Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\begin{aligned} &6.47+8.604\\\ &3.37+90.8 \end{aligned}$$

Short answer

The properly rounded sums are 15.07 and 94.2.

Step-by-step solution

Understand the Concept of Significant Figures

The number of significant figures in a result is based on the numbers in the original expression. When adding or subtracting, the result must have the same number of decimal places as the number with the fewest decimal places in the expression.

Add 6.47 and 8.604

6.47 has two decimal places, and 8.604 has three. The result should be rounded to two decimal places, the same as the number with fewer decimal places.

Perform the Addition of the First Expression

Add 6.47 and 8.604 to get 15.074. Round this to two decimal places to get 15.07.

Add 3.37 and 90.8

3.37 has two decimal places, and 90.8 has one. The result should be rounded to one decimal place, the same as the number with fewer decimal places.

Perform the Addition of the Second Expression

Add 3.37 and 90.8 to get 94.17. Round this to one decimal place to get 94.2.

Key concepts

Combined Operations with Approximate Numbers

When performing operations such as addition or subtraction with approximate numbers, it's crucial to pay attention to the precision of the numbers involved. Approximate numbers often arise from measurements or calculations that inherently carry some level of uncertainty—this is why significant figures come into play.

Consider the operation of adding 6.47 and 8.604. Before we combine these numbers, we need to understand that although both are close approximations of quantities, they do not carry the same level of precision due to their differing number of decimal places. The operation must respect the least precise number, in this case, 6.47, which has two decimal places. When we complete the calculation, we maintain this level of precision in our final answer by rounding accordingly.

Rounding in Mathematics

Rounding is a fundamental concept in mathematics used to reduce the number of digits in a number while preserving its overall value within an acceptable range. In the context of combined operations with approximate numbers, rounding becomes a tool to maintain consistency in the level of precision.

When rounding, if the digit immediately right of the last retained place is five or more, we increase the last remaining digit by one. If it is less than five, we leave it as it is. For example, when you see the number 15.074 from our addition, and you're asked to round to two decimal places, you would round up to 15.07 because the third decimal digit, 4, is less than five.

Significant Figures Rules

Understanding the rules of significant figures is imperative for working with approximate numbers. Significant figures convey the precision of a measurement. The rules are straightforward yet crucial:

• Non-zero digits are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros are not significant as they only indicate the position of the decimal point.
• Trailing zeros in the decimal portion are significant.

Applying these rules involves first determining which number in your operation has the least number of significant decimal places. For instance, in 3.37 and 90.8, despite 90.8 having only one decimal place, both numbers have significant digits. Hence, we round the sum, 94.17, to one decimal place, becoming 94.2, aligning with the rule that the result has the same number of decimal places as the number with the fewest decimal places.