Problem 19 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The sum is \(11 \frac{4}{5}\).

Step 1: Rewrite Mixed Numbers

Convert the mixed numbers into improper fractions. To do this for each mixed number, multiply the whole number part by the denominator of the fractional part and then add the numerator of the fractional part. For \(7 \frac{2}{5}\): \(7 \times 5 + 2 = 35 + 2 = 37\), so \(7 \frac{2}{5} = \frac{37}{5}\). Similarly, for \(4 \frac{2}{5}\): \(4 \times 5 + 2 = 20 + 2 = 22\), so \(4 \frac{2}{5} = \frac{22}{5}\).

Step 2: Add the Improper Fractions

Now that both numbers are in the same format with a common denominator, add the two improper fractions. \(\frac{37}{5} + \frac{22}{5} = \frac{37 + 22}{5} = \frac{59}{5}\).

Step 3: Simplify the Result

Convert the resulting improper fraction back into a mixed number. Divide the numerator by the denominator: \(59 \div 5 = 11\) with a remainder of \(4\). Therefore, \(\frac{59}{5} = 11 \frac{4}{5}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (top number) is greater than or equal to the denominator (bottom number). This occurs when a mixed number is converted entirely into fraction form. For example, when we transform a mixed number like \(7 \frac{2}{5}\) into an improper fraction, we apply the formula: multiply the whole number part by the denominator and add the numerator. Thus, for \(7 \frac{2}{5}\), we calculate:

• Multiply 7 (whole number) by 5 (denominator).
• Add 2 (numerator) to the result.

This computation results in an improper fraction of \(\frac{37}{5}\). Improper fractions are useful because they allow us to easily perform operations such as addition or subtraction with fractions having like denominators.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical expression into its simplest form. When we add fractions, especially improper ones, the initial result might not seem straightforward. For instance, in the example of adding \(\frac{37}{5}\) and \(\frac{22}{5}\), we get:

• \(\frac{37 + 22}{5} = \frac{59}{5}\).

To simplify this result, observe the fraction's numerator and denominator. If possible, divide them by a common factor to further reduce the fraction. If the fraction is improper, as in this case, the simplest form could involve converting it back to a mixed number.

Mixed Number Conversion

Converting an improper fraction to a mixed number involves dividing the numerator by the denominator. This process helps in expressing the fraction into a combined form of whole numbers and proper fractions. Take, for instance, \(\frac{59}{5}\):

• Divide 59 by 5, which equals 11 with a remainder of 4.
• This answer signifies that 59 is 5 times 11 with 4 remaining.

Thus, \(\frac{59}{5}\) can be converted into the mixed number \(11 \frac{4}{5}\). Mixed numbers make it easy to understand the representation of fractions within the context of whole quantities with additional parts.