Problem 62 - Prealgebra

Question

Work Problems \(61-68\) mentally, without pencil and paper or a calculator. The product \(32(-522)\) is closest to which of the following numbers? a. \(15,000\) b. \(-500\) c. \(-1,500\) d. \(-15,000\)

Step-by-Step Solution

Answer

The product is closest to option d: -15,000.

Step 1: Estimate the Product

First, understand that we need to find the product of two numbers: 32 and -522. Since one of the numbers is negative, the product will also be negative. Next, approximate 522 to a nearby round number that is easier to multiply. Let's approximate it to 500 to simplify the calculation.

Step 2: Calculate Using Approximation

Now that 522 is approximated to 500, compute the product using this rounded number: \[ 32 imes (-500) = -16000 \] By using simple multiplication, we identify that the estimated product of 32 and -500 is -16000.

Step 3: Compare with the Options

Compare the estimated product of -16000 with the available choices. The options provided are a) 15,000, b) -500, c) -1,500, and d) -15,000. Based on the estimates, -16,000 is closest to -15,000, which is option d.

Key Concepts

Integer Multiplication

Integer multiplication is a basic arithmetic operation involving two whole numbers. Whole numbers, otherwise known as integers, can be positive, negative, or zero. When multiplying integers, it is important to keep in mind the rules regarding signs:

• If both integers are positive or both are negative, the product is positive.
• If one integer is positive and the other is negative, the product is negative.

For example, in the problem, when we multiply 32 (a positive integer) with -522 (a negative integer), the result is negative. Understanding these basic principles is crucial as it helps predict whether the outcome of a multiplication will be positive or negative. Thus, integer multiplication forms a foundation for more complex math problems.

Negative Numbers

Dealing with negative numbers can sometimes be tricky, but it's essential for various math operations, especially in algebra. When considering negative numbers, always remember:

• Negative numbers are less than zero and are on the left side of zero on the number line.
• Adding two negative numbers yields a more negative number.
• Subtracting a negative number is the same as adding its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive product or quotient.

In the given exercise, understanding the multiplication involving a negative number (-522) is crucial. The rule stating that a positive number multiplied by a negative number yields a negative product was applied here. That's why the product of 32 and -522 is negative, indicating direction, not quantity, on a number line.

Problem-Solving Strategies

Effective problem-solving strategies often involve approximation and estimation, especially when working mentally. This approach allows you to simplify complex multiplication problems by rounding to the nearest easily computable numbers. Here are some tips:

• Identify the numbers to approximate to simplify calculations.
• Round numbers to the nearest easy-to-handle figures, such as multiples of 10.
• Perform the calculation with the rounded numbers to estimate.
• Compare your estimate to available options if given.

In this specific exercise, we approximated -522 to -500 for smoother mental calculation. This technique is powerful as it quickly determines an answer close enough to identify the most likely option among available choices. By practicing these strategies, you'll enhance your speed and accuracy in mental math estimation.