Problem 66 - Prealgebra

Question

Simplify the given expression. \(-\frac{7}{5}-3.34\)

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{237}{50}\).

Step 1: Convert Decimal to Fraction

To simplify the expression, the first step is to convert the decimal into a fraction. The decimal \(3.34\) can be written as \(\frac{334}{100}\). Now you have an equation with fractions: \(-\frac{7}{5} - \frac{334}{100}\).

Step 2: Find a Common Denominator

The next step is to find a common denominator for the fractions. The denominators are \(5\) and \(100\). The least common multiple of \(5\) and \(100\) is \(100\), so that will be our common denominator.

Step 3: Convert Fractions to Common Denominator

Convert \(-\frac{7}{5}\) so it has a denominator of 100. Multiply both the numerator and denominator by 20 to get \(-\frac{140}{100}\). Now, the expression is \(-\frac{140}{100} - \frac{334}{100}\).

Step 4: Combine the Fractions

Now that the fractions have the same denominator, you can combine them by subtracting the numerators: \(-140 - 334 = -474\). This gives \(-\frac{474}{100}\).

Step 5: Simplify the Fraction

Simplify \(-\frac{474}{100}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This results in \(-\frac{237}{50}\).

Step 6: Check Your Answer

Ensure your answer is correct by comparing it to the decimal form. \(-\frac{237}{50} \approx -4.74\) which matches the calculation, confirming that \(-\frac{7}{5} - 3.34 = -\frac{237}{50}\).

Key Concepts

Fraction Conversion

Fraction conversion is an essential skill when dealing with expressions that mix decimals and fractions. Sometimes, you'll encounter numbers in decimal form that need to be expressed as fractions. For example, to convert the decimal 3.34 into a fraction, observe the number's place value. Here, 3.34 means 3 and 34 hundredths. You can represent this as \(\frac{334}{100}\). Use this straightforward approach:

• Count the digits after the decimal point to determine the denominator. In 3.34, there are two digits (34).
• Write the number without the decimal as the numerator (334).
• The denominator is '1' followed by as many zeros as there are digits in the decimal portion (100 in this case).
• Simplify the fraction if possible.

Converting decimals into fractions allows you to seamlessly transition between the two forms, making it easier to perform operations like addition, subtraction, or comparison.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions with different denominators. It simplifies the operation and allows you to work with fractions more effectively. Consider two fractions: \(-\frac{7}{5}\) and \(\frac{334}{100}\).

• The denominators are 5 and 100, respectively.
• You need a common multiple of these numbers to combine the fractions. The easiest way to find one is to determine the least common multiple (LCM).
• In this case, the LCM of 5 and 100 is 100 because 100 is already a multiple of 5.

With a common denominator, each fraction can be converted to have a matching base, simplifying the subtraction process. This process is essential in fraction arithmetic, ensuring precision and simplicity.

Decimal to Fraction

Converting a decimal to a fraction involves understanding the relationship between decimals and fractions. Each position in a decimal number represents a fractional part of a whole. When transforming decimals to fractions:

• Identify the decimal part and how many places it extends. For instance, in 3.34, the fractional part (0.34) stretches two places to the right of the decimal point.
• Translate this into a fraction: 0.34 can be written as \(\frac{34}{100}\), since it reflects 34 hundredths.
• The whole number 3 becomes the numerator when combined with 34 to establish \(\frac{334}{100}\).

This conversion is important for handling diverse mathematical problems, especially when expressions mix different numeric forms. It offers a structured method to enable clean operation execution.

Subtracting Fractions

Subtracting fractions involves a few straightforward steps once both have a common denominator. For the problem \(-\frac{7}{5} - 3.34\), follow these steps:

• Convert the decimal to a fraction, \(3.34 = \frac{334}{100}\), and find a common denominator. We've already established 100 as our common denominator.
• Convert \(-\frac{7}{5}\) to \(-\frac{140}{100}\).
• Subtract the two fractions: subtract the numerators (\(-140 - 334 = -474\)).
• The result \(-\frac{474}{100}\) can be simplified further if necessary. In this exercise, we simplify by dividing by 2 to get \(-\frac{237}{50}\).

This simplification results in cleaner, more concise mathematical expressions, while maintaining accuracy in representation and operations.