Problem 14 - Basic Engineering Mathematics

Question

Express \(25 \mathrm{p}\) as a ratio of \(£ 4.251\)

Step-by-Step Solution

Answer

25p is approximately in a ratio of 1:17 to £4.251.

Step 1: Convert Pounds to Pence

Since the amounts need to be in the same units to calculate the ratio, convert the value in pounds (£4.251) to pence. There are 100 pence in a pound, so multiply the amount in pounds by 100: \[£4.251 \times 100 = 425.1 \text{ pence}\]

Step 2: Write the Ratio

Now that both amounts are in pence, you can express them as a ratio. We have 25p and 425.1p.Thus, the ratio is:\[\frac{25}{425.1}\]

Step 3: Simplify the Ratio

To simplify this ratio, divide both the numerator and the denominator by their greatest common divisor (GCD). Since the GCD of 25 and 425.1 is 25, divide both by 25:\[\frac{25}{425.1} = \frac{25 / 25}{425.1 / 25} = \frac{1}{17.004}\] This ratio can be represented as approximately 1:17.

Key Concepts

Currency Conversion

Currency conversion is the process of exchanging one type of currency for another at a specific rate. This is especially useful when dealing with different units in financial calculations.
For instance, in our exercise, we're converting British pounds (£) to pence (p) to ensure both amounts are in the same unit before calculating a ratio.

To convert from pounds to pence, multiply the amount in pounds by 100 since there are 100 pence in a pound. Here is how you do it for £4.251:

• Multiply by 100: £4.251 x 100 = 425.1 pence

Understanding how to properly convert currency is crucial for making accurate financial comparisons.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means finding an equivalent fraction where the numerator and the denominator have no common factors other than 1.
In our example, we have the fraction \( \frac{25}{425.1} \).

The first step in simplifying is to identify any common factors in the numerator and denominator. Dividing both by the greatest common divisor (GCD), we can transform the fraction into its simplest form. This simplification helps to make the numbers easier to work with and understand visually. It's akin to finding a simpler ratio between two quantities.
For this fraction:

• Identify the common factors: GCD of 25 and 425.1 is 25
• Divide both parts of the fraction: \( \frac{25}{425.1} = \frac{1}{17.004} \)

This makes the fraction easier to interpret: approximately \( 1:17 \).

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest positive integer that divides two or more numbers without leaving a remainder. It is crucial for simplifying fractions effectively.
Using the GCD ensures that both parts of a fraction are divided by the same number, maintaining the proportional relationship between them.

Here's how it works:

• Identify two numbers, such as 25 and 425.1
• Determine the largest number that divides both without a remainder (GCD here is 25)

Using the GCD for simplification in our example, we divide both the numerator and the denominator of \( \frac{25}{425.1} \) by 25, yielding \( \frac{1}{17.004} \). This approach guarantees that the fraction is reduced to its most straightforward form, making further calculations or interpretations more straightforward.