Problem 10 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(7 \frac{9}{10}-6 \frac{3}{5}\)

Step-by-Step Solution

Answer

The answer is \(1 \frac{3}{10}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions for easier calculation. For the mixed number \(7 \frac{9}{10}\), convert as follows:\[7 \frac{9}{10} = \frac{7 \times 10 + 9}{10} = \frac{70 + 9}{10} = \frac{79}{10}\]Similarly, for \(6 \frac{3}{5}\):\[6 \frac{3}{5} = \frac{6 \times 5 + 3}{5} = \frac{30 + 3}{5} = \frac{33}{5}\]

Step 2: Find a Common Denominator

Next, find a common denominator for the fractions \(\frac{79}{10}\) and \(\frac{33}{5}\). The least common multiple of 10 and 5 is 10. Fraction \(\frac{33}{5}\) needs to be converted:\[\frac{33}{5} = \frac{33 \times 2}{5 \times 2} = \frac{66}{10}\]

Step 3: Perform the Subtraction

Subtract the fractions with the same denominator:\[\frac{79}{10} - \frac{66}{10} = \frac{79 - 66}{10} = \frac{13}{10}\]

Step 4: Convert the Result to a Mixed Number

Finally, convert the improper fraction \(\frac{13}{10}\) back to a mixed number:\[\frac{13}{10} = 1 \frac{3}{10}\], since 13 divided by 10 is 1 with a remainder of 3, so the fraction part is \(\frac{3}{10}\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator is larger than or equal to the denominator. In other words, the top number is bigger than or equal to the bottom number. This can make fractions look a bit top-heavy.
Converting mixed numbers to improper fractions is a standard step when performing operations like addition or subtraction. For example, take the mixed number 7 \(\frac{9}{10}\). Here, you convert it to an improper fraction:

• Multiply the whole number (7) by the denominator (10): 7 x 10 = 70.
• Add the result to the numerator (9): 70 + 9 = 79.
• Now, write it over the original denominator to get \(\frac{79}{10}\).

Improper fractions may initially look more complex, but they simplify the math process significantly by removing the added complexities of mixed numbers.

Common Denominator

When subtracting fractions, it's important for them to share the same bottom number, known as the denominator. This makes it easier to line up numbers for subtraction, much like ensuring everyone in a game uses the same rules.
To achieve this, we often convert denominators to show a common denominator. Suppose we have the fractions \(\frac{79}{10}\) and \(\frac{33}{5}\). The least common multiple of 10 and 5 is 10, meaning both denominators can easily become 10.

• The fraction \(\frac{33}{5}\) does not initially have a denominator of 10, so we multiply both the numerator and the denominator by 2:
• \(\frac{33}{5} = \frac{33 \times 2}{5 \times 2} = \frac{66}{10}\).

Now, both fractions \(\frac{79}{10}\) and \(\frac{66}{10}\) share a common denominator, allowing for easy subtraction.

Subtraction of Fractions

Subtracting fractions can be simple once a common denominator is ensured. You can think of this like having the same type of measurement units before comparing values.
To subtract, keep the denominators the same, and just subtract the numerators.

• For example, with common denominators, subtract \(\frac{79}{10} - \frac{66}{10}\).
• Simply subtract the numerators: 79 - 66 = 13, and keep the denominator as 10.

This subtraction gives you \(\frac{13}{10}\), which is still an improper fraction. The process of subtraction is straightforward once the fractions are ready with a common denominator.

Conversion to Mixed Numbers

After getting your result from the subtraction, you might need to switch it back into a more familiar format, like a mixed number. Especially when final answers are usually expected in this form.
Take \(\frac{13}{10}\) from the subtraction. To convert it back to a mixed number:

• Divide the numerator by the denominator, 13 divided by 10 equals 1 with a remainder of 3.
• The whole number of the division becomes the whole part of your mixed number.
• The remainder becomes your new numerator: \(\frac{3}{10}\), staying over the same denominator.

So, \(\frac{13}{10}\) is expressed as 1 \(\frac{3}{10}\), making it much easier to understand and relate to quantities, just like piecing together whole pieces and fractions of a pie.