Barbie And Robert's Sculpture Puzzle: Solve For Robert's Time
This article dives into a classic work-rate problem involving Barbie and Robert, two sculptors collaborating on a human statue. These types of mathematical puzzles often appear in standardized tests and provide a great way to exercise your problem-solving skills. Our main goal is to determine how long it would take Robert to complete the sculpture entirely on his own, given that Barbie and Robert, when working in tandem, finish the project in 6 days and Barbie is faster than Robert. This problem requires careful setup of equations and some algebraic manipulation to isolate the desired variable. We will walk through the solution step-by-step, ensuring clarity in every step. Understanding the concepts of individual work rates and combined work rates is crucial here. Let’s get started unraveling this fascinating sculpting dilemma. To accurately tackle the problem, we will begin by defining the variables and forming the relevant equations. This approach will guide us to the correct solution in a systematic manner.
Barbie and Robert, when collaborating, complete a human statue in 6 days. Barbie, working independently, can finish the same sculpture 5 days faster than Robert. The question is: how many days will it take Robert to complete the sculpture if he works alone? The options given are:
a. 3 days
b. 8 days
c. 10 days
d. 15 days
This problem statement provides all the necessary information to solve the puzzle. The core challenge lies in converting the word problem into mathematical equations and then solving them. We need to carefully consider the relationships between the work rates of Barbie and Robert, both individually and when they work together. Identifying the key variables and their connections is paramount to success. The problem's structure hints at a system of equations approach, where we define variables for the time each person takes to complete the work and then formulate equations based on their work rates. Furthermore, understanding the concept of combined work rate is crucial, as it dictates how their individual contributions merge to achieve the task collaboratively.
To solve this problem, we'll first define our variables:
- Let 'x' represent the number of days it takes Robert to finish the sculpture alone.
- Then, Barbie can finish it in 'x - 5' days.
Now, we need to express their work rates. Work rate is defined as the amount of work completed per unit of time (in this case, per day). Thus:
- Robert's work rate is 1/x (he completes 1/x of the sculpture each day).
- Barbie's work rate is 1/(x - 5) (she completes 1/(x - 5) of the sculpture each day).
When they work together, their work rates add up. Since they complete the sculpture in 6 days, their combined work rate is 1/6. This gives us the equation:
1/x + 1/(x - 5) = 1/6
This equation is the cornerstone of our solution. It captures the essence of the problem, representing the combined effort of Barbie and Robert in terms of their individual work rates. Solving this equation will provide the value of 'x', which is the time Robert takes to finish the sculpture alone. The process of setting up equations is fundamental in mathematical problem-solving. It transforms a word problem into a symbolic representation, which can then be manipulated using algebraic techniques. Understanding how to convert real-world scenarios into mathematical equations is a crucial skill. This specific equation showcases the additive nature of work rates when individuals collaborate on a task. It’s important to meticulously define each variable and ensure the equation accurately reflects the relationships described in the problem statement.
Now, let’s solve the equation we established: 1/x + 1/(x - 5) = 1/6.
First, we need to find a common denominator to combine the fractions on the left side of the equation. The common denominator is x(x - 5). Rewriting the equation with the common denominator gives us:
[(x - 5) + x] / [x(x - 5)] = 1/6
Simplifying the numerator, we get:
(2x - 5) / [x(x - 5)] = 1/6
Next, we cross-multiply to eliminate the fractions:
6(2x - 5) = x(x - 5)
Expanding both sides, we have:
12x - 30 = x^2 - 5x
Rearrange the equation to form a quadratic equation:
x^2 - 17x + 30 = 0
Now, we need to factor this quadratic equation. We are looking for two numbers that multiply to 30 and add up to -17. These numbers are -15 and -2. So, we can factor the equation as:
(x - 15)(x - 2) = 0
This gives us two possible solutions for x:
- x = 15
- x = 2
However, we must consider the context of the problem. If x = 2, then Barbie would finish the sculpture in x - 5 = -3 days, which is not possible. Therefore, the only valid solution is x = 15. The process of solving an equation often involves multiple steps, each requiring careful attention to algebraic rules. In this case, we progressed from combining fractions to forming a quadratic equation and finally factoring it to find the possible solutions. It’s crucial to remember that in word problems, the mathematical solutions must also make sense in the context of the real-world situation. This is why we had to discard one of the solutions, as it led to a nonsensical result. Understanding how to handle quadratic equations and interpret their solutions is a vital mathematical skill.
We found that x = 15, which means it would take Robert 15 days to finish the sculpture alone. Barbie would take x - 5 = 15 - 5 = 10 days to finish the sculpture alone.
Let's verify if this solution satisfies the given conditions:
- Robert's work rate: 1/15
- Barbie's work rate: 1/10
Their combined work rate should be 1/6. Let's check:
1/15 + 1/10 = (2 + 3) / 30 = 5/30 = 1/6
This confirms that our solution is correct. When they work together, they indeed complete 1/6 of the sculpture each day, which means they finish it in 6 days.
Thus, it will take Robert 15 days to finish the sculpture alone. Verifying the solution is a critical step in problem-solving. It ensures that the answer we've obtained not only satisfies the mathematical equation but also aligns with the original problem statement. By plugging our solution back into the initial conditions, we can confirm its accuracy and gain confidence in our result. This process also helps in identifying any potential errors made during the solution process. In this instance, we verified that the individual work rates of Robert and Barbie, based on our solution, do indeed add up to the combined work rate, thereby validating our answer. Solution verification is a hallmark of thorough and effective problem-solving.
The correct answer is:
d. 15 days
This final answer is the culmination of our step-by-step problem-solving approach. We systematically converted the word problem into a mathematical equation, solved the equation using algebraic techniques, and then verified our solution to ensure its accuracy. Understanding the underlying concepts of work rate, combined work rate, and the formation and solution of equations is essential for tackling similar problems. This exercise showcases the power of mathematics in solving real-world problems and highlights the importance of careful analysis and methodical execution. The process we followed can be applied to a wide range of similar problems, making it a valuable tool for anyone looking to enhance their problem-solving skills. The key takeaways from this problem include the ability to translate word problems into mathematical expressions, solve equations, and validate the results within the context of the problem.
Barbie and Robert, working together, can finish sculpting a human statue in 6 days. Barbie, working alone, can finish the same sculpture 5 days faster than Robert. How long will it take Robert to finish the job alone?
Barbie and Robert's Sculpture Puzzle Solve for Robert's Time