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Question

Difficulty: MediumSolving Linear Equations

A storage tank contains 1818 gallons of water and is being filled at a constant rate of 23\frac{2}{3} gallons per minute. A second storage tank contains 3333 gallons of water and is being drained at a constant rate of 56\frac{5}{6} gallons per minute. If both tanks begin their processes at the same time, they will contain the same amount of water after mm minutes. What is the value of 2m+32m + 3?

  1. 23Answer
  2. B
    26
  3. C
    33
  4. D
    71
  5. E
    183

Answer

23
The correct answer is 23. By representing the volume of the first tank as 18+23m18 + \frac{2}{3}m and the second tank as 33−56m33 - \frac{5}{6}m, setting them equal gives 18+23m=33−56m18 + \frac{2}{3}m = 33 - \frac{5}{6}m. Solving this equation by finding a common denominator for the fraction coefficients yields 96m=15\frac{9}{6}m = 15, which simplifies to 32m=15\frac{3}{2}m = 15, and thus m=10m = 10. Substituting m=10m = 10 into the expression 2m+32m + 3 results in 2(10)+3=232(10) + 3 = 23.

Step-by-Step Solution

1
Set up the equation representing the water volume in both tanks over time.
18+23m=33−56m18 + \frac{2}{3}m = 33 - \frac{5}{6}m
The first tank starts with 1818 gallons and increases by 23\frac{2}{3} gallons per minute, while the second starts with 3333 gallons and decreases by 56\frac{5}{6} gallons per minute. We set their volumes equal to find when they contain the same amount.
2
Isolate the variable terms on one side and the constant terms on the other side.
23m+56m=33−18\frac{2}{3}m + \frac{5}{6}m = 33 - 18
Adding 56m\frac{5}{6}m to both sides and subtracting 1818 from both sides groups like terms together.
3
Find a common denominator to add the fraction coefficients.
46m+56m=15⇒96m=15⇒32m=15\frac{4}{6}m + \frac{5}{6}m = 15 \Rightarrow \frac{9}{6}m = 15 \Rightarrow \frac{3}{2}m = 15
A common denominator of 66 is used to add the fractions 23\frac{2}{3} and 56\frac{5}{6}.
4
Solve for mm by multiplying both sides by the reciprocal of the coefficient.
m=15×23=10m = 15 \times \frac{2}{3} = 10
Multiplying by 23\frac{2}{3} isolates mm on the left side.
5
Evaluate the expression 2m+32m + 3 using the value of mm.
2(10)+3=20+3=232(10) + 3 = 20 + 3 = 23
Substitute m=10m = 10 into the expression and follow the correct order of operations (multiply first, then add).

Key Concept

Solving linear equations with fractional coefficients and translating word problems into algebraic equations.
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Topics

Solving Linear Equations

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