Problem 50 - Prealgebra

Question

Order each group of numbers from least to greatest. $$\frac{7}{8}, 0.8, \frac{7}{9}$$

Step-by-Step Solution

Answer

\(\frac{7}{9}, 0.8, \frac{7}{8}\)

Step 1: Convert Fractions to Decimals

First, convert each fraction to its decimal form to make comparison easier.\- For \(\frac{7}{8}\), divide 7 by 8: \[\frac{7}{8} = 0.875\]- For \(\frac{7}{9}\), divide 7 by 9:\[\frac{7}{9} \approx 0.7778\]

Step 2: List All Decimal Numbers

Gather the decimal forms of all numbers to be compared:- \(0.875\) (from \(\frac{7}{8}\))- \(0.8\)- \(0.7778\) (from \(\frac{7}{9}\))

Step 3: Order from Least to Greatest

Now, compare the decimal numbers and list them in order from the smallest to the largest:- Smallest: \(0.7778\) (equivalent to \(\frac{7}{9}\))- Next: \(0.8\)- Largest: \(0.875\) (equivalent to \(\frac{7}{8}\))Therefore, the order from least to greatest is: \[\frac{7}{9}, 0.8, \frac{7}{8}\]

Key Concepts

Comparing Fractions

When we compare fractions, it helps to simplify the process by finding a common way to express them, often using decimals, so that they become easier to compare. Fractions represent parts of a whole, but they can sometimes be tricky to compare when they have different denominators.
When comparing two fractions, such as \( \frac{7}{8} \) and \( \frac{7}{9} \), ask, which is larger? Since they have the same numerator, the fraction with the smaller denominator is larger because it divides the whole into fewer, therefore larger, pieces.
This can lead to quick conclusions in simple cases, but converting fractions to decimals allows for direct comparison without estimation or extra calculations. It's like speaking the same language for easy communication. For instance:

• \( \frac{7}{8} \) becomes \( 0.875 \)
• \( \frac{7}{9} \) becomes approximately \( 0.7778 \)

Once we convert fractions to decimals, comparing becomes straightforward: just look at who is bigger or smaller, like regular numbers!

Converting Fractions to Decimals

The process of converting fractions to decimals is about finding a decimal equivalent to a fraction, which helps in comparing and performing calculations easily. This involves simply dividing the numerator (top number) by the denominator (bottom number).
On your calculator or through long division, the fraction \( \frac{7}{8} \) becomes a decimal by dividing 7 by 8, resulting in \( 0.875 \). Similarly, for \( \frac{7}{9} \), divide 7 by 9 to get approximately \( 0.7778 \).
This conversion gives us a clearer and often quicker way to manage numbers that were in fractional form. It avoids cumbersome fraction arithmetic and aligns the numbers on a similar field, making further operations like addition or comparison a breeze.
Sometimes decimals can be repeating or rounded, like our \( \frac{7}{9} \), which shows as \( 0.777 ext{...} \). Understanding this can help with accuracy in calculations and recognizing patterns in numbers.

Decimal Representation

Decimal representation is a crucial concept when working with numbers, as it provides a straightforward way of expressing and comparing values. Decimals are essentially fractions written in a base-10 system, which is familiar and manageable.
This representation allows us to easily assess the size of numbers without additional calculations. For example, with \( 0.8 \), \( 0.875 \), and \( 0.7778 \), we instantly see the order: \( 0.7778, 0.8, 0.875 \).
Where things can get intriguing is with repeating decimals, such as \( 0.7778 \), which is derived by rounding. Recognizing when a decimal extends can help in maintaining precision, allowing us to verify precision or round accordingly based on the context.
Using decimals simplifies operations that might be more complex in their fractional forms, such as addition, subtraction, and especially comparison. By placing everything in decimal form, problems and solutions become clearer and more straightforward to solve.