Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$(7.37)(3.28)+(8.36)(2.64)$$

Short answer

24.22 + 22.07 = 46.29

Step-by-step solution

Multiply the first set of numbers

Multiply 7.37 by 3.28 to get their product. Use the proper significant figures based on the least number of decimal places in the given numbers.

Multiply the second set of numbers

Multiply 8.36 by 2.64 to get their product. Again, use the proper significant figures based on the least number of decimal places in the given numbers.

Add the products

Add the two products obtained from Step 1 and Step 2 together. The sum should also respect the significant figures, based on the least number of significant figures in the added numbers.

Round to the proper number of significant figures

Round the final answer to the correct number of significant figures if necessary. The final result must reflect the precision of the least precise number used in the calculations.

Key concepts

Significant Figures

When working with measurements in science and mathematics, we often encounter numbers that are not exact, known as approximate numbers. The precision of such numbers is indicated by the number of significant figures they contain. Significant figures are the digits in a number that carry meaning contributing to its precision, excluding any leading zeros.

In the context of the exercise, when multiplying the numbers 7.37 and 3.28, each having three significant figures, the result should be reported with the same level of precision. This is because the precision of the final answer should not exceed the least precise measurement involved in the calculation. It's important to remember that all non-zero digits, and any zeros between them or after a decimal point following a non-zero are considered significant.

Rounding Numbers

Rounding numbers is a technique used to reduce the digits in a number while keeping its value close to the original number. In our exercise, after performing the arithmetic operations, we might end up with a result that has more significant figures or decimal places than the numbers we started with. To maintain consistency, we round the result.

When rounding a number, we look at the digit to the right of our target place value: if it's five or greater, we round up; if it's less than five, we round down. For instance, if we need to round a sum to three significant figures and our result is 123.456, we would round it to 123 based on this rule.

Arithmetic Operations in Mathematics

Performing arithmetic operations with approximate numbers follows the same basic principles as operations with exact numbers: addition, subtraction, multiplication, and division. However, when working with approximate numbers, we must be extra careful to ensure the precision of our results. This entails adhering to the rules of significant figures throughout our calculations, whether we're adding products or subtracting sums.

In the given exercise, we perform multiplication before addition, adhering to the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We then apply significant figure rules and rounding to ensure that our final answer correctly reflects the precision of the numbers we used. This approach not only gives us a mathematically sound result but also teaches the importance of accuracy in scientific and mathematical computations.