Problem 51 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{6}{5}-7.65\)

Step-by-Step Solution

Answer

The simplified expression is \( -6.45 \).

Step 1: Convert the Fraction to Decimal

To simplify the expression \( \frac{6}{5} - 7.65 \), first convert \( \frac{6}{5} \) into a decimal. Divide 6 by 5 to get \( 1.2 \).

Step 2: Substitute the Decimal Value

Replace \( \frac{6}{5} \) in the original expression with its decimal equivalent. The expression is now \( 1.2 - 7.65 \).

Step 3: Perform the Subtraction

Subtract 7.65 from 1.2. Remember to align the decimal points: \(1.2 - 7.65 = -6.45.\)

Step 4: Check the Result

Double-check your subtraction to ensure there are no mistakes: \( 1.2 - 7.65 = -6.45.\)

Key Concepts

Fractions to Decimals

Converting fractions into decimals is a useful skill in mathematics. It helps simplify and solve problems involving fractions, especially when performing arithmetic operations.
To convert a fraction into a decimal, you need to perform a simple division of the numerator by the denominator.

• For the fraction \(\frac{6}{5}\), divide 6 (the numerator) by 5 (the denominator).
• This gives you 1.2, which is a terminating decimal. Terminating decimals are decimals that have a finite number of digits after the decimal point.

Converting fractions to decimals can often make addition, subtraction, multiplication, or division of fractions more straightforward. Keep practicing to get comfortable with this process, as it will aid you in many mathematical tasks.

Decimal Subtraction

Subtracting decimals might seem tricky at first, but once you understand the process, it's quite straightforward. The key is to align the decimal points of the numbers involved.

• Write the numbers in a column, ensuring the decimal points are in a straight line.
• If necessary, add zeros to make the decimal places equal, e.g., write 1.20 instead of 1.2 to match the two decimal places of 7.65.
• Then, subtract as usual, starting from the rightmost digit and moving left, borrowing from the next column when needed.

In the original problem, subtracting 7.65 from 1.20 results in \(-6.45\). It's helpful to double-check your work by repeating the subtraction. This ensures that you didn't make any errors, especially if borrowing was necessary.

Arithmetic Operations

Arithmetic operations are the building blocks of math, crucial for solving everyday problems.

• The basic operations include addition, subtraction, multiplication, and division. Each plays a vital role in performing calculations.
• In simplifying expressions, we often need to use several operations together, as seen in the problem of \(\frac{6}{5} - 7.65\).
• Start by converting fractions (through division, as described earlier) to make calculations more manageable.
• Next, perform the necessary arithmetic operation—in our example, subtraction is needed once the fraction is converted into a decimal.

Understanding and practicing these operations will give you a solid foundation in handling more complex mathematical concepts. Simplification, like in this problem, becomes easier when you master these basics.