Problem 64 - 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.

Step 1: Understand the Problem

We are asked to find which number the product of egative 47 and egative 470 is closest to. The numbers provided are (a) 25,000, (b) 420, (c) -2,500, and (d) -25,000.

Step 2: Simplify the Expression

Let's compute (-47) imes (-470) . Since multiplying two negative numbers results in a positive number, we know the product will be positive.

Step 3: Estimate the Numbers

Breakdown the multiplication: (-47) imes (-470) = 47 imes 470 . We can approximate: 47 imes 470 which is roughly equal to 50 imes 500 = 25,000 . That's a good estimation.

Step 4: Match to Closest Number

Comparing our estimated result, 25,000, to the options provided, we see that option (a) 25,000 is most closely aligned with our calculated value.

Key Concepts

Multiplication of Negative Numbers

When multiplying two numbers, where both are negative, the product is always positive. This might seem surprising at first because negative numbers represent values less than zero. However, when you multiply two negatives, they essentially cancel each other out.
For instance, if you think of multiplication as repeated addition,

• negative numbers represent a flip in direction.
• Thus, multiplying (-47) with (-470) is essentially flipping direction twice, resulting in a positive value.

Approximation Techniques

Approximating numbers is a valuable skill, especially when you need quick calculations. This technique allows us to simplify complex numbers into easily manageable figures that are close enough to give us a reliable estimate.
In our original problem, we approximated (-47) and (-470) to 50 and 500 respectively. These numbers are close and easier to multiply mentally.

• Multiplying 50 by 500 gives an approximate product of 25,000.
• Although this is an estimate, it provides a quick check that helps in many different settings.
• This kind of rounding and approximation is a cornerstone in mental math techniques.

Education Problem Solving

Problem-solving in education often involves breaking down complex problems into simpler parts. First, we understand what is being asked; in this case, determining the closest approximate product of (-47) and (-470). Next, we identify the properties of the numbers, such as their negativity and effect on the product direction. Finally, we apply simplification techniques, such as approximation, to ease calculations.
Education problem-solving has these key steps:

• Understanding the problem and identifying objectives.
• Breaking the problem into manageable segments.
• Using strategic techniques to approach the solution effectively.
• Reflecting on the solution and matching it against given options.

Mental Arithmetic Skills

The ability to perform calculations in your head is a crucial skill, not just in academic settings but in everyday life. Mental arithmetic involves quickly understanding and manipulating numbers without writing them down.
Consider how breaking numbers apart, like seeing 470 as 500, and multiplying mentally facilitates faster calculations:

• Focus on simple multipliers, like 50 and 500, to make the multiplication faster.
• Understand relationships, such as multiplying large numbers by first estimating and then adjusting slightly to account for the rounding.
• Consider practicing this skill regularly to improve speed and accuracy.
• Mental arithmetic can enhance problem-solving skills and allow for quick decision-making.