Problem 21 - Basic Engineering Mathematics

Question

Evaluate \(37.81 \div 1.7\), correct to (i) 4 significant figures and (ii) 4 decimal places.

Step-by-Step Solution

Answer

(i) 22.24; (ii) 22.2412.

Step 1: Understand Significant Figures

Significant figures are the numbers that carry meaning contributing to a figure's precision. Non-zero digits and any zeros between them are always significant. Trailing zeros in the decimal part are also significant. In this problem, we need to express the result to 4 significant figures.

Step 2: Perform the Division

Use a calculator to divide 37.81 by 1.7. The exact result is approximately 22.24117647.

Step 3: Round to 4 Significant Figures

We need the result to have 4 significant figures. The number 22.24117647 rounded to 4 significant figures is 22.24. We keep the digits 2, 2, 2, and 4.

Step 4: Understand Decimal Places

Decimal places refer to the number of digits to the right of the decimal point. We need to express the result with 4 decimal places.

Step 5: Round to 4 Decimal Places

To round 22.24117647 to 4 decimal places, we look at the fifth decimal place (a '7'), which tells us to round up. Thus, the result becomes 22.2412.

Key Concepts

Significant Figures

Significant figures are crucial for indicating the precision of a number. These are digits in a number that contribute to its measurement resolution, starting from the first non-zero digit. Here’s how to look at significant figures:

• All non-zero digits (1 through 9) are always significant.
• Any zeros between non-zero digits are significant as they indicate the level of precision in a measurement.
• Trailing zeros in the decimal part are significant because they show accuracy in the number.

For the division of 37.81 by 1.7, you want the result expressed with 4 significant figures. This means you need to select the first four digits of the quotient that carry meaningful value. In our example, once you perform the division, you get approximately 22.24117647, where you consider the first four digits: 2, 2, 2, and 4, leading to 22.24.

Decimal Places

When working with decimal places, you focus on the digits that follow the decimal point. These numbers represent fractions of ten and are crucial for expressing exact amounts in mathematics. Here’s a breakdown of understanding decimal places:

• Decimal places help in detailed numerical representation by showing tiny portions of whole units.
• To denote accuracy, count the digits starting right after the decimal point.
• Adjusting decimal places allows balancing precision with simplicity depending on the context.

In our exercise, after dividing 37.81 by 1.7, you need the quotient to be shown with 4 decimal places: 22.24117647 becomes 22.2412. You focus on rewriting the number so it contains exactly four digits after the decimal, emphasizing precision.

Rounding Numbers

Rounding numbers involves adjusting a number to make it simpler, while keeping its value close to what it originally was. This process helps in making numbers easier to work with, especially when exact precision isn’t necessary, or when comparing numbers is required. Here’s how rounding works:

• Identify which place you want to round the number to. It could be a particular decimal place or significant figure.
• Look at the digit immediately following that place. If it is 5 or higher, round up by adding one to the number you're rounding to; if it’s less than 5, round down by keeping the original number as is.
• Rounding can simplify calculations and make numerical values more comprehensible.

In our division problem, rounding to 4 significant figures resulted in 22.24, because we observed the digit after the fourth figure in 22.24117647 was less than 5. Similarly, rounding to 4 decimal places required noticing the digit in the fifth decimal place was a 7, meaning you round up: therefore, 22.2412.