Problem 2 - Basic Engineering Mathematics

Question

State which type of error has been made in the following statements: (a) \(72 \times 31.429=2262.9\) (b) \(16 \times 0.08 \times 7=89.6\) (c) \(11.714 \times 0.0088=0.3247\), correct to 4 decimal places. (d) \(\frac{29.74 \times 0.0512}{11.89}=0.12\), correct to 2 significant figures.

Step-by-Step Solution

Answer

Statement (b): Decimal error. (c): Calculation error. (d): Rounding error.

Step 1: Verify statement (a)

Calculate the product of the numbers in the statement (a) to check for accuracy: \ \(72 \times 31.429 = 2262.888\) \ This rounds to 2262.9 at one decimal place, so statement (a) is correct.

Step 2: Verify statement (b)

Perform the multiplication to check if it equals 89.6: \ \(16 \times 0.08 \times 7 = 8.96\) \ The original statement gives 89.6, which is incorrect as the accurate calculation gives 8.96. The error here is a misplacement of the decimal point.

Step 3: Verify statement (c)

Calculate the product and round the result to four decimal places: \ \(11.714 \times 0.0088 = 0.1030832\) \ Rounded to four decimal places this becomes 0.1031, not 0.3247. The mistake here is the incorrect calculation.

Step 4: Verify statement (d)

Perform the operation and round the result to two significant figures: \ \(\frac{29.74 \times 0.0512}{11.89} = 0.128088\) \ Rounded to two significant figures, this becomes 0.13, not 0.12. The error is in the incorrect rounding to significant figures.

Key Concepts

Understanding Decimal Point Error

A decimal point error occurs when the decimal point is placed incorrectly, leading to an inaccurate result. For example, when calculating

1. perform one step at a time: 16 \times 0.08 \times 7, the expected result: \(8.96\). The error arose from misplacing the decimal, transforming it to 89.6 inaccurately.

Decimal point errors are common when converting or handling large figures, or switching between units. Always check your calculations to ensure the decimal point aligns vertically in each stage. Double-checking your results with a calculator or performing a reverse calculation can help you avoid such mistakes.

Insight Into Rounding Errors

Rounding errors happen when numbers are rounded off to a certain number of decimal places or significant figures, impacting the final outcome. While using figures like

• 0.1030832 rounded to four decimal places results in \(0.1031\), not \(0.3247\).

Rounding can simplify complex numbers and calculations, but must be approached with caution. It’s important to keep track of how much rounding is used during tight-margin calculations. Always take note of the instructions provided on how many decimal points or significant figures to round to ensure precision in results. Remember, incorrect rounding is often a silent error in calculations.

Exploring Calculation Errors

Calculation errors occur during the process of computation, resulting from various factors like incorrect arithmetic operations, mistyped numbers, or logical errors. In the exercise, multiplying

• 11.714 \times 0.0088produced a wrong result \(0.3247\) when it should have been \(0.1030832\) rounded appropriately.

To avoid these, systematic checking of all mathematical operations is crucial. Double-entry method is useful, wherein calculations are done independently twice to verify consistency. Moreover, always have a habit of reviewing your math operations step-by-step, and using reliable calculators or software can significantly reduce the chance of calculation mishaps.

Grasping Significant Figures

Significant figures are digits within a number that contribute to its precision. They are important in representing numbers accurately, especially in scientific calculations.

• For example, when rounding\frac{29.74 \times 0.0512}{11.89} to two significant figures, one should get \(0.13\) and not \(0.12\).

Handling significant figures correctly means paying close attention to instructions regarding how many figures to use. Precision is essential here, as mishandling leads to round-off errors, affecting the dependent calculations. Practicing on diverse scenarios where significant figures apply can aid understanding and improve confidence in using them accurately.