Problem 25 - Basic Engineering Mathematics

Question

A hire tool firm finds that their net return from hiring tools is decreasing by \(10 \%\) per annum. If their net gain on a certain tool this year is \(£ 400\), find the possible total of all future profits from this tool (assuming the tool. lasts for ever).

Step-by-Step Solution

Answer

The total future profit from the tool is £4000.

Step 1: Understand the Problem

The problem describes a scenario where the net return from hiring tools is decreasing annually by a certain percentage. We need to determine the total future profits from the tool, which implies calculating the sum of an infinite series.

Step 2: Identify the Series Type

The given problem involves a situation where returns are decreasing by a constant percentage every year, depicting a geometric series. Each year's profit is less than the previous one by a certain factor.

Step 3: Define the Geometric Series

For a geometric series, the nth term can be expressed as: \( a_n = a_1 \times r^{n-1} \)where \( a_1 \) is the first term (\( \text{£}400 \)) and \( r \) is the common ratio. Here, \( r = 1 - 0.10 = 0.9 \) because the profit decreases by \( 10\% \) each year.

Step 4: Formula for the Sum of an Infinite Geometric Series

The sum \( S \) of an infinite geometric series with first term \( a_1 \) and common ratio \( r \) (where \( |r| < 1 \)) is given by:\[ S = \frac{a_1}{1 - r} \]

Step 5: Calculate the Total Future Profit

Using the formula from Step 4, substitute \( a_1 = 400 \) and \( r = 0.9 \): \[ S = \frac{400}{1 - 0.9} = \frac{400}{0.1} = 4000 \]Thus, the total future profit from the tool is \( \text{£}4000 \).

Key Concepts

Infinite Series

An infinite series is a summation of an infinite sequence of terms. When studying geometric series, you often encounter the concept of infinite series, particularly when determining the long-term behavior of an investment or profit, like in our tool hire example.
An infinite geometric series is written as:
\[ a_1 + a_1r + a_1r^2 + a_1r^3 + \ldots \] This indicates that the sequence continues indefinitely. In mathematics, we are often interested in whether the series converges to a specific value, rather than diverging.
To qualify as convergent, the common ratio \( r \) of an infinite geometric series must satisfy \(|r| < 1\). When this is the case, the sum of the series doesn't become infinitely large, but instead stabilizes to a fixed value, which we can calculate using a specific formula. This formula will help determine the cumulative future profits from the tool hire firm as profits become gradually smaller each year, approaching zero without ever quite reaching it.

Common Ratio

In a geometric series, the common ratio is a crucial component that determines how the series progresses. It is the factor by which each term is multiplied to get the next term.
In the given problem, the common ratio \( r \) is derived from the rate at which the net returns decrease. Since the profits decrease by \(10\%\) yearly, the common ratio turns out to be \(1 - 0.10 = 0.9\). This means each following year's return is only \(90\%\) of the prior year's profit.

• If \(|r| < 1\) – the series converges and has a finite sum, indicated in our scenario.
• If \(|r| = 1\) or \(|r| > 1\) – the series does not converge, implying there isn't a finite sum.

Understanding the common ratio helps us compute the series' total sum, allowing us to predict the complete future profits even if they decline yearly.

Sum of Series

The sum of a geometric series, especially an infinite series, is an essential point in calculations when forecasting trends over a long period. An infinite geometric series is particularly useful for understanding consistent changes, like decreasing annual profits.
When the condition \(|r| < 1\) is met, the formula to calculate the sum \( S \) of the infinite series is:
\[ S = \frac{a_1}{1 - r} \] In our scenario, the first term \( a_1 \) is \(\text{£}400\) and the common ratio \( r \) is \(0.9\). Plugging these values into the formula gives:
\[ S = \frac{400}{1 - 0.9} = \frac{400}{0.1} = 4000 \] Thus, the sum of all future profits from hiring this tool, even while decreasing annually, is \(\text{£}4000\). This calculation is crucial for any business trying to anticipate long-term earnings from steadily declining annual revenue.