Problem 78 - Prealgebra

Question

Find each of the following sums and differences. (Add or subtract.) $$5 \frac{35}{100}+2 \frac{3}{10}$$

Step-by-Step Solution

Answer

The sum is \(7 \frac{13}{20}\).

Step 1: Convert the Mixed Numbers to Improper Fractions

To solve the sum, first convert each mixed number into an improper fraction. For \(5 \frac{35}{100}\), multiply the whole number by the denominator and add the numerator: \(5 \times 100 + 35 = 535\). So, \(5 \frac{35}{100} = \frac{535}{100}\). For \(2 \frac{3}{10}\), multiply \(2\) by \(10\) and add \(3\): \(2 \times 10 + 3 = 23\). Therefore, \(2 \frac{3}{10} = \frac{23}{10}\).

Step 2: Find a Common Denominator

Find a common denominator for the fractions \(\frac{535}{100}\) and \(\frac{23}{10}\). The least common denominator (LCD) of \(100\) and \(10\) is \(100\). Convert \(\frac{23}{10}\) to \(\frac{230}{100}\) by multiplying both the numerator and the denominator by 10.

Step 3: Add the Fractions

Now, add the fractions \(\frac{535}{100}\) and \(\frac{230}{100}\). Since they have the same denominator, simply add the numerators: \(535 + 230 = 765\). The resulting fraction is \(\frac{765}{100}\).

Step 4: Simplify the Resulting Fraction

To simplify \(\frac{765}{100}\), divide the numerator and denominator by their greatest common divisor (GCD), which is 5: \(\frac{765 \div 5}{100 \div 5} = \frac{153}{20}\).

Step 5: Convert Back to a Mixed Number

To express \(\frac{153}{20}\) as a mixed number, divide 153 by 20. The quotient is 7 with a remainder of 13, so \(\frac{153}{20} = 7 \frac{13}{20}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are composed of two parts: a whole number and a proper fraction. Proper fractions have numerators smaller than their denominators. When you read a mixed number like 5\(\frac{35}{100}\), it indicates that there are 5 whole units, plus a fraction that represents a smaller part than a whole. These numbers are useful for expressing quantities that exceed whole units but are not quite complete multiples. For example, imagine needing precisely 5 and 35/100 meters of rope – expressing this as a mixed number makes it clearer in context compared to using just a fraction.

• Components: Whole number + Fraction
• Useful for expressing quantities exceeding whole numbers

When performing arithmetic operations like addition or subtraction with mixed numbers, they are often converted into improper fractions for easier calculations. Converting involves multiplying the whole number by the fraction's denominator and adding the numerator, which transitions the mixed number into a single fractional form.

Working with Improper Fractions

An improper fraction is where the numerator is greater than or equal to the denominator. For instance, when converting the mixed number 5\(\frac{35}{100}\), you get the improper fraction \(\frac{535}{100}\). Even though improper fractions might seem unusual, they are incredibly handy in arithmetic operations.

• Fractions can be easily added, subtracted, multiplied, and divided
• No separate parts, makes computation straightforward

In natural numbers, operations are straightforward, and when we use improper fractions, it’s a similar approach since they don't require additional steps for handling whole numbers and fractions separately. Thus, converting mixed numbers to improper fractions simplifies the process of finding sums or differences. Just combine the numerators accordingly once you have a common denominator.

Finding the Least Common Denominator

The least common denominator (LCD) is essential when adding or subtracting fractions because it is the smallest number that is a multiple of each denominator in the given fractions. For the example \(\frac{535}{100}\) and \(\frac{23}{10}\), the LCD is 100.

• Ensures fractions have the same denominators for easy addition or subtraction
• Is the smallest shared multiple between denominators

To align these two fractions, \(\frac{23}{10}\) is converted by multiplying both its numerator and denominator by 10, resulting in \(\frac{230}{100}\), making it straightforward to add with \(\frac{535}{100}\). The LCD serves as the unifier, turning fractions into comparable forms, which is vital for successful calculations without altering the values they represent.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form while preserving their original value by dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, simplifying \(\frac{765}{100}\) involves finding the GCD of 765 and 100, which is 5.

• Reduces fractions to their simplest form
• Is efficient for both readability and further calculations

After division, the result is \(\frac{153}{20}\), which is less cumbersome and easier to comprehend or use in further mathematical equations. For everyday utility, it might also be beneficial to convert this simplified fraction back into a mixed number format, such as 7\(\frac{13}{20}\), for clear communication of the value it represents. Simplified forms are not only pristine in appearance but are practically easier to interpret, share, and further manipulate in mathematical contexts.