Problem 41 - Prealgebra

Question

Simplify each expression. $$\frac{3}{5}\left(2 \frac{1}{5}-1 \frac{1}{10}\right)$$

Step-by-Step Solution

Answer

\(\frac{33}{50}\)

Step 1: Convert Mixed Numbers to Improper Fractions

The mixed number \(2 \frac{1}{5}\) can be converted to an improper fraction by multiplying the whole number 2 by the denominator 5 and then adding the numerator 1. This gives \(2 \times 5 + 1 = 11\), so \(2 \frac{1}{5} = \frac{11}{5}\). Similarly, \(1 \frac{1}{10}\) is converted to \(\frac{11}{10}\) using \(1 \times 10 + 1 = 11\).

Step 2: Grt a Common Denominator

The fractions \(\frac{11}{5}\) and \(\frac{11}{10}\) need a common denominator to be subtracted. The least common denominator of 5 and 10 is 10. Convert \(\frac{11}{5}\) to \(\frac{22}{10}\) by multiplying the numerator and the denominator by 2.

Step 3: Perform the Subtraction

Subtract \(\frac{11}{10}\) from \(\frac{22}{10}\): \(\frac{22}{10} - \frac{11}{10} = \frac{11}{10}\).

Step 4: Multiply by \(\frac{3}{5}\)

Now multiply \(\frac{3}{5}\) by \(\frac{11}{10}\). This is done by multiplying the numerators and the denominators: \(\frac{3}{5} \times \frac{11}{10} = \frac{33}{50}\).

Step 5: Check for Further Simplification

The fraction \(\frac{33}{50}\) cannot be simplified further as 33 and 50 have no common factors other than 1.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top part) is larger than or equal to the denominator (the bottom part). This means that the fraction is in a state where it is actually bigger than a whole number. To work with these, it is often useful to first convert any mixed numbers (which have both whole numbers and fractions) into improper fractions. Here's how you can do it:
- Multiply the whole number part by the denominator of the fractional part.- Add this result to the numerator of the fractional part to get the new numerator.- Keep the original denominator the same.
For example, converting the mixed number \(2\frac{1}{5}\) to an improper fraction involves calculating:

• \(2\times 5 = 10\)
• Adding the 1: \(10 + 1 = 11\)
• The improper fraction becomes \(\frac{11}{5}\)

This makes calculations more straightforward, especially when you need to add, subtract, or multiply fractions.

Common Denominator

In order to perform arithmetic operations like addition or subtraction with fractions, they need to have the same denominator. This is known as the "common denominator."
Finding a common denominator is like finding a common language for fractions so we can easily combine or compare them. Here's how you can find the least common denominator (LCD):

• Identify the denominators of the fractions in question.
• Find the least common multiple (LCM) of these denominators.This LCM will be your common denominator.

Let's look at our fractions \(\frac{11}{5}\) and \(\frac{11}{10}\):

• Denominators are 5 and 10.
• The LCM of 5 and 10 is 10.
• We convert \(\frac{11}{5}\) to \(\frac{22}{10}\).

Once the denominators match, addition or subtraction can proceed smoothly.

Mixed Numbers

A mixed number is a whole number and a fraction combined into one "mixed" number. Mixed numbers are useful because they can easily show quantities greater than one. However, when doing arithmetic with them, it can be handy to convert them to improper fractions.
Once arithmetic is done, you might want to convert the result back to a mixed number for ease of understanding. Here's a quick guide to do so:

• Divide the numerator by the denominator.
• The integer part of the result is the whole number.
• The remainder becomes the numerator of the fractional part.
• The denominator stays the same.

For example, if you have the improper fraction \(\frac{11}{5}\) and you want to convert it back to a mixed number:

• 11 divided by 5 is 2 with a remainder of 1.
• This gives us the mixed number \(2\frac{1}{5}\).

Converting back to mixed numbers can often make it simpler to interpret the size or value of the fraction in practical terms.