Problem 29 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(7 \frac{4}{7}-2 \frac{5}{7}\)

Step-by-Step Solution

Answer

The result is \(4 \frac{6}{7}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The first step is to convert the mixed numbers to improper fractions for easier subtraction. For the number \(7 \frac{4}{7}\): multiply 7 (whole number) by 7 (denominator) and add 4 (numerator), giving us \( \frac{7 \times 7 + 4}{7} = \frac{53}{7} \). Similarly, for \(2 \frac{5}{7}\): multiply 2 (whole number) by 7 (denominator) and add 5 (numerator), yielding \( \frac{2 \times 7 + 5}{7} = \frac{19}{7} \).

Step 2: Subtract the Improper Fractions

Now that both numbers are improper fractions with the same denominator, you can subtract them by subtracting their numerators. Thus, \( \frac{53}{7} - \frac{19}{7} = \frac{53 - 19}{7} = \frac{34}{7} \).

Step 3: Simplify the Improper Fraction

Check if the result can be simplified. Since 34 and 7 have no common factors other than 1, \( \frac{34}{7} \) is already in its simplest form. However, since it's an improper fraction, you can convert it back to a mixed number. Divide 34 by 7, which gives 4 with a remainder of 6. So, \( \frac{34}{7} \) can be written as \(4 \frac{6}{7}\).

Key Concepts

Mixed Numbers

A mixed number is a useful way to express numbers that lie between whole numbers. It combines a whole number and a fraction. For example, in the mixed number \(7 \frac{4}{7}\), "7" is the whole number and "\(\frac{4}{7}\)" is the fraction. This representation helps students visualize numbers that aren't whole. When dealing with operations like addition or subtraction involving mixed numbers, it's often helpful to first convert them into improper fractions. This makes mathematical operations easier to handle and reduces the potential for error.

Improper Fractions

Improper fractions have a numerator larger than or equal to the denominator, such as \(\frac{53}{7}\). They are the result of converting mixed numbers, where you multiply the whole number by the fraction's denominator and then add the original numerator. For example:

• Multiply the whole number by the denominator: \(7 \times 7\)
• Add the numerator: \(49 + 4\)
• Write the result over the original denominator: \(\frac{53}{7}\)

Converting mixed numbers to improper fractions standardizes the format, making subtraction straightforward, as seen in subtraction operations like \(\frac{53}{7} - \frac{19}{7}\). By subtracting numerators directly, the result can easily be expressed as an improper fraction.

Simplification of Fractions

Simplification involves breaking down a fraction to its most reduced form. After performing arithmetic operations, like the subtraction example \(\frac{53}{7} - \frac{19}{7} = \frac{34}{7}\), it's crucial to check if the fraction can be simplified further. This is done by finding common factors of the numerator and denominator. In our example, 34 and 7 only share the factor 1, meaning \(\frac{34}{7}\) is simplified.
However, you might choose to convert it back to a mixed number format for clarity. To do this:

• Divide the numerator by the denominator: 34 ÷ 7 = 4
• Remainder becomes the new numerator: remainder 6
• Mixed number: \(4 \frac{6}{7}\)

Converting improper fractions back to mixed numbers can make them easier to understand within certain contexts, especially when ensuring the results are presented in the simplest way possible.