Problem 80
Question
Perform the operations and, if possible, simplify. $$ 3 \frac{4}{5}-3 \frac{1}{10} $$
Step-by-Step Solution
Verified Answer
The result is \(\frac{7}{10}\).
1Step 1: Convert Mixed Numbers to Improper Fractions
Both numbers are in mixed number form. To convert them to improper fractions:The first number: \(3 \frac{4}{5} = 3 + \frac{4}{5} = \frac{3 imes 5 + 4}{5} = \frac{19}{5}\).The second number:\(3 \frac{1}{10} = 3 + \frac{1}{10} = \frac{3 \times 10 + 1}{10} = \frac{31}{10}\).
2Step 2: Find a Common Denominator
To subtract the fractions, we need a common denominator. The denominators are 5 and 10, so the least common denominator is 10.
3Step 3: Convert Fractions to Common Denominator
We convert \(\frac{19}{5}\) to a denominator of 10:\(\frac{19}{5} = \frac{19 \times 2}{5 \times 2} = \frac{38}{10}\).
4Step 4: Subtract the Fractions
Now, we subtract \(\frac{31}{10}\) from \(\frac{38}{10}\):\(\frac{38}{10} - \frac{31}{10} = \frac{38 - 31}{10} = \frac{7}{10}\).
5Step 5: Simplify the Result
The fraction \(\frac{7}{10}\) is already in its simplest form as 7 and 10 have no common factors other than 1.
Key Concepts
Improper FractionsCommon DenominatorMixed Numbers
Improper Fractions
When working with fractions, understanding improper fractions is crucial. An improper fraction is a type of fraction where the numerator, which is the top number, is greater than or equal to the denominator (the bottom number). For instance, if you have a fraction like \(\frac{19}{5}\), this is an improper fraction. This is because 19, the numerator, is larger than 5, the denominator.
Converting mixed numbers into improper fractions is often needed to perform operations like addition or subtraction. For instance, if you have \(3 \frac{4}{5}\), you can convert it to an improper fraction by multiplying the whole number part (3) by the denominator (5) and then adding the numerator (4). So it becomes \(\frac{19}{5}\). This makes it easier to manage fractions that previously included whole numbers.
Remember, it's essential to be comfortable with converting between mixed numbers and improper fractions to simplify mathematical expressions and solutions.
Converting mixed numbers into improper fractions is often needed to perform operations like addition or subtraction. For instance, if you have \(3 \frac{4}{5}\), you can convert it to an improper fraction by multiplying the whole number part (3) by the denominator (5) and then adding the numerator (4). So it becomes \(\frac{19}{5}\). This makes it easier to manage fractions that previously included whole numbers.
Remember, it's essential to be comfortable with converting between mixed numbers and improper fractions to simplify mathematical expressions and solutions.
Common Denominator
A common denominator is needed when you want to add or subtract fractions. It refers to converting fractions so that their denominators are the same. This common value allows you to directly add or subtract the numerators. For example, if you have fractions \(\frac{19}{5}\) and \(\frac{31}{10}\), you can't simply subtract them because their denominators are different. You need a common denominator to do this easily.
In our exercise, we found the least common denominator, which was 10. To convert a fraction like \(\frac{19}{5}\) to have this common denominator, you multiply the numerator and the denominator by 2, resulting in \(\frac{38}{10}\).
Achieving a common denominator aligns fractions under a shared framework, making them easy to operate on. This step is critical when combining or comparing fractions, as it ensures consistency across different fraction operations.
In our exercise, we found the least common denominator, which was 10. To convert a fraction like \(\frac{19}{5}\) to have this common denominator, you multiply the numerator and the denominator by 2, resulting in \(\frac{38}{10}\).
Achieving a common denominator aligns fractions under a shared framework, making them easy to operate on. This step is critical when combining or comparing fractions, as it ensures consistency across different fraction operations.
Mixed Numbers
Mixed numbers are a combination of whole numbers and fractions. They are useful when you want to express a quantity that's not quite whole or a little more than a whole number. An example of a mixed number is \(3 \frac{4}{5}\), where 3 is the whole number and \(\frac{4}{5}\) is the fractional part.
To perform mathematical operations on mixed numbers, it often helps to convert them into improper fractions. Doing so allows for straightforward arithmetic manipulations, as improper fractions provide a singular expression, free from a whole number part.
Understanding mixed numbers and how to convert them is useful in everyday math problems, especially those involving measurement, where quantities often do not divide evenly into whole numbers.
To perform mathematical operations on mixed numbers, it often helps to convert them into improper fractions. Doing so allows for straightforward arithmetic manipulations, as improper fractions provide a singular expression, free from a whole number part.
Understanding mixed numbers and how to convert them is useful in everyday math problems, especially those involving measurement, where quantities often do not divide evenly into whole numbers.