Problem 63 - Prealgebra

Question

Work Problems \(61-68\) mentally, without pencil and paper or a calculator. The product \(-47(470)\) is closest to which of the following numbers? a. \(25,000\) b. 420 c. \(-2,500\) d. \(-25,000\)

Step-by-Step Solution

Answer

The product is closest to \ -25,000 (option d).

Step 1: Estimate the Product

Start by estimating the product of the given numbers: \[-47 \times 470\]To make mental calculations easier, round the numbers to simpler forms. Round \-47 to \-50 and 470 to 500. This gives:\[-50 \times 500 = -25,000\]

Step 2: Compare Closest Options

Now, compare this estimated product, \-25,000, with the given options: \a. 25,000 \b. 420 \c. \-2,500 \d. \-25,000.The option closest to \-25,000 is option \d: \(-25,000\).

Key Concepts

Estimation

When we talk about estimation, we refer to the process of finding an approximate value that makes calculations more manageable. Estimation is a valuable skill in mental math because it allows us to simplify complex problems by transforming them into easier ones. For instance, in the exercise, where you are calculating the product of \(-47\) and \(470\), exact calculation could be challenging to do mentally. But estimating transforms this complex multiplication into a simpler one by changing the numbers to \(-50\) and \(500\). Both being round numbers makes the arithmetic straightforward.

Here are some steps to estimate effectively:

• Identify the key numbers involved in the problem.
• Round these numbers to nearest ten or hundred for easier computation.
• Calculate the new, simplified problem.

It's important to remember that while the estimate isn't exact, it's close enough to guide you in choosing between the multiple choice options.

Rounding Numbers

Rounding numbers is an essential part of estimation, especially when dealing with large figures like \(-47\) and \(470\). Rounding makes numbers easier to work with by replacing them with approximate number values. For example, rounding \(-47\) to \(-50\) and \(470\) to \(500\) allows us to bypass the complexity of feeling around awkward digits.

Here's how you approach rounding:

• Look at the significant digits in the number (those numbers that are not zero).
• Decide which place value to round to (often tens, hundreds, etc.).
• If the digit after your rounding place is 5 or higher, round up by adding 1 to your place; if it's lower than 5, you leave your rounding place as is.

By rounding these to \(-50\) and \(500\), it makes mental math operations like multiplication much simpler and helps to generate an easy-to-use product.

Negative Numbers

Negative numbers can initially be tricky, especially in the context of multiplication, because the rules do not appear as intuitively as they do with positive numbers. However, the basic rules are simple and very critical to correct arithmetic operations. In negative number multiplication, the following rules apply:

• A negative number multiplied by a positive number results in a negative product.
• Two negative numbers multiplied together result in a positive product (not the case in our exercise, but useful to remember).

In our example, multiplying \(-50\) by \(500\) yields a negative product, \(-25,000\). This is indeed expected—one factor is negative, and the other is positive.

Multiplication

Multiplication is foundational in math, which combines numbers to produce a product. Long before calculators, understanding and performing basic multiplication was crucial for problem solving. Within the scope of estimation and rounding, multiplication demands that we comprehend how better approximations lead to a sounder mental estimate.

In this exercise, after rounding \(-47\) to \(-50\) and \(470\) to \(500\), you perform the multiplication:

• Multiplying \(-50\) by \(500\), you recognize that you multiply the absolute values and then apply the sign rule for one negative and one positive factor.
• The actual multiplication, \(50 \times 500\), proceeds as follows: \(50 \times 5 = 250\) then append the two zeros, resulting in 25,000.

Finally, applying the sign gives you \(-25,000\). This thought process and breakdown help reinforce both multiplication skills and confident use of mental math strategies.