Problem 28 - Fundamentals of Mathematics

Question

Estimate each calculation using the method of rounding. After you have made an estimate, find the exact value and compare this to the estimated result to see if your estimated value is reasonable. Results may vary. $$ 67 \cdot 42 $$

Step-by-Step Solution

Answer

The estimate is 2800, and the exact product is 2814; the estimate is reasonable.

Step 1: Round the Numbers

First, round each number to the nearest ten to make the calculation easier. 67 rounds to 70 and 42 rounds to 40.

Step 2: Calculate the Estimated Product

Multiply the rounded numbers to get an estimated product: \[ 70 \times 40 = 2800 \]

Step 3: Calculate the Exact Product

Now, calculate the exact product using the numbers 67 and 42: \[ 67 \times 42 = 2814 \]

Step 4: Compare the Estimated and Exact Values

Compare the estimated product (2800) with the exact product (2814). The estimated value is close to the exact value.

Key Concepts

Estimation

Estimation is a valuable mathematical skill that helps simplify complex calculations. It's especially useful when you need a quick approximation or when performing calculations without a calculator.

• To estimate a product, the first step usually involves rounding each number to a simpler, more manageable value.
• In this exercise, the numbers 67 and 42 are rounded to 70 and 40, respectively.
• Rounding to the nearest ten makes mental calculations much more straightforward.

After rounding, you multiply the new values to get an estimated product. Here, that product is 2800. This estimate gives you an idea of the size of the exact answer and helps in checking the reasonableness of your final calculations. Estimation is not about getting the exact number but rather finding a number that is close enough to give insight into the scale of the required product.

Multiplication

Multiplication is a fundamental arithmetic operation where one number is added to itself a certain number of times. It forms the base for more advanced math concepts like algebra and calculus.

• In the original problem, you multiply the numbers 67 and 42 to find the exact product.
• The multiplication of two numbers involves taking one number and adding it to itself the number of times equaled to the other number.

This calculation results in 2814, giving the precise amount you get when both numbers are mathfully interacted with each other. Being proficient in multiplication is essential, as it's applied in various fields such as science, engineering, and economics. Understanding multiplication ensures accuracy when precise calculation is necessary.

Exact value

The exact value of a multiplication problem is the precise result obtained by calculating the numbers without rounding or estimation.

• For the problem \(67 \times 42\), the exact value is 2814.
• Finding the exact value involves carefully performing the mathematical operation as intended with no alterations to the original numbers.

Comparing this with the estimated product, you can see how close the estimate was to the true value. This comparison helps in understanding the effectiveness of the estimation technique applied and provides insight into whether the rounded numbers gave a good approximation. Knowing the exact value is crucial when accuracy is of utmost importance.