answer:To solve this problem, we need to determine the value of x, which represents the number of medical schools Mandy researched. Let's break down the information given: Number of medical schools Mandy applied to: 1/3 * x Number of medical schools Mandy was accepted to: 1/2 * (1/3 * x) Number of medical schools Mandy was accepted to: 7 We can set up the equation as follows: 1/2 * (1/3 * x) = 7 Let's simplify and solve for x: 1/6 * x = 7 To isolate x, we multiply both sides of the equation by 6: 1/6 * x * 6 = 7 * 6 x = 42 The value of x is 42. 42 The answer is: 42

answer:We can rewrite each logarithm using the change of base formula: begin{align*} log_{y^6}{x} &= frac{log_x{x}}{log_x{y^6}} = frac{1}{log_x{y^6}} log_{x^5}{y^2} &= frac{log_y{y^2}}{log_y{x^5}} = frac{2}{log_y{x^5}} log_{y^4}{x^3} &= frac{log_x{x^3}}{log_x{y^4}} = frac{3}{log_x{y^4}} log_{x^3}{y^4} &= frac{log_y{y^4}}{log_y{x^3}} = frac{4}{log_y{x^3}} log_{y^2}{x^5} &= frac{log_x{x^5}}{log_x{y^2}} = frac{5}{log_x{y^2}} end{align*} Multiplying all these expressions together, we get: begin{align*} left(frac{1}{log_x{y^6}}right) cdot left(frac{2}{log_y{x^5}}right) cdot left(frac{3}{log_x{y^4}}right) cdot left(frac{4}{log_y{x^3}}right) cdot left(frac{5}{log_x{y^2}}right) = frac{1cdot2cdot3cdot4cdot5}{log_x{y^6}cdotlog_y{x^5}cdotlog_x{y^4}cdotlog_y{x^3}cdotlog_x{y^2}} = frac{120}{log_x{y^{6cdot5cdot4cdot3cdot2}}} = frac{120}{log_x{y^{720}}} end{align*} Since log_x{y^{720}} = 720log_x{y}, we have: begin{align*} frac{120}{log_x{y^{720}}} = frac{120}{720log_x{y}} = frac{1}{6log_x{y}} = frac{1}{log_y{x^6}} = frac{1}{log_y{x^6}} cdot frac{log_y{x}}{log_y{x}} = frac{log_y{x}}{6log_y{x}} = boxed{frac{1}{6}} end{align*} The answer is: frac{1}{6}

answer:To solve this problem, we need to determine the value of x, which represents the number of pieces of yarn. Let's break down the information given: Length of green yarn: 156 cm Length of red yarn: 3 * 156 cm + 8 cm Total length of all yarn: 632 cm We can set up the equation as follows: Length of green yarn + Length of red yarn + (Length of x-2 pieces of yarn) = Total length of all yarn 156 cm + (3 * 156 cm + 8 cm) + (Length of x-2 pieces of yarn) = 632 cm Let's simplify and solve for x: 156 cm + (468 cm + 8 cm) + (Length of x-2 pieces of yarn) = 632 cm 156 cm + 476 cm + (Length of x-2 pieces of yarn) = 632 cm 632 cm + (Length of x-2 pieces of yarn) = 632 cm To isolate x, we subtract 632 cm from both sides of the equation: 632 cm - 632 cm + (Length of x-2 pieces of yarn) = 632 cm - 632 cm Length of x-2 pieces of yarn = 0 Finally, we add 2 to both sides of the equation to solve for x: Length of x-2 pieces of yarn + 2 = 0 + 2 Length of x pieces of yarn = 2 The value of x is 2. 2 The answer is: 2

answer:To solve this problem, we need to determine the value of x, which represents the number of red macaroons Fran baked. Let's break down the information given: Number of red macaroons: x Number of green macaroons: 40 Number of green macaroons Fran ate: 15 Number of red macaroons Fran ate: 2 * Number of green macaroons Number of macaroons remaining: 45 We can set up the equation as follows: Number of red macaroons - Number of red macaroons Fran ate + Number of green macaroons - Number of green macaroons Fran ate = Number of macaroons remaining x - (2 * 15) + 40 - 15 = 45 Let's simplify and solve for x: x - 30 + 40 - 15 = 45 x - 5 = 45 To isolate x, we add 5 to both sides of the equation: x - 5 + 5 = 45 + 5 x = 50 The value of x is 50. 50 The answer is: 50