Two water taps together can fill a tank in 1⅞ hours. The tap with a longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

To solve the problem of how long each tap takes to fill the tank separately, we can start by defining the variables and using the information given.

Defining Variables

Let:

• x = time taken by the smaller tap to fill the tank (in hours)
• x - 2 = time taken by the larger tap to fill the tank (in hours)

Combined Rate of Filling

When both taps are used together, they can fill the tank in 1⅞ hours, which is equivalent to 1.875 hours. The rate of filling for each tap can be expressed as:

• Rate of smaller tap = 1/x (tank per hour)
• Rate of larger tap = 1/(x - 2) (tank per hour)

Setting Up the Equation

The combined rate of both taps filling the tank can be written as:

1/x + 1/(x - 2) = 1/1.875

Solving the Equation

First, simplify the right side:

1/1.875 = 0.5333

Now, we can rewrite the equation:

1/x + 1/(x - 2) = 0.5333

To eliminate the fractions, multiply through by x(x - 2):

(x - 2) + x = 0.5333x(x - 2)

This simplifies to:

2x - 2 = 0.5333x^2 - 1.0666x

Rearranging the Equation

Bringing all terms to one side gives:

0.5333x^2 - 3.0666x + 2 = 0

Using the Quadratic Formula

Now, we can apply the quadratic formula:

x = [3.0666 ± √(3.0666² - 4 * 0.5333 * 2)] / (2 * 0.5333)

Calculating the discriminant:

3.0666² - 4 * 0.5333 * 2 = 9.4

Now, substituting back into the formula gives:

x = [3.0666 ± √9.4] / 1.0666

Finding the Values

Calculating the square root and solving for x will yield two possible values. However, we only need the positive value that makes sense in this context.

Final Results

After solving, we find:

• The smaller tap takes approximately 3 hours to fill the tank.
• The larger tap takes approximately 1 hour to fill the tank.

Thus, the smaller tap fills the tank in 3 hours, while the larger tap does it in 1 hour.