Problem 59 - Prealgebra

Question

Without pencil and paper or a calculator. Which number is closest to the product \(-151(-49) ?\) a. \(-200\) b. \(-100\) c. 3 d. \(7,500\)

Step-by-Step Solution

Answer

The closest number is \(7500\), option d.

Step 1: Recognize the Sign of the Product

The product of two negative numbers is always positive. Since both \(-151\) and \(-49\) are negative, their product will be positive.

Step 2: Estimate the Product

We need to find the numerical value of the product. Estimate \(-151 \times -49\) by rounding \(-151\) to \(-150\) and \(-49\) to \(-50\). The product becomes approximately \(150 \times 50\).

Step 3: Calculate the Estimated Product

Calculate \(150 \times 50\) by using simple multiplication. Multiply \(150\) times \(50\) which gives you \(150 \times 50 = 7500\).

Step 4: Compare with the Options

Given the approximate product is \(7500\), compare this value to the options provided: a. \(-200\), b. \(-100\), c. 3, d. \(7500\). It is clear that option d. \(7500\) is the closest to our estimated product.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero and represented with a minus sign, such as -3, -151, or -49. These numbers play a crucial role in mathematics, especially when it comes to multiplication and estimation. When you multiply two negative numbers, the product is always positive. This might sound a bit counterintuitive at first, but here's a simple way to think about it: when you have two negatives, they effectively "cancel" each other out. For instance, if you multiply -2 by -3, the product is 6, not -6. This is because reversing a negative direction twice brings you back to a positive direction.
This fundamental property is helpful in estimating product values without using a calculator, ensuring a more intuitive understanding of negative number operations. Understanding this concept helps you solve more complex equations and better grasp number interactions.

Multiplication

Multiplication can be thought of as repeated addition. If you multiply 3 by 5, it is the same as adding three 5s together: 5 + 5 + 5 = 15. When dealing with larger numbers, like -151 and -49, multiplication simplifies adding -151 repeatedly 49 times.
However, the multiplication of large numbers isn't just about repeated addition. It's essential to understand the principles behind it, especially when working with negative numbers. While it doesn’t visually appear as repetitions with large numbers, the underlying concept remains similar.
Multiplication can also be split into smaller, manageable parts using distributive properties. For instance, multiplying 150 by 50 can be easier when you break it down to smaller numbers and multiply in parts. This strategy is often used for multiplication problems requiring mental math, easing the process.

Rounding

Rounding is a technique used to simplify numbers, making mental computations or estimations easier. When we round, we adjust numbers to the nearest practical value. For example, -151 can be rounded to -150, and -49 can be rounded to -50. This makes multiplication simpler because dealing with neat numbers like 150 and 50 is more straightforward than 151 and 49.
The general rule for rounding is:

• If the digit to the right is 5 or greater, round up.
• If it's less than 5, round down.

When estimating products, rounding helps you quickly identify a rough value that is close enough to the real result. This is why rounding is a perfect tool for estimating product values, allowing you to compare against multiple-choice options and find the closest match without performing detailed calculations.