Question

Solve Epigram 130: Of the four spouts, one filled the whole tank in a day, the second in two days, the third in three days, and the fourth in four days. What time will all four take to fill it?

Short answer

Short answer: When all four spouts are working together, it will take them 12/25 days to fill the tank.

Step-by-step solution

Find the rate of each spout

The first spout fills the tank in 1 day, the second spout in 2 days, the third in 3 days, and the fourth in 4 days. To find the rate at which each spout fills the tank, we can use the formula: rate = work / time. Since each spout is filling the whole tank, the work is equal to 1 for each spout. For the first spout: \[\frac{1}{1} = \frac{1\text{ tank}}{1\text{ day}}\] For the second spout: \[\frac{1}{2} = \frac{1\text{ tank}}{2\text{ days}}\] For the third spout: \[\frac{1}{3} = \frac{1\text{ tank}}{3\text{ days}}\] For the fourth spout: \[\frac{1}{4} = \frac{1\text{ tank}}{4\text{ days}}\]

Calculate the combined rate

Next, we need to determine the combined rate when all four spouts are working together. To do this, add the rates of all four spouts. Combined rate = (rate of first spout) + (rate of second spout) + (rate of third spout) + (rate of fourth spout) \[R_\text{combined} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\]

Simplify the combined rate

To simplify the combined rate, find a common denominator for the fractions and add them together. The common denominator of 1, 2, 3, and 4 is 12, so convert each fraction to have a denominator of 12: \[\frac{1*12}{1*12} + \frac{1*6}{2*6} + \frac{1*4}{3*4} + \frac{1*3}{4*3} = \frac{12}{12} + \frac{6}{12} + \frac{4}{12} + \frac{3}{12}\] Then, add the numerators together: \[R_\text{combined} = \frac{12+6+4+3}{12} = \frac{25}{12}\]

Find the time it takes to fill the tank together

Finally, to find the time it takes for all four spouts to fill the tank together, use the formula: time = work / rate. Since they are filling the whole tank, the work is equal to 1. \[\text{Time} = \frac{1\text{ tank}}{\frac{25}{12}\text{ tank per day}}\] To find the time, multiply both sides by \[\frac{12}{25}\] and simplify: \[\text{Time} = \frac{1}{(25/12)} = 1*\frac{12}{25} = \frac{12}{25} days\] Therefore, all four spouts will take 12/25 days to fill the tank together.

Key concepts

Rate of Work

The concept of 'rate of work' is fundamental in solving problems related to time and efficiency. It is defined as the amount of work done per unit of time. In mathematical terms, we express it as a fraction where the numerator indicates the total work completed and the denominator represents the time taken to complete that work. For instance, if a spout fills a tank in one day, its rate of work is \[ \frac{1\text{ tank}}{1\text{ day}} \], which signifies a full tank per day.

Understanding this rate allows us to calculate how long it will take various workers—or in this case, spouts—to complete a job when working together. Important to note is that when multiple workers or mechanisms work simultaneously, their individual rates of work are additive, leading to the combined rate of work. This combined rate can then be used to find out the total time taken to complete a task by multiple entities.

Fraction Addition

Fraction addition is a mathematical operation where two or more fractions are added together to form a single fraction. The key to adding fractions is to have a common denominator. Once the common denominator is found, the numerators of the fractions are added, while the common denominator remains the same.

For example, adding \[\frac{1}{2}\] and \[\frac{1}{4}\] requires changing them so they both have the same denominator. This is usually achieved by finding the least common multiple (LCM) of the original denominators. After conversion, the calculation would be: \[\frac{1 \times 2}{2 \times 2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}\].

Finding a Common Denominator

In our tank filling problem, we converted all rates to have a denominator of 12 before adding.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It is a crucial concept for fraction addition, as it allows us to sum fractions that initially have different denominators. To add fractions effectively, they must be expressed with this common multiple to simplify the process.

The least common multiple (LCM) of the denominators typically becomes the common denominator, as it’s the smallest number that all denominators divide evenly into. Finding the LCM is a two-step process: first, list the multiples of each denominator, and then identify the smallest multiple common to all denominators. For our exercise, since we dealt with denominators 1, 2, 3, and 4, our LCM was 12, which is the smallest number divisible by each of 1, 2, 3, and 4.

Algebraic Manipulation

Algebraic manipulation involves rearranging algebraic equations or expressions to solve for a particular variable or to simplify the expression. It's a set of tools and methods that allow mathematicians to rewrite algebraic statements in equivalent forms, often making them easier to understand or solve. Key techniques include factoring, expanding, grouping, and isolating variables.

In the context of our problem, we used algebraic manipulation to rearrange the equation so we could find the time all four spouts take to fill the tank. This involved inverting the combined rate of work to find the time, which is a typical algebraic operation. After the addition of the fractions, we manipulated the equation to find \[\text{Time} = \frac{1}{(25/12)} = 1 \times \frac{12}{25} = \frac{12}{25}\text{ days}\], which is the final step to solving this real-world problem using algebra.