Problem 60 - Prealgebra

Question

Work Problems \(55-60\) mentally, without pencil and paper or a calculator. Is \(-553-50\) closer to \(-600\) or \(-500 ?\)

Step-by-Step Solution

Answer

-603 is closer to -600.

Step 1: Understand the Problem

We're tasked with finding whether the number \(-553-50\) is closer to \(-600\) or \(-500\).Let's begin by calculating the result of the expression.

Step 2: Calculate the Expression

First, compute \(-553 - 50\). Subtract 50 from \(-553\):\[-553 - 50 = -603\]This gives us \(-603\).

Step 3: Determine Proximity to -600

Next, evaluate how close \(-603\) is to \(-600\). Calculate the absolute difference:\[|-603 - (-600)| = | -603 + 600 | = |-3| = 3\]

Step 4: Determine Proximity to -500

Now, assess how close \(-603\) is to \(-500\). Calculate the absolute difference:\[|-603 - (-500)| = |-603 + 500| = |-103| = 103\]

Step 5: Compare Distances

Compare the two differences: 3 (distance to \(-600\)) vs. 103 (distance to \(-500\)).Since 3 is less than 103, \(-603\) is closer to \(-600\).

Key Concepts

Mental Math

Mental math is the art of performing calculations in your head, without the aid of calculators, paper, or other tools. This skill is incredibly useful for quick estimation and problem-solving in daily life. In our given problem, you need to determine the result of

• -553 - 50

You start with

• -553

and simply subtract 50.
Using mental math, recall that subtracting a positive number like 50 means you are moving further into the negatives.
Imagine the number line: moving left from -553 by 50 steps lands you on

• -603

This quick mental calculation gives you a significant advantage in determining the proximity to

• -600

versus

• -500.

Even if these numbers sound daunting, practice can significantly enhance your mental math abilities. Think of them as skills you develop over time, making more spontaneous math tasks easier and more intuitive.

Integer Operations

Integer operations involve the addition, subtraction, multiplication, and division of whole numbers.
For this exercise, subtracting integers is particularly critical as we calculated

• -553 - 50

In operations with integers, keep in mind:

• Subtracting a smaller number from a larger negative number yields an even larger negative number, moving further left on the number line.
• Use the rules of addition: adding a negative (subtraction) makes the resultant number smaller.

Applying these principles to our example, subtracting 50 from

• -553

effectively lowers the number, resulting in

• -603.

It’s essential to be comfortable with integer operations as they form the foundational basis for advanced mathematical concepts. By mastering these, you will find yourself better equipped for more complex problem-solving scenarios.

Proximity and Comparison

Understanding proximity and comparison is about assessing how close one number is to others. Given the numbers

• -600
• -500

in comparison with

• -603,

we need to determine which it is closer to.
We calculate the differences,

• |-603 - (-600)| = 3
• |-603 - (-500)| = 103

These differences tell how far

• -603

is from other values and allow us to compare proximity easily.
The smaller difference (3, in this case) indicates greater closeness to

• -600,

proving that

• -603

forms a number closer to

• -600

than to

• -500

.
Such comparisons are commonly applied in real-world situations as well as standardized tests, assisting in comprehending numerical relationships and making educated decisions.