Problem 48 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(-\frac{7}{4}+3.3\)

Step-by-Step Solution

Answer

The simplified expression is 1.55.

Step 1: Convert the Fraction to a Decimal

The fraction \(-\frac{7}{4}\) needs to be converted into a decimal. Divide 7 by 4 to get 1.75. Since the fraction is negative, the result will be \(-1.75\).

Step 2: Add the Decimal to the Given Number

Now, take the decimal \(-1.75\) and add it to the number 3.3. This is done as follows: \(3.3 + (-1.75) = 3.3 - 1.75\).

Step 3: Calculate the Result of the Addition

Perform the subtraction: \(3.3 - 1.75\). Align the decimals and subtract 5 from 0 and borrow, then 7 from 12, and finally 1 from 2. The result is 1.55.

Key Concepts

Fraction Simplification

Fraction simplification is the process of transforming a fraction to its simplest form. The simplest form of a fraction has a numerator and a denominator that are both whole numbers and have no common factors other than 1.
For example, simplifying \( \frac{7}{4} \) doesn't mean changing the numbers but it's about reconverting the result or keeping the fraction as it is if no common factors exist.
To simplify a fraction, you follow these basic steps:

• Identify the factors of the numerator and the denominator.
• Find the greatest common factor (GCF).
• Divide both the numerator and the denominator by the GCF.

Fractions like \( \frac{7}{4} \) are improper fractions (numerator is greater than the denominator) and can be converted to a decimal instead of simplifying further as a fraction.
This is where decimal conversion becomes a tool to simplify calculations, especially when performing arithmetic operations.

Terminating Decimals

Terminating decimals are decimals that have a definitive ending after a finite number of digits, meaning they don't go on forever.
A fraction converts into a terminating decimal when its denominator in its simplest fractional form is composed of factors 2 and/or 5 only.
For instance, converting the fraction \( \frac{7}{4} \) involves determining whether the denominator 4 (2 x 2) fulfills this condition. It does, so the division of 7 by 4 results in the terminating decimal 1.75.
Some practical tips for dealing with terminating decimals include:

• Identify if a decimal is terminating by checking its finite number of digits.
• Recognize the pattern when repeating decimals occur.
• Practice converting between fractions and decimals to gain confidence.

Understanding when and how a decimal terminates can greatly simplify arithmetic operations and help in making calculations less cumbersome.

Arithmetic Operations

Arithmetic operations encompass basic mathematical actions such as addition, subtraction, multiplication, and division.
In the context of decimal and fraction conversion, understanding arithmetic operations is vital for accurate calculations.
For instance, blending fractions and decimals requires clear knowledge of both operations. Here's how you can manage it:

• Use addition and subtraction for comparing and computing with decimals and fractions.
• Convert all numbers into decimals or fractions when combining the two forms.
• Align decimal points carefully for accurate results.

Moreover, practice breaking down expressions step-by-step. As shown in the exercise, converting \(-\frac{7}{4}\) to the decimal \(-1.75\) simplifies the arithmetic operation of adding it to 3.3. By subtracting \(1.75\) from \(3.3\), the addition becomes easier to handle.
Remaining organized in the steps ensures precision, and understanding each operation's mechanics leads to a successful outcome.