Two playful kittens working together can unwind all the balls of yarn in a basket 9 minutes faster than if the first kitten is working alone, and 4 minutes faster than if the second kitten is working alone. How long does it take each kitten to unwind all the balls of yarn alone?
The correct answer and explanation is:
Let’s denote:
- xx = time (in minutes) it takes the first kitten to unwind all the yarn alone.
- yy = time (in minutes) it takes the second kitten to unwind all the yarn alone.
- tt = time (in minutes) it takes both kittens working together to unwind all the yarn.
Given information:
- Both kittens working together take tt minutes.
- Together they are 9 minutes faster than the first kitten alone: t=x−9t = x – 9
- Together they are 4 minutes faster than the second kitten alone: t=y−4t = y – 4
Step 1: Express xx and yy in terms of tt
From the two equations above: x=t+9x = t + 9 y=t+4y = t + 4
Step 2: Work rate setup
- First kitten’s work rate = 1x=1t+9\frac{1}{x} = \frac{1}{t+9} (jobs per minute)
- Second kitten’s work rate = 1y=1t+4\frac{1}{y} = \frac{1}{t+4} (jobs per minute)
- Combined work rate = 1t\frac{1}{t}
Since their rates add when working together: 1t+9+1t+4=1t\frac{1}{t+9} + \frac{1}{t+4} = \frac{1}{t}
Step 3: Solve the equation
Multiply through by t(t+9)(t+4)t(t+9)(t+4) to clear denominators: t(t+4)+t(t+9)=(t+9)(t+4)t(t+4) + t(t+9) = (t+9)(t+4)
Expand each term: t2+4t+t2+9t=t2+4t+9t+36t^2 + 4t + t^2 + 9t = t^2 + 4t + 9t + 36
Simplify: 2t2+13t=t2+13t+362t^2 + 13t = t^2 + 13t + 36
Subtract t2+13tt^2 + 13t from both sides: 2t2+13t−t2−13t=362t^2 + 13t – t^2 – 13t = 36 t2=36t^2 = 36
So, t=6(discard negative root t=−6)t = 6 \quad (\text{discard negative root } t = -6)
Step 4: Find xx and yy
x=t+9=6+9=15 minutesx = t + 9 = 6 + 9 = 15 \text{ minutes} y=t+4=6+4=10 minutesy = t + 4 = 6 + 4 = 10 \text{ minutes}
Answer:
- First kitten alone takes 15 minutes.
- Second kitten alone takes 10 minutes.
- Together they take 6 minutes.
Explanation:
This problem involves rates of work. Each kitten has a rate of unwinding yarn defined as one job divided by the time it takes (jobs per minute). When working together, their rates add up to complete the job faster. The key was translating the problem into equations that describe their rates and how their combined work time compares to their individual times.
By setting the time both kittens take together as tt, and expressing the individual times in terms of tt, we created an equation to solve for tt. The problem nicely shows how teamwork (combined rates) speeds up the process, and once tt was found, it was straightforward to find the individual times for each kitten.
This approach can be used in many similar problems involving combined work, such as filling tanks, painting walls, or any situation where two or more workers (or machines) work together and finish faster than alone.