON| = frac{1}{2}|MF_2|).Wait, is that correct? Let me verify. If (N) is the midpoint of (MF_1), then vectorially, (N = frac{M + F_1}{2}). So, (ON = frac{OM + OF_1}{2}). Since (F_1) is at ((-4, 0)), (OF_1) is just the vector ((-4, 0)). Therefore, (ON = frac{OM + (-4, 0)}{2}).But I need to relate this to (MF_2). Let me think about the coordinates again. If (M) is ((x, y)), then (F_2) is ((4, 0)). The distance (|MF_2|) is 8, as we found earlier.Is there a way to express (|ON|) in terms of (|MF_2|)? Let's see. If (N) is the midpoint of (MF_1), then (N = left(frac{x - 4}{2}, frac{y}{2}right)). The distance from (O) to (N) is (sqrt{left(frac{x - 4}{2}right)^2 + left(frac{y}{2}right)^2}).Alternatively, maybe I can express this in terms of vectors. Let me denote vectors for simplicity. Let (vec{OM} = (x, y)) and (vec{OF_1} = (-4, 0)). Then, (vec{ON} = frac{vec{OM} + vec{OF_1}}{2}).So, (vec{ON} = frac{(x, y) + (-4, 0)}{2} = left(frac{x - 4}{2}, frac{y}{2}right)). The magnitude of (vec{ON}) is (sqrt{left(frac{x - 4}{2}right)^2 + left(frac{y}{2}right)^2}).But I need to find this magnitude. Maybe I can relate it to (|MF_2|). Let's compute (|MF_2|):(|MF_2| = sqrt{(x - 4)^2 + y^2} = 8).So, ((x - 4)^2 + y^2 = 64).Now, let's compute (|ON|^2):(|ON|^2 = left(frac{x - 4}{2}right)^2 + left(frac{y}{2}right)^2 = frac{(x - 4)^2 + y^2}{4} = frac{64}{4} = 16).Therefore, (|ON| = sqrt{16} = 4).Wait, that's a much simpler way! Instead of trying to find the coordinates of (M), I can use the fact that (|MF_2| = 8) and relate it directly to (|ON|) through the midpoint formula.So, by calculating (|ON|^2) as (frac{(x - 4)^2 + y^2}{4}), and knowing that ((x - 4)^2 + y^2 = 64), I can substitute and find that (|ON| = 4).That makes sense! So, the length of (|ON|) is 4 units.**Final Answer**boxed{B}

💡Okay, so I have this problem here: Suppose ( y = f(x) ) has an inverse function ( y = f^{-1}(x) ). Also, ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverse functions of each other. I need to find the value of ( f^{-1}(2004) - f^{-1}(1) ).Hmm, let me try to break this down step by step. First, I know that if two functions are inverses of each other, then applying one after the other should get me back to where I started. So, if ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverses, then:1. ( f^{-1}(f(x + 2) - 1) = x )2. ( f(f^{-1}(x - 1) + 2) = x )Wait, let me make sure I got that right. If ( f ) and ( f^{-1} ) are inverses, then ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ). So, if ( f(x + 2) ) and ( f^{-1}(x - 1) ) are inverses, then:- Applying ( f^{-1}(x - 1) ) after ( f(x + 2) ) should give me ( x ).- Similarly, applying ( f(x + 2) ) after ( f^{-1}(x - 1) ) should also give me ( x ).Let me write that out more formally.First, let's take ( f(x + 2) ) and apply ( f^{-1}(x - 1) ) to it:( f^{-1}(f(x + 2) - 1) = x )Similarly, take ( f^{-1}(x - 1) ) and apply ( f(x + 2) ) to it:( f(f^{-1}(x - 1) + 2) = x )Okay, so both of these equations should hold true because they are inverses.Let me try to simplify the first equation:( f^{-1}(f(x + 2) - 1) = x )Since ( f^{-1}(f(z)) = z ), if I let ( z = x + 2 ), then ( f^{-1}(f(z)) = z ). But here, it's ( f^{-1}(f(z) - 1) ). Hmm, that's a bit different. Maybe I can manipulate it.Let me denote ( w = f(x + 2) ). Then the equation becomes:( f^{-1}(w - 1) = x )But ( w = f(x + 2) ), so substituting back:( f^{-1}(f(x + 2) - 1) = x )I'm not sure if that helps directly. Maybe I should look at the second equation:( f(f^{-1}(x - 1) + 2) = x )Again, let me let ( u = f^{-1}(x - 1) ). Then the equation becomes:( f(u + 2) = x )But ( u = f^{-1}(x - 1) ), so substituting back:( f(f^{-1}(x - 1) + 2) = x )Hmm, similar structure. Maybe I can relate these two equations somehow.Wait, let me think about the relationship between ( f ) and ( f^{-1} ). If I have ( f(x + 2) ) and ( f^{-1}(x - 1) ) as inverses, then perhaps there's a shift involved in the inverse function.Let me recall that if ( f ) is shifted horizontally, its inverse will be shifted vertically, or something like that. Maybe I can express ( f^{-1}(x - 1) ) in terms of ( f^{-1}(x) ).Alternatively, let's try to find a relationship between ( f^{-1}(x) ) and ( f^{-1}(x - 1) ). Maybe there's a functional equation here.From the first equation:( f^{-1}(f(x + 2) - 1) = x )Let me denote ( y = f(x + 2) ). Then, ( f^{-1}(y - 1) = x ). But ( y = f(x + 2) ), so ( f^{-1}(y - 1) = x ). Therefore, ( f^{-1}(y - 1) = x ) implies that ( y - 1 = f(x) ), because ( f^{-1}(z) = x ) implies ( z = f(x) ).So, ( y - 1 = f(x) ), but ( y = f(x + 2) ), so substituting back:( f(x + 2) - 1 = f(x) )Ah! That's a functional equation: ( f(x + 2) = f(x) + 1 ). So, this tells me that shifting the input by 2 increases the output by 1. That suggests that ( f ) is a linear function with a slope of 1/2. Because if you increase x by 2, f(x) increases by 1, so the rate of change is 1/2.Wait, is that necessarily the case? Let me think. If ( f(x + 2) = f(x) + 1 ), then for any x, adding 2 to x increases f(x) by 1. So, the function has a constant rate of change, which is 1/2. Therefore, f is a linear function with slope 1/2.So, ( f(x) = frac{1}{2}x + c ), where c is a constant. Let me check if that works.If ( f(x) = frac{1}{2}x + c ), then ( f(x + 2) = frac{1}{2}(x + 2) + c = frac{1}{2}x + 1 + c ). On the other hand, ( f(x) + 1 = frac{1}{2}x + c + 1 ). So yes, they are equal. So that works.Therefore, ( f(x) = frac{1}{2}x + c ). Now, let's find its inverse function ( f^{-1}(x) ).To find the inverse, we can swap x and y and solve for y:( x = frac{1}{2}y + c )Multiply both sides by 2:( 2x = y + 2c )So,( y = 2x - 2c )Therefore, ( f^{-1}(x) = 2x - 2c ).Now, let's see if this fits with the other condition that ( f(x + 2) ) and ( f^{-1}(x - 1) ) are inverses.We already have ( f(x + 2) = frac{1}{2}(x + 2) + c = frac{1}{2}x + 1 + c ).And ( f^{-1}(x - 1) = 2(x - 1) - 2c = 2x - 2 - 2c ).Now, let's check if these are inverses. Let's apply ( f^{-1}(x - 1) ) to ( f(x + 2) ):( f^{-1}(f(x + 2) - 1) = f^{-1}left( frac{1}{2}x + 1 + c - 1 right) = f^{-1}left( frac{1}{2}x + c right) )Substitute into ( f^{-1} ):( 2left( frac{1}{2}x + c right) - 2c = x + 2c - 2c = x ). Perfect, that works.Similarly, applying ( f(x + 2) ) to ( f^{-1}(x - 1) ):( f(f^{-1}(x - 1) + 2) = f(2(x - 1) - 2c + 2) = f(2x - 2 - 2c + 2) = f(2x - 2c) )Compute ( f(2x - 2c) ):( frac{1}{2}(2x - 2c) + c = x - c + c = x ). Also works.So, everything checks out. Therefore, ( f(x) = frac{1}{2}x + c ) and ( f^{-1}(x) = 2x - 2c ).Now, the problem asks for ( f^{-1}(2004) - f^{-1}(1) ).Let's compute each term:( f^{-1}(2004) = 2 times 2004 - 2c = 4008 - 2c )( f^{-1}(1) = 2 times 1 - 2c = 2 - 2c )Subtracting them:( f^{-1}(2004) - f^{-1}(1) = (4008 - 2c) - (2 - 2c) = 4008 - 2c - 2 + 2c = 4006 )So, the value is 4006.Wait, that's interesting. The constant c cancels out, so regardless of the value of c, the difference is always 4006. That makes sense because the constant term in the inverse function is just a vertical shift, and when we take the difference, the shifts cancel each other out.So, the answer is 4006.**Final Answer**boxed{4006}

💡First, I need to identify the three consecutive odd numbers. Given that the first number is 4.2, I can denote the numbers as:- First number: ( x = 4.2 )- Second number: ( x + 2 = 4.2 + 2 = 6.2 )- Third number: ( x + 4 = 4.2 + 4 = 8.2 )Next, according to the problem, 9 times the first number is equal to a combination of multiples of the second and third numbers plus 9. This can be expressed as:[ 9x = k(x + 4) + m(x + 2) + 9 ]Substituting ( x ) with 4.2:[ 9(4.2) = k(8.2) + m(6.2) + 9 ]Simplifying:[ 37.8 = 8.2k + 6.2m + 9 ]Subtracting 9 from both sides:[ 28.8 = 8.2k + 6.2m ]At this point, I realize that there are two variables (( k ) and ( m )) but only one equation, making it impossible to determine unique values for ( k ) and ( m ) without additional information.However, the question specifically asks for the ratio of the third number to the second number, which is:[ frac{x + 4}{x + 2} = frac{8.2}{6.2} ]Simplifying this ratio:[ frac{8.2}{6.2} = frac{41}{31} ]Therefore, the ratio of the third number to the second number is ( frac{41}{31} ).

💡Okay, so I have this math problem here: "What is the divisor if the dividend is 139, the quotient is 7, and the remainder is 6?" Hmm, let me try to figure this out step by step. First, I remember that in division, there's a relationship between the dividend, divisor, quotient, and remainder. The basic formula is:Dividend = (Divisor × Quotient) + RemainderSo, in this case, the dividend is 139, the quotient is 7, and the remainder is 6. I need to find the divisor. Let me write down what I know:Dividend = 139 Quotient = 7 Remainder = 6 Divisor = ?Plugging these values into the formula:139 = (Divisor × 7) + 6Okay, so I need to solve for the divisor. Let me rearrange the equation to isolate the divisor. First, subtract the remainder from both sides:139 - 6 = Divisor × 7 133 = Divisor × 7Now, to find the divisor, I need to divide both sides by 7:Divisor = 133 / 7Hmm, let's do that division. 7 goes into 13 once, with a remainder of 6. Then, bring down the 3 to make 63. 7 goes into 63 nine times exactly. So, 133 divided by 7 is 19.Wait, does that make sense? Let me check by plugging it back into the original formula:Dividend = (Divisor × Quotient) + Remainder 139 = (19 × 7) + 6 139 = 133 + 6 139 = 139Yes, that checks out. So, the divisor is 19.I think I got it right. But just to be sure, let me think about it another way. If I divide 139 by 19, what do I get? Let's see:19 × 7 = 133 139 - 133 = 6So, the quotient is 7 and the remainder is 6, which matches the problem statement. That confirms that 19 is indeed the correct divisor.I guess I was a bit nervous because sometimes when dealing with division, especially with remainders, it's easy to mix up the numbers. But by carefully applying the formula and double-checking my work, I feel confident that 19 is the right answer.