We are given the following information:

• Two taps together can fill a tank in $9 \frac{3}{8}$ hours.
• The larger tap takes 10 hours less than the smaller one to fill the tank.

We need to find the time it takes for each tap (larger and smaller) to fill the tank separately.

Step 1: Convert the mixed fraction

$9 \frac{3}{8}$ hours can be converted to an improper fraction: $9 \frac{3}{8} = \frac{75}{8} \text{ hours}$

Step 2: Define variables

Let:

• $t$ be the time taken by the smaller tap to fill the tank on its own.
• The larger tap takes $t - 10$ hours to fill the tank.

The rate of work (portion of the tank filled per hour) for each tap is:

• Smaller tap: $\frac{1}{t}$ of the tank per hour.
• Larger tap: $\frac{1}{t - 10}$ of the tank per hour.

Together, they fill $\frac{1}{t} + \frac{1}{t - 10}$ of the tank per hour. According to the problem, they fill the tank in $\frac{75}{8}$ hours together, so the rate of work is: $\frac{1}{t} + \frac{1}{t - 10} = \frac{8}{75}$

Step 3: Set up the equation

Now, let's solve the equation: $\frac{1}{t} + \frac{1}{t - 10} = \frac{8}{75}$

Multiply both sides by $75t(t - 10)$ to eliminate the denominators: $75(t - 10) + 75t = 8t(t - 10)$

Simplify: $75t - 750 + 75t = 8t^2 - 80t$ $150t - 750 = 8t^2 - 80t$ $8t^2 - 230t + 750 = 0$

Step 4: Solve the quadratic equation

We can solve the quadratic equation $8t^2 - 230t + 750 = 0$ using the quadratic formula: $t = \frac{-(-230) \pm \sqrt{(-230)^2 - 4(8)(750)}}{2(8)}$ $t = \frac{230 \pm \sqrt{52900 - 24000}}{16}$ $t = \frac{230 \pm \sqrt{28900}}{16}$ $t = \frac{230 \pm 170}{16}$

So we have two possible solutions: $t = \frac{230 + 170}{16} = \frac{400}{16} = 25 \quad \text{or} \quad t = \frac{230 - 170}{16} = \frac{60}{16} = 3.75$

Step 5: Interpret the results

The time $t = 3.75$ hours is not valid because the larger tap would take negative time. Therefore, the valid solution is: $t = 25 \text{ hours}$

Thus:

• The smaller tap takes 25 hours to fill the tank.
• The larger tap takes $25 - 10 = 15$ hours to fill the tank.

Final Answer:

• The smaller tap fills the tank in 25 hours.
• The larger tap fills the tank in 15 hours.

Would you like further clarification on any part of the solution?

Here are 5 related questions:

1. What is the formula for solving work rate problems?
2. How would the solution change if both taps worked at the same rate?
3. Can this problem be solved graphically? If so, how?
4. How do you solve quadratic equations by factoring instead of the quadratic formula?
5. How can you verify the solution once the times are found?

Tip: For work problems, always start by defining the rates for each worker and combine their rates for joint work.