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Ben likes to go in the covered dome when he feels overexcited. The dome covers an area of 1(2)/(3) square meters, which is (5)/(840) of the total area of the playground. What is the total area of the playground? square meters

Ben likes to go in the covered dome when he feels overexcited. The dome covers an area of 123 1 \frac{2}{3} square meters, which is 5840 \frac{5}{840} of the total area of the playground.\newlineWhat is the total area of the playground?\newlinesquare meters

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Full solution

Q. Ben likes to go in the covered dome when he feels overexcited. The dome covers an area of 123 1 \frac{2}{3} square meters, which is 5840 \frac{5}{840} of the total area of the playground.\newlineWhat is the total area of the playground?\newlinesquare meters
  1. Understanding the Relationship: Understand the relationship between the area of the dome and the total area of the playground.\newlineThe problem states that the area of the dome is 1(23)1\left(\frac{2}{3}\right) square meters, which is also given as (5840)\left(\frac{5}{840}\right) of the total area of the playground. This means that if we let AA represent the total area of the playground, then the area of the dome is a fraction of the total area.
  2. Converting to Improper Fraction: Convert the mixed number to an improper fraction to represent the area of the dome.\newline1231\frac{2}{3} can be converted to an improper fraction by multiplying the whole number by the denominator of the fraction and adding the numerator, then placing that result over the original denominator.\newline123=(1×3+2)/3=(3+2)/3=531\frac{2}{3} = (1 \times 3 + 2)/3 = (3 + 2)/3 = \frac{5}{3} square meters
  3. Setting up the Equation: Set up the equation to find the total area of the playground. Since the area of the dome is (5840)(\frac{5}{840}) of the total area of the playground, we can write the equation: (5840)×A=53(\frac{5}{840}) \times A = \frac{5}{3}
  4. Solving for Total Area: Solve for AA, the total area of the playground.\newlineTo find AA, we need to divide both sides of the equation by (5/840)(5/840):\newlineA=5/35/840A = \frac{5/3}{5/840}
  5. Simplifying the Equation: Simplify the equation by multiplying by the reciprocal of (5840)(\frac{5}{840}).A=(53)×(8405)A = (\frac{5}{3}) \times (\frac{840}{5})
  6. Canceling Common Factors: Cancel out the common factors of 55 in the numerator and denominator.\newlineA=(13)×(8401)A = \left(\frac{1}{3}\right) \times \left(\frac{840}{1}\right)
  7. Finding the Total Area: Multiply the remaining fractions to find the total area.\newlineA=8403A = \frac{840}{3}\newlineA=280A = 280 square meters

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