Problem 96 - Beginning and Intermediate Algebra

Question

Without calculating, determine whether each answer is positive or negative. Then use a calculator to find the exact difference. \(56.875-87.262\)

Step-by-Step Solution

Answer

The answer is negative, specifically -30.387.

Step 1: Analyze the Subtraction

We need to find the difference between 56.875 and 87.262. Notice that 56.875 is less than 87.262.

Step 2: Determine Sign of Difference

Since the number we are subtracting from (56.875) is smaller than the number we are subtracting (87.262), the result will be negative.

Step 3: Calculate the Exact Difference Using a Calculator

Using a calculator, subtract 56.875 from 87.262. The calculation will be 56.875 - 87.262 = -30.387.

Key Concepts

Negative Numbers

Negative numbers might seem a bit confusing at first, but they are simply numbers that are less than zero. Imagine a number line - zero is in the middle, positive numbers are on the right, and negative numbers are on the left.
In subtraction, if the starting number is less than the number being subtracted, your answer will be negative. Just like when you have $5, but you owe $8, you are $3 in debt; or rather, you are at negative -3.
In mathematics, we use a minus sign (-) in front of a number to show that it is negative. Negative numbers are very useful in representing situations where values decrease, like debts, losses, or drops in temperature. Remember, whenever the subtraction tail (the number you are subtracting from) is smaller than the head (the number you are subtracting), expect a negative outcome.

Difference Calculation

Calculating the difference between two numbers essentially means finding how much one number varies from another. This is a fundamental concept in subtraction.
To determine this difference, you subtract the smaller number from the larger one. The formula is simple: \( \text{Difference} = \text{Larger number} - \text{Smaller number} \). If the number you start subtracting from is smaller, then the result will end in the negative.
For the given numbers, 56.875 and 87.262, subtracting the first from the second gives us \(-30.387\). This negative result shows how much more 87.262 is compared to 56.875.

Number Comparison

Comparing numbers means looking at their sizes and understanding how they relate to each other in terms of value. This concept is handy when solving subtraction problems, as it helps us anticipate whether a result will be positive or negative.
To easily compare, line up the numbers vertically and check the digits from left to right. The first location where they differ determines which is larger.
In this case, comparing 56.875 and 87.262 immediately reveals that 56.875 is less than 87.262, since comparing the first digits 5 and 8 already shows which is the larger number.
When doing such comparisons, remember:

• If the first number is smaller, subtracting will yield a negative result.
• If the first number is larger, the result will be positive.

This simple method helps in quickly determining outcomes when dealing with subtraction.