Understanding Tyre Puncture Rates

This problem involves calculating the combined effect of two separate processes (punctures leaking air) that contribute to a single outcome (the tyre becoming flat). This is similar to 'work rate' problems where multiple agents complete a task together.

The key idea is that if something completes a task in time \(T\), its rate of work (or leakage rate in this case) is \( \frac{1}{T} \) of the task per unit of time.

Calculating Individual Puncture Rates

• The first puncture can flatten the tyre in 45 minutes.
• So, the rate of the first puncture is \( \frac{1}{45} \) of the tyre per minute.

• The second puncture can flatten the tyre in 90 minutes.
• So, the rate of the second puncture is \( \frac{1}{90} \) of the tyre per minute.

Calculating Combined Puncture Rate

When both punctures are leaking air simultaneously, their rates add up. The combined rate is the sum of the individual rates.

Combined rate = Rate of first puncture + Rate of second puncture

Combined rate \( = \frac{1}{45} + \frac{1}{90} \)

To add these fractions, we need a common denominator, which is 90.

\( \frac{1}{45} = \frac{1 \times 2}{45 \times 2} = \frac{2}{90} \)

So, Combined rate \( = \frac{2}{90} + \frac{1}{90} = \frac{2 + 1}{90} = \frac{3}{90} \)

Simplifying the fraction: \( \frac{3}{90} = \frac{1}{30} \)

The combined rate of both punctures together is \( \frac{1}{30} \) of the tyre per minute.

Calculating Time for Both Punctures Together

If the combined rate is \( \frac{1}{30} \) of the tyre per minute, it means that in one minute, \( \frac{1}{30} \) of the tyre becomes flat. To flatten the entire tyre (which is 1 whole), the time taken is the reciprocal of the combined rate.

Time taken together = \( \frac{1}{\text{Combined rate}} \)

Time taken together \( = \frac{1}{\frac{1}{30}} = 1 \times \frac{30}{1} = 30 \)

So, it takes 30 minutes for both the punctures together to make the tyre flat.

Step-by-Step Solution Summary

Here’s a quick summary of the steps to solve this tyre puncture problem:

1. Identify the time taken by each puncture individually.
2. Calculate the individual rate for each puncture (1/time).
3. Add the individual rates to find the combined rate.
4. Calculate the time taken together by taking the reciprocal of the combined rate.

Since the combined rate is \( \frac{1}{30} \) tyre per minute, the time taken is 30 minutes.

Revision Table: Tyre Puncture Problem

Additional Information: Work and Rate Problems

Problems like the tyre puncture scenario are common examples of 'work and rate' problems. In these problems, a certain amount of 'work' needs to be done (like filling a tank, completing a job, or flattening a tyre).

• The 'rate' is the amount of work done per unit of time.
• If an agent completes the work in time \(T\), their rate is \(1/T\).
• If multiple agents work together, and their work contributes to the same goal, their rates are typically added.
• If one agent's work counteracts another's (like one pipe filling and another draining a tank), their rates might be subtracted.
• The total time taken when working together is the reciprocal of the combined rate.

Understanding the relationship between work, rate, and time (Work = Rate × Time) is crucial for solving these types of problems.