Solving Tyre Puncture Time Problems

This problem involves calculating the combined effect of multiple independent processes happening simultaneously, specifically air leaking from a tyre through multiple punctures. When dealing with rates of work (or in this case, rates of deflation), we can determine the individual rates and then combine them to find the total rate.

Understanding Individual Puncture Rates

The rate at which a single puncture flattens the tyre is the inverse of the time it takes for that puncture to flatten the tyre alone. If a puncture can flatten the tyre in $T$ minutes, its rate is $\frac{1}{T}$ of the tyre per minute.

• The first puncture takes 9 minutes alone. Its rate is $\frac{1}{9}$ of the tyre per minute.
• The second puncture takes 18 minutes alone. Its rate is $\frac{1}{18}$ of the tyre per minute.
• The third puncture takes 6 minutes alone. Its rate is $\frac{1}{6}$ of the tyre per minute.

Calculating the Combined Leak Rate

When all punctures are leaking air simultaneously, their rates add up. The total rate of air leakage is the sum of the individual rates of each puncture. Let $R_{total}$ be the combined rate.

$$R_{total} = R_1 + R_2 + R_3$$

Substituting the individual rates:

$$R_{total} = \frac{1}{9} + \frac{1}{18} + \frac{1}{6}$$

To add these fractions, we need a common denominator, which is 18. Convert each fraction to have a denominator of 18:

• $\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18}$
• $\frac{1}{18}$ remains $\frac{1}{18}$
• $\frac{1}{6} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18}$

Now, add the fractions:

$$R_{total} = \frac{2}{18} + \frac{1}{18} + \frac{3}{18} = \frac{2 + 1 + 3}{18} = \frac{6}{18}$$

Simplify the total rate:

$$R_{total} = \frac{6}{18} = \frac{1}{3}$$

The combined rate of air leakage from all three punctures is $\frac{1}{3}$ of the tyre per minute.

Determining the Total Time to Flatten the Tyre

The time it takes for all punctures together to flatten the tyre is the inverse of the combined rate. If the combined rate is $R_{total}$, the time $T_{total}$ is $\frac{1}{R_{total}}$.

$$T_{total} = \frac{1}{R_{total}}$$

Substitute the calculated combined rate:

$$T_{total} = \frac{1}{1/3}$$

Dividing by a fraction is the same as multiplying by its reciprocal:

$$T_{total} = 1 \times 3 = 3$$

Therefore, it takes 3 minutes for all three punctures together to make the tyre flat.

Revision Table: Tyre Puncture Calculation

Here's a summary of the steps:

Additional Information: Work and Rate Concepts

This type of problem is a classic example of "work and rate" problems. The general principle is that if a task can be completed in time $T$, the rate at which the task is completed is $\frac{1}{T}$ per unit of time. When multiple agents (like punctures) work together to complete the same task (flattening the tyre), their individual rates are added to find the combined rate, assuming they work independently and simultaneously.

Key concepts:

• Rate of Work: Amount of work done per unit of time. If a task takes $T$ time, the rate is $1/T$.
• Total Work: In these problems, the total work is usually completing one whole task (e.g., filling a tank, finishing a job, flattening a tyre). We often represent this as '1 unit' of work.
• Combined Rate: When multiple agents work together, their individual rates add up: $R_{total} = R_1 + R_2 + R_3 + ...$
• Time Taken Together: The time taken to complete the task together is the inverse of the combined rate: $T_{total} = \frac{1}{R_{total}}$.

This principle applies to various scenarios, such as pipes filling a tank, people completing a job, or multiple processes working in parallel.