Appearance
question:A jar on the family's counter contains change they've been saving a trip to the ice cream shop. There are 123 pennies, 85 nickels, 35 dimes, and a number of quarters. All x family members get a double scoop, which costs 3 each. After the trip, they have 48 cents left over. There were 26 quarters in the jar. What is the value of unknown variable x? What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the number of family members. Let's break down the information given: Number of pennies: 123 Number of nickels: 85 Number of dimes: 35 Number of quarters: 26 Cost of a double scoop: 3 Amount of money left over: 48 cents We can set up the equation as follows: Total amount of money from pennies + Total amount of money from nickels + Total amount of money from dimes + Total amount of money from quarters - Cost of double scoops = Amount of money left over (123 * 0.01) + (85 * 0.05) + (35 * 0.10) + (26 * 0.25) - (3 * x) = 0.48 Let's simplify and solve for x: 1.23 + 4.25 + 3.50 + 6.50 - 3x = 0.48 15.48 - 3x = 0.48 To isolate x, we subtract 15.48 from both sides of the equation: 15.48 - 15.48 - 3x = 0.48 - 15.48 -3x = -15 Finally, we divide both sides of the equation by -3 to solve for x: x = -15 / -3 x = 5 The value of x is 5. 5 The answer is: 5
question:f Christopher, Jameson, and June each purchased a toy sword, and Christopher's sword is 15 inches long, Jameson's sword is 3 inches longer than twice the length of Christopher's sword, and June's sword is 5 inches longer than Jameson's sword, how many inches longer is June's sword compared to Christopher's sword?
answer:Twice the length of Christopher's sword is 15 x 2 = 30 inches. Jameson's sword is 3 inches longer than twice the length of Christopher's sword, so it is 30 + 3 = 33 inches. June's sword is 5 inches longer than Jameson's sword, so it is 33 + 5 = 38 inches. To find out how many inches longer June's sword is compared to Christopher's sword, we subtract their lengths: 38 - 15 = 23 inches. June's sword is 23 inches longer than Christopher's sword. 23 The answer is: 23
question:Let x, y, and z be positive real numbers such that x + y + z = 1. Find the minimum value of [frac{1}{x} + frac{1}{y} + frac{1}{z}.]
answer:By Cauchy-Schwarz, [(x + y + z) left( frac{1}{x} + frac{1}{y} + frac{1}{z} right) ge (1 + 1 + 1)^2 = 9,]so [frac{1}{x} + frac{1}{y} + frac{1}{z} ge 9(x + y + z) = 9.]Equality occurs when x = y = z = frac{1}{3}, so the minimum value is boxed{9}.The answer is: 9
question:f 5 out of 10 students in the math class scored a 92 on the test and 4 students scored an 80, what score does the last student need to achieve in order to have a class average test score of at least 85?
answer:There are 10 students in total in the math class. 5 out of 10 students scored a 92 on the test, so the total score for those 5 students is 5 * 92 = 460. 4 students scored an 80 on the test, so the total score for those 4 students is 4 * 80 = 320. To find the score the last student needs to achieve, we need to find the total score required for an average of at least 85 for the whole class. The total score required for an average of at least 85 for the whole class is (10 * 85) = 850. To find the score the last student needs to achieve, we subtract the total score of the other students from the total required score: 850 - (460 + 320) = 70. Therefore, the last student needs to achieve a score of 70 in order to have a class average test score of at least 85. 70 The answer is: 70