Problem 29 - Fundamentals of Mathematics

Question

Estimate each value using the method of rounding. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. \(27 \cdot 473\)

Step-by-Step Solution

Answer

Estimated product: 15000. Exact product: 12771; estimated value is higher.

Step 1: Understand the Problem

We need to estimate the product of two numbers, 27 and 473, by rounding them. Then, we find the exact product and compare it with the estimated product.

Step 2: Round the Numbers

For estimation, round 27 to the nearest ten, which is 30, and round 473 to the nearest hundred, which is 500.

Step 3: Calculate the Estimated Value

Use the rounded numbers to estimate the product: \[ 30 \times 500 = 15000 \]

Step 4: Calculate the Exact Value

Multiply the original numbers to find the exact product: \[ 27 \times 473 = 12771 \]

Step 5: Compare Estimated and Exact Values

The estimated value (15000) is greater than the exact value (12771) by 2229. This shows the estimate is an over-approximation.

Key Concepts

Multiplication

When we multiply two numbers, we are essentially adding a number to itself a certain number of times. In the given exercise, we have to find the product of 27 and 473. The product is the result you get after multiplication. To understand it easily, think of it as:

• 27 groups of 473, or
• 473 groups of 27

To multiply these numbers, it's generally easier to break the calculation into smaller parts. You can use methods like traditional multiplication, the distributive property, or even simple multiplication algorithms.
When you multiply directly, you align the numbers and start multiplying one digit at a time, adding the intermediate results properly. In some cases, it might be useful to break numbers into their component parts—such as writing 473 as 470 + 3—and then multiplying each part by 27 separately, before adding the results together.

Product Comparison

Comparing products involves looking at both the estimated and exact values obtained through multiplication. Let's say we round the numbers to make the multiplication easier, as with rounding 27 to 30 and 473 to 500 here. This simplification allows us to quickly calculate the estimated product:\[30 \times 500 = 15000\]Compare this with the exact product of the calculation:\[27 \times 473 = 12771\]As we can see, the estimated product of 15000 is larger than the real product of 12771. This difference arises because our rounding led to using numbers that were larger than the originals, thus inflating the estimate. Though it's an approximation that helps us quickly get an idea of the result, comparing it with the exact product gives us a clear understanding of any over-approximation or under-approximation we might have encountered.

Exact Value Calculation

The exact value calculation gives us the precise answer to a multiplication problem without rounding. This involves working with the numbers as they are:\[ 27 \times 473\]To find the exact value, you follow a systematic process of multiplication under arithmetic rules; this may involve:

• Using paper and pencil calculations to align and multiply each digit sequentially.
• Adding intermediate steps often found in traditional long multiplication.

In this example, the exact calculation results in a product of 12771, a crucial answer that is free from estimation errors. The importance of calculating the exact value lies in providing an accurate solution necessary for contexts where precision is critical. While estimated calculations save time, they can sometimes be misleading if used alone without checking the precise outcome.