Problem 45 - Prealgebra

Question

In Exercises 45-52, simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{7}{4}-7.4\)

Step-by-Step Solution

Answer

The simplified expression is -5.65.

Step 1: Convert Fraction to Decimal

Start by converting the fraction \( \frac{7}{4} \) into a decimal. To do this, divide the numerator 7 by the denominator 4. When you divide 7 by 4, you get 1.75. Thus, \( \frac{7}{4} = 1.75 \).

Step 2: Perform the Subtraction

Now, subtract the decimal obtained from the fraction from the decimal number given in the expression. We have to calculate: \( 1.75 - 7.4 \).

Step 3: Solve the Expression

Carry out the subtraction: \( 1.75 - 7.4 \). To make the subtraction easier, align the decimal points and perform the operation as follows: \\[ \ 1.75 \ \-7.40 \ \------ \ -5.65 \ \] This gives -5.65.

Key Concepts

Fractions

Fractions are a way to express numbers that are not whole. They are composed of two numbers: the numerator and the denominator. The numerator is the top number and represents how many parts of a whole are being considered. The denominator is the bottom number, showing into how many equal parts the whole is divided.
For example, the fraction \( \frac{7}{4} \) means 7 parts of a whole that is divided into 4 equal parts. Fractions can also be converted into decimals to make certain calculations easier. This involves dividing the numerator by the denominator.

Decimal Conversion

Decimal conversion is the process of transforming a fraction into a decimal number. This is particularly useful in calculations such as addition or subtraction, where decimals may often be easier to work with.
To convert a fraction like \( \frac{7}{4} \) to a decimal, perform a division of the numerator by the denominator. When you divide 7 by 4, you get 1.75. This means that the fraction \( \frac{7}{4} \) is equal to the decimal 1.75.
This step allows you to perform arithmetic operations more smoothly by aligning the decimal points.

Subtraction of Decimals

Subtraction of decimals can be done easily if you align the numbers by their decimal points. It's a simple process of ensuring each digit is in its correct place value during the calculation.
To subtract 1.75 from 7.4, align the decimals:

• Write 1.75 under 7.40, ensuring the decimals line up vertically for accuracy.
• Subtract each column, starting from the rightmost digit and moving to the left. Borrow from the next column as needed.
• The result of this operation will be -5.65.

Understanding the alignment and place value of each digit is crucial to arriving at the correct answer.