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Difficulty: MediumSolving Linear Equations

A local community garden allocates a plot of land for different types of vegetables. Tomatoes occupy 25\frac{2}{5} of the total area, peppers occupy 13\frac{1}{3} of the total area, and the remaining 80 square feet are planted with herbs. What is the total area, in square feet, of the community garden plot?

  1. A
    100
  2. B
    120
  3. C
    128
  4. D
    200
  5. 300Answer

Answer

300 square feet
Let xx be the total area of the garden plot. Since the garden is divided into tomatoes (25\frac{2}{5} of the total area), peppers (13\frac{1}{3} of the total area), and herbs (8080 square feet), the sum of these parts equals the total area: 25x+13x+80=x\frac{2}{5}x + \frac{1}{3}x + 80 = x. Finding a common denominator of 15 for the fractions gives 615x+515x+80=x\frac{6}{15}x + \frac{5}{15}x + 80 = x, which simplifies to 1115x+80=x\frac{11}{15}x + 80 = x. Subtracting 1115x\frac{11}{15}x from both sides yields 80=415x80 = \frac{4}{15}x. Multiplying both sides by the reciprocal 154\frac{15}{4} gives x=80×154=300x = 80 \times \frac{15}{4} = 300 square feet.

Step-by-Step Solution

1
Define the variable and translate the given fractions and numbers into an equation. Let xx represent the total area of the garden plot.
25x+13x+80=x\frac{2}{5}x + \frac{1}{3}x + 80 = x
The sum of the areas of the tomato plot, the pepper plot, and the herb plot must equal the total area of the garden.
2
Find a common denominator to combine the fractional coefficients of xx. The least common multiple of 5 and 3 is 15. Convert 25\frac{2}{5} and 13\frac{1}{3} to fractions with a denominator of 15.
615x+515x+80=x  ⟹  1115x+80=x\frac{6}{15}x + \frac{5}{15}x + 80 = x \implies \frac{11}{15}x + 80 = x
Combining like terms simplifies the expression to help isolate the variable.
3
Subtract 1115x\frac{11}{15}x from both sides of the equation to group all xx terms on one side.
80=x−1115x  ⟹  80=415x80 = x - \frac{11}{15}x \implies 80 = \frac{4}{15}x
This isolates the constant term on one side of the equation.
4
Multiply both sides of the equation by the reciprocal of 415\frac{4}{15}, which is 154\frac{15}{4}, to solve for xx.
x=80×154=20×15=300x = 80 \times \frac{15}{4} = 20 \times 15 = 300
This yields the value of the total area of the garden plot.

Key Concept

Solving linear equations in one variable with fractional coefficients
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Topics

Solving Linear Equations

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