Problem 28 - Algebra 1: Concepts and Skills

Question

Find the difference. $$ -\frac{5}{8}-\left(-\frac{3}{8}\right) $$

Step-by-Step Solution

Answer

-1/4

Step 1 - Identify the components

Here is the main operation, subtraction. But there is a catch. The second part comprises a minus sign followed by a negative fraction (-1 * -3/8), which effectively becomes addition.

Step 2 - Apply operation

So, perform the operation as follows: -5/8 - (-3/8) = -5/8 + 3/8.

Step 3 - Perform the addition

The addition will result as -5/8 + 3/8 = -2/8. As these fractions share the same denominator, simple arithmetic operations can be done on the numerators.

Step 4 - Simplify the fraction

The fraction -2/8 can be simplified to -1/4 by dividing both the numerator and denominator by the highest common factor, which is 2.

Key Concepts

Understanding Negative Numbers

Negative numbers are those that are less than zero. They are often used to represent a lack of something, like a debt or a drop in temperature. In math, they are denoted with a minus sign (-) before the number. When handling operations involving negative numbers, such as subtraction, understanding how the signs interact is crucial.

Consider two negative numbers:

• A negative number subtracted from another negative number can turn into an addition problem. For example: - When we see \(-(-3)\), we should read this as \'subtracting a negative number,\' which becomes a positive operation.
• Think about removing debt: Subtracting a negative (like removing debt) adds to what you have.

If you start off with \(-\frac{5}{8}\) and want to subtract \(-\frac{3}{8}\), all you really do is add \(\frac{3}{8}\) instead.

Simplifying Fractions Made Easy

Simplifying fractions means reducing them to their simplest form, which makes them easier to work with or understand. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

To simplify fractions, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.
• The result is a simplified fraction.

For example, let's simplify \(-\frac{2}{8}\):

1. Here, the GCF of 2 and 8 is 2.
2. Dividing the numerator (\(-2\)) and the denominator (8) by 2 gives you \(-\frac{1}{4}\).
3. This is the simplest form of the fraction, making it easier to use in any future calculations.

Finding Common Denominators

When adding or subtracting fractions, a common denominator is necessary. A common denominator is a shared multiple of the denominators of the fractions involved.

Here's how you find a common denominator:

• Identify the denominators of the fractions you want to add or subtract.
• If they are already the same, like in \-\frac{5}{8}\ and \-\frac{3}{8}\, you can perform the operation immediately.
• If they are different, find a number that both denominators can divide into without a remainder — this is the least common denominator (LCD).

In our specific exercise, since both fractions \-\frac{5}{8}\ and \(-\frac{3}{8}\) share a common denominator of 8, we simply performed the operation on the numerators, leading to a straightforward solution. This concept simplifies the process of working with fractions greatly, making addition and subtraction easier and faster.