Problem 28 - Technical Mathematics with Calculus

Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\frac{6.73}{8.38}-\frac{5.97}{8.06}+\frac{8.63}{1.91}$$

Step-by-Step Solution

Answer

4.58

Step 1: Perform the Division for Each Fraction

Divide the numerators by their respective denominators. This gives us the initial decimal results for each fraction. Calculate each division to several decimal places to maintain accuracy throughout the process:\[\frac{6.73}{8.38} = 0.803343...\text{, }\frac{5.97}{8.06} = 0.740694...\text{, and }\frac{8.63}{1.91} = 4.518324...\text{.}\]

Step 2: Round Each Result According to Significant Figures

The question requires that we keep the proper number of digits. Typically, when dividing, the number of significant figures in the result should be the same as the number in the least precise measurement (the one with the fewest figures after the decimal point). In this case, each number has two figures after the decimal point, so each result should be rounded to two decimal places:\[\frac{6.73}{8.38} \rightarrow 0.80\text{, }\frac{5.97}{8.06} \rightarrow 0.74\text{, }\frac{8.63}{1.91} \rightarrow 4.52.\]

Step 3: Combine the Rounded Results

Now combine the rounded results. Perform the subtraction and addition using these rounded numbers to ensure that the proper number of digits is maintained in your answer:\[0.80-0.74+4.52 = 0.06+4.52 = 4.58.\]

Step 4: State the Final Answer

Express the combined result to the proper number of significant figures, which in this case is two decimal places:\[4.58\text{.}\]

Key Concepts

Significant Figures

When performing calculations, particularly in science and engineering, the precision of the numbers involved is very important. Significant figures (or 'sig figs') are the digits in a number that carry meaning contributing to its measurement resolution. These include all non-zero digits, any zeros between significant digits, and trailing zeros in a decimal number. For instance, in the number 0.004560, there are three significant figures: 4, 5, and 6; the trailing zeros are also significant because they are after the decimal point.

When combining operations with approximate numbers, like in the exercise given, it's important to consider significant figures to maintain the correct level of precision in your final result. This means that when you divide numbers, the quantity of significant figures in the result should match the smallest number of significant figures of any of the numbers you are dividing. In our exercise, each number in the operations had two significant figures after the decimal, which set the standard for the final answer.

Decimal Division

The process of dividing one decimal by another, known as decimal division, can sometimes be intimidating, but it's quite similar to dividing whole numbers. Take the numbers expressed in the exercise: You line them up by the decimal point and perform division as usual. It's also crucial to extend the division to enough places beyond the decimal point to ensure accuracy before rounding. Remember, you carry out the division exactly as you would with whole numbers, except you must account for the decimal points.

Different exercises might require different levels of precision, which is where significant figures come into play. After division, results can have many decimal places, but they should be rounded according to the rules of significant figures. If you're solving an exercise and get a result with more significant figures than the original numbers, like in the given problem, you'll need to round off your results to match the precision dictated by the original numbers.

Rounded Results

Rounding is the process of reducing the digits in a number while trying to keep its value similar. The rules of rounding are essential when working with approximate numbers because they directly influence the precision of your results. To round properly, look at the digit immediately after the place value to which you're rounding: if it's five or higher, increase the rounding digit by one; if it's less than five, leave the rounding digit as is.

In calculations, rounded results ensure that the answer stays significant and reliable, as seen in our original exercise. After dividing and obtaining the initial decimals, each outcome is rounded to maintain consistency with the least precise measurement used in the calculation. This prevents the false precision, which could suggest a greater degree of accuracy than the original data actually supports. In essence, keeping the rounded results to the appropriate number of significant figures as dictated by the calculation rules ensures that the final answer respects the data's original precision.