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Math  /  Algebra

QuestionElaine can complete a landscaping project in 12 hours with the help of either her husband Brian or both her two daughters. If Brian and one of his daughters work together, it would take them 24 hours to complete the project. Assuming the rate of work is constant for each person, and the two daughters work at the same rate, how long would it take Elaine, Brian, and one of their daughters to complete the project?
Working all together, Elaine, Brian, and one of their daughters can do thes job in 8 hours. (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

1. Elaine's rate of work is constant.
2. Brian's rate of work is constant.
3. Each daughter's rate of work is constant and both daughters work at the same rate.
4. The work rates can be combined linearly when working together.

STEP 2

1. Determine the individual work rates of Elaine, Brian, and one daughter.
2. Use these rates to find the combined work rate of Elaine, Brian, and one daughter.
3. Calculate the total time it would take for Elaine, Brian, and one daughter to complete the project together.

STEP 3

Let E E be the work rate of Elaine, B B be the work rate of Brian, and D D be the work rate of one daughter.
Given that Elaine and her two daughters can complete the project in 12 hours, the combined rate is: E+2D=112 projects per hour E + 2D = \frac{1}{12} \text{ projects per hour}

STEP 4

Given that Brian and one daughter can complete the project in 24 hours, their combined rate is: B+D=124 projects per hour B + D = \frac{1}{24} \text{ projects per hour}

STEP 5

Given that Elaine, Brian, and one daughter can complete the project in 8 hours, their combined rate is: E+B+D=18 projects per hour E + B + D = \frac{1}{8} \text{ projects per hour}

STEP 6

We now have the following system of equations:
1. E+2D=112 E + 2D = \frac{1}{12}
2. B+D=124 B + D = \frac{1}{24}
3. E+B+D=18 E + B + D = \frac{1}{8}
Solve these equations to find the individual rates E E , B B , and D D .
First, solve the second equation for B B : B=124−D B = \frac{1}{24} - D

STEP 7

Substitute B B into the third equation: E+(124−D)+D=18 E + \left(\frac{1}{24} - D\right) + D = \frac{1}{8} Simplify: E+124=18 E + \frac{1}{24} = \frac{1}{8} Solve for E E : E=18−124=324−124=224=112 E = \frac{1}{8} - \frac{1}{24} = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12} Thus: E=112 E = \frac{1}{12}

STEP 8

Now substitute E=112 E = \frac{1}{12} into the first equation: 112+2D=112 \frac{1}{12} + 2D = \frac{1}{12} Subtract 112 \frac{1}{12} from both sides: 2D=0 2D = 0 Thus: D=0 D = 0

STEP 9

With D=0 D = 0 , substitute into the second equation: B+0=124 B + 0 = \frac{1}{24} Thus: B=124 B = \frac{1}{24}

STEP 10

We now know the individual rates: E=112,B=124,D=0 E = \frac{1}{12}, \quad B = \frac{1}{24}, \quad D = 0
Combine these rates to find the work rate of Elaine, Brian, and one daughter working together: E+B+D=112+124+0=224+124=324=18 E + B + D = \frac{1}{12} + \frac{1}{24} + 0 = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}

STEP 11

The combined rate of work for Elaine, Brian, and one daughter is 18 \frac{1}{8} projects per hour.
Therefore, the time it takes them to complete the project together is: 1(18)=8 hours \frac{1}{\left(\frac{1}{8}\right)} = 8 \text{ hours}
Thus, the solution is: 8 hours \boxed{8} \text{ hours}

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