💡Okay, so I have this problem here where I need to find the sum of the reciprocals of two nonzero real numbers, x and y. The problem states that the sum of x and y is 5 times their product, and also that x is equal to twice y. Hmm, let me try to unpack this step by step.First, let's write down what we know. The problem gives us two pieces of information:1. The sum of x and y is 5 times their product. So, mathematically, that would be: [ x + y = 5xy ] 2. It also tells us that x is twice y, which can be written as: [ x = 2y ] Alright, so we have two equations here. The second equation directly relates x and y, which is helpful because it allows us to substitute one variable in terms of the other into the first equation. That way, we can solve for one variable first and then find the other.Let me substitute x with 2y in the first equation. So, replacing x in the equation (x + y = 5xy) with 2y gives me:[2y + y = 5 times 2y times y]Simplifying the left side, 2y + y is 3y. On the right side, 5 times 2y times y is 10y². So now the equation looks like:[3y = 10y²]Hmm, okay. So this is a quadratic equation in terms of y. Let me rearrange it to make it easier to solve. I'll bring all terms to one side:[10y² - 3y = 0]Now, I can factor out a y from both terms:[y(10y - 3) = 0]So, this gives me two possible solutions for y:1. ( y = 0 )2. ( 10y - 3 = 0 ) which implies ( y = frac{3}{10} )But wait, the problem states that x and y are nonzero real numbers. So, y can't be zero because that would make x also zero (since x = 2y), and we can't have zero in the denominators when we take reciprocals later. Therefore, we discard y = 0 and take the other solution:[y = frac{3}{10}]Now that we have y, we can find x using the equation x = 2y:[x = 2 times frac{3}{10} = frac{6}{10} = frac{3}{5}]So, x is 3/5 and y is 3/10.The next part of the problem asks for the sum of the reciprocals of x and y. The reciprocal of a number is 1 divided by that number. So, the reciprocal of x is ( frac{1}{x} ) and the reciprocal of y is ( frac{1}{y} ). Therefore, the sum of the reciprocals is:[frac{1}{x} + frac{1}{y}]Substituting the values of x and y we found:[frac{1}{frac{3}{5}} + frac{1}{frac{3}{10}}]Dividing by a fraction is the same as multiplying by its reciprocal, so:[frac{5}{3} + frac{10}{3}]Now, adding these two fractions together:[frac{5}{3} + frac{10}{3} = frac{15}{3} = 5]So, the sum of the reciprocals of x and y is 5.Let me just double-check my steps to make sure I didn't make any mistakes. I substituted x = 2y into the equation x + y = 5xy, which gave me 3y = 10y². Factoring out y, I found y = 0 or y = 3/10. Since y can't be zero, I took y = 3/10 and found x = 3/5. Then, I calculated the reciprocals and added them to get 5. Everything seems to check out.I think I've got it right. The sum of the reciprocals is 5, which corresponds to option B.**Final Answer**boxed{B}

💡First, I need to determine the final salary after a 60% decrease followed by a 60% increase.Let’s assume Robert's original salary is ( S ).1. **Decrease by 60%:** - The decrease amount is ( 60% ) of ( S ), which is ( 0.6S ). - The new salary after the decrease is ( S - 0.6S = 0.4S ).2. **Increase by 60%:** - The increase is ( 60% ) of the decreased salary ( 0.4S ), which is ( 0.24S ). - The final salary after the increase is ( 0.4S + 0.24S = 0.64S ).3. **Calculate the percentage loss:** - The loss in salary is ( S - 0.64S = 0.36S ). - The percentage loss is ( frac{0.36S}{S} times 100 = 36% ).Therefore, Robert loses 36% of his original salary after the decrease and subsequent increase.

💡Okay, so I have this problem about triangle PQR. Points M and N are on sides PQ and PR respectively. M is the midpoint of PQ, which means it splits PQ into two equal parts. N divides PR in the ratio 1:2 from P to R, so PN is one part and NR is two parts. The area of triangle PQR is given as 36 square units, and I need to find the area of triangle PMN.Hmm, let me visualize this. Triangle PQR with points M and N. M is halfway along PQ, and N is closer to P on PR, dividing it into a 1:2 ratio. So, PN is one-third of PR, and NR is two-thirds. I think I can use area ratios here because the points divide the sides proportionally.Maybe I can use coordinate geometry to solve this. Let me assign coordinates to the triangle to make it easier. Let's place point P at the origin (0,0). Let me assume that PQ is along the x-axis for simplicity. So, point Q would be at (2a, 0) since M is the midpoint, making M at (a, 0). Point R can be somewhere in the plane; let's say R is at (0, 3b) because PR is divided in a 1:2 ratio, so N would be at (0, b).Wait, is that right? If PR is from (0,0) to (0,3b), then PN would be from (0,0) to (0,b), which is one-third of PR, and NR would be from (0,b) to (0,3b), which is two-thirds. That makes sense.Now, triangle PQR has vertices at P(0,0), Q(2a,0), and R(0,3b). The area of triangle PQR can be calculated using the formula for the area of a triangle with base and height. The base PQ is 2a, and the height is 3b. So, area = (1/2)*base*height = (1/2)*(2a)*(3b) = 3ab. We know the area is 36, so 3ab = 36, which means ab = 12.Now, I need to find the area of triangle PMN. Points P, M, and N have coordinates P(0,0), M(a,0), and N(0,b). So, triangle PMN is a right triangle with vertices at (0,0), (a,0), and (0,b). The area of this triangle can be calculated as (1/2)*base*height = (1/2)*a*b.Since we know ab = 12, the area of triangle PMN is (1/2)*12 = 6. So, the area is 6 square units.Wait, let me double-check. If I use vectors or another method, would I get the same result? Maybe using the concept of similar triangles or area ratios.Since M is the midpoint of PQ, the area of triangle PMR should be half of the area of triangle PQR, which would be 18. Then, since N divides PR in a 1:2 ratio, the area of triangle PMN would be one-third of 18, which is 6. Yeah, that confirms it.Alternatively, using mass point geometry or barycentric coordinates might also lead to the same conclusion, but coordinate geometry seems straightforward here. I think I'm confident that the area is 6.

💡First, I need to determine the number of elements in the set ( A = {1, 2} ). By examining the set, I see that it contains two distinct elements: 1 and 2.Next, I recall the formula for finding the number of subsets of a set. The formula states that if a set has ( n ) elements, it has ( 2^n ) subsets. This is because each element can either be included or excluded from a subset, resulting in two choices per element.Applying this formula to the set ( A ), where ( n = 2 ), I calculate the number of subsets as ( 2^2 = 4 ).Therefore, the set ( A = {1, 2} ) has 4 subsets.

💡First, I need to determine how much money Sara has already saved. Let's denote this amount as ( S ).Sara saves 10 each week, and Jim saves 15 each week. After 820 weeks, both will have saved the same total amount.The total amount Sara will have saved after 820 weeks is her initial savings plus her weekly savings over 820 weeks:[S + 10 times 820]Jim's total savings after 820 weeks is:[15 times 820]Since their total savings are equal after 820 weeks, I can set up the equation:[S + 10 times 820 = 15 times 820]Calculating the weekly savings:[10 times 820 = 8200][15 times 820 = 12300]Substituting these values back into the equation:[S + 8200 = 12300]To find ( S ), I subtract 8200 from both sides of the equation:[S = 12300 - 8200][S = 4100]Therefore, Sara has already saved 4100.

💡First, I need to determine the breadth of the rectangle. Let's denote the breadth as ( b ) meters. Since the length is three times the breadth, the length will be ( 3b ) meters.Next, I'll use the area formula for a rectangle, which is the product of its length and breadth. Given that the area is 147 m², I can set up the equation:[ 3b times b = 147 ][ 3b^2 = 147 ]To find ( b ), I'll divide both sides of the equation by 3:[ b^2 = 49 ]Taking the square root of both sides gives:[ b = 7 ]Now that I have the breadth, I can find the length:[ text{Length} = 3b = 3 times 7 = 21 ]Finally, to calculate the perimeter of the rectangle, I'll use the formula:[ text{Perimeter} = 2 times (text{Length} + text{Breadth}) ][ text{Perimeter} = 2 times (21 + 7) = 2 times 28 = 56 ]Therefore, the perimeter of the rectangle is 56 meters.