Please analyze the following json for appropriate latex expression:

[
  {
    "text": "A certain virus infects a computer system, causing the number of corrupted files to increase by a factor of 3 every 2 hours. If there were initially 5 corrupted files, how many corrupted files will there be after 8 hours?",
    "type": "multiple-choice-single-answer",
    "id": 38,
    "correctAnswer": "d",
    "quiztype": "satmath",
    "explanation": "The number of corrupted files increases by a factor of 3 every 2 hours. First, determine the number of times the files will triple in 8 hours: $$\\text{Number of periods} = \\frac{8 \\text{ hours}}{2 \\text{ hours/period}} = 4.$$ The number of corrupted files after 8 hours is calculated using the formula: $$\\text{Final amount} = \\text{Initial amount} \\times (\\text{Multiplier})^{\\text{Number of periods}}.$$ Here, the multiplier is 3, so: $$\\text{Final amount} = 5 \\times 3^4 = 5 \\times 81 = 405.$$",
    "options": [
      {
        "text": "135",
        "id": "a"
      },
      {
        "text": "243",
        "id": "b"
      },
      {
        "text": "324",
        "id": "c"
      },
      {
        "text": "405",
        "id": "d"
      }
    ]
  },
  {
    "text": "An investment of $10,000 appreciates by 20% every year. What is the value of the investment after 3 years?",
    "type": "multiple-choice-single-answer",
    "id": 39,
    "correctAnswer": "b",
    "quiztype": "satmath",
    "explanation": "The investment grows by 20% each year. The multiplier for a 20% increase is $$1 + 0.20 = 1.20.$$ The value after 3 years is: $$10,000 \\times (1.20)^3 = 10,000 \\times 1.728 = 17,280.$$",
    "options": [
      {
        "text": "14,400",
        "id": "a"
      },
      {
        "text": "17,280",
        "id": "b"
      },
      {
        "text": "20,736",
        "id": "c"
      },
      {
        "text": "24,000",
        "id": "d"
      }
    ]
  },
  {
    "text": "A radioactive substance decays at a rate of 25% every hour. If the initial mass of the substance is 80 grams, how much remains after 6 hours?",
    "type": "multiple-choice-single-answer",
    "id": 40,
    "correctAnswer": "a",
    "quiztype": "satmath",
    "explanation": "The substance decays by 25% each hour, so the multiplier is $$1 - 0.25 = 0.75.$$ The amount remaining after 6 hours is: $$80 \\times (0.75)^6.$$ Calculate: $$0.75^6 \\approx 0.178.$$ Then, $$80 \\times 0.178 \\approx 14.22.$$ Therefore, approximately 14 grams remain.",
    "options": [
      {
        "text": "Approximately 14 grams",
        "id": "a"
      },
      {
        "text": "Approximately 18 grams",
        "id": "b"
      },
      {
        "text": "Approximately 24 grams",
        "id": "c"
      },
      {
        "text": "Approximately 30 grams",
        "id": "d"
      }
    ]
  },
  {
    "text": "The population of a town increases by 10% every year. If the current population is 50,000, what will be the population after 5 years?",
    "type": "multiple-choice-single-answer",
    "id": 41,
    "correctAnswer": "c",
    "quiztype": "satmath",
    "explanation": "The population grows by 10% annually. The growth multiplier is $$1 + 0.10 = 1.10.$$ After 5 years, the population will be: $$50,000 \\times (1.10)^5.$$ Calculate: $$ (1.10)^5 \\approx 1.61051.$$ So, $$50,000 \\times 1.61051 \\approx 80,525.$$ Therefore, the population will be approximately 80,525.",
    "options": [
      {
        "text": "Approximately 70,000",
        "id": "a"
      },
      {
        "text": "Approximately 75,000",
        "id": "b"
      },
      {
        "text": "Approximately 80,525",
        "id": "c"
      },
      {
        "text": "Approximately 85,000",
        "id": "d"
      }
    ]
  }
]