Problem 25 - Introductory Algebra for College Students

Question

Express each rational number as a decimal. $$\frac{7}{8}$$

Step-by-Step Solution

Answer

The decimal representation of \(\frac{7}{8}\) is 0.875.

Step 1: Perform the Division

We divide 7 by 8 using long division. 7 divided by 8 equals 0.875.

Step 2: Conversion to Decimal

The result of the division is the decimal representation of our fraction. So our original fraction, \( \frac{7}{8} \) , as a decimal is now 0.875.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal is a common mathematical operation and is incredibly useful in various applications. A fraction like \( \frac{7}{8} \) can be understood as "7 divided by 8." Performing this division will give us a decimal number. This helps in comparing the fraction with other numbers which might already be in decimal form.

• The numerator (the top number in a fraction) is the number you will divide.
• The denominator (the bottom number) is the number you divide by.

For our fraction \( \frac{7}{8} \), we take 7 and divide it by 8. This division will result in a decimal number, making it easier to use in various calculations, such as addition or subtraction with other decimals.
The conversion process allows us to express ratios in a different form, simplifying mathematical operations! Making sure this conversion is understood is crucial for doing more complex math problems.

Long Division

Long division is a method that can solve division problems step-by-step and is especially useful when dealing with numbers that don't divide evenly. This method helps in converting fractions to decimals when you don't have a calculator at hand. In the case of the fraction \( \frac{7}{8} \), we use long division to find its decimal form.
First, see how many times 8 can go into 7. Since 8 can't go into 7, we use 70 instead by adding a decimal point and a zero, noting it will be part of a decimal result.

• Determine how many times 8 fits into 70 — this is 8 times (\(8 \times 8 = 64\)). Put this 8 as the first decimal digit.
• Subtract 64 from 70 to get 6, then bring down another 0 to make 60.
• See how many times 8 fits into 60, which is 7 times (\(8 \times 7 = 56\)). Add 7 as the next decimal digit.
• Subtract 56 from 60 to get 4, bring down another 0 to make 40.
• Finally, check how many times 8 goes into 40. It goes 5 times (\(8 \times 5 = 40\)), which is the final decimal digit.

As a result, using long division, \( \frac{7}{8} = 0.875 \). This shows the utility of long division in providing an accurate and complete answer, even when complex fractions are involved.

Decimal Representation

Decimal representation serves as a straightforward way to express fractions. Once a fraction is converted into a decimal, it becomes easier to visualize and use for computations. The decimal representation of \( \frac{7}{8} \) is 0.875, and here's what that means:

• Decimals express parts of a whole, using the base-10 numerical system that aligns well with our arithmetic system.
• They make it simpler to perform arithmetic operations like addition, subtraction, multiplication, and division compared to fractions.
• A terminating decimal, such as 0.875, clearly shows the end number when a fraction is divided completely.

Decimals illustrate numbers in a more tangible and accessible way. For example, if you need to compare \( \frac{7}{8} \) to another number, converting it to a decimal ensures you're using the most directly comparable form. Furthermore, decimal representation helps in various fields such as measurement, finance, and everyday counting, indicating its versatility.