Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<,>,\) or \(=\) in the shaded area to make a true statement. $$\frac{30}{40}-\frac{3}{4} \quad \square\quad\frac{14}{15} \cdot \frac{15}{14}$$

Step-by-Step Solution

Answer

The correct symbol to insert is \( < \)

Step 1: Simplify the First Expression

The given expression is subtracting two fractions, \(30/40\) - \(3/4\). We can simplify both fractions. \(30/40\) simplifies to \(3/4\), therefore the whole expression simplifies to \(3/4 - 3/4\), which equals 0.

Step 2: Simplify the Second Expression

The other expression is multiplying two fractions, \(14/15\) * \(15/14\). Numbers in the numerator of the first fraction match the numbers in the denominator of the second fraction. They cancel each other out when multiplication is performed giving a result of 1.

Step 3: Insert the Correct Comparison Symbol

Now, comparing the values, we have 0 and 1. 0 is less than 1, therefore the correct symbol to insert is \(<\)

Key Concepts

Simplifying Fractions

Simplifying fractions is like tidying up your room. You want to make things as neat as possible. When we simplify a fraction, we make it as simple as it can get without changing its value.

For example, look at the fraction \( \frac{30}{40} \). Both the numerator and the denominator have a common factor, which is 10. So, when we divide both the top and bottom by 10, we simplify \( \frac{30}{40} \) to \( \frac{3}{4} \).

This process is crucial because it makes comparing and working with fractions easier. Remember, always look for the greatest common factor between the numerator and the denominator to simplify efficiently.

• Find the greatest common factor (GCF)
• Divide both the numerator and the denominator by the GCF
• Rewrite the fraction in simplest form

Keep in mind that simplifying does not alter the fraction's value, so \( \frac{30}{40} \) and \( \frac{3}{4} \) represent the same number, they are just more straightforward to handle in the latter form.

Fraction Subtraction

Subtracting fractions can be trickier than adding them. Especially when the denominators are different. Fortunately, when the denominators are the same, like with \( \frac{3}{4} - \frac{3}{4} \), the process becomes simple.

First, always ensure the fractions have the same denominator. If not, you'll need to find a common denominator before you can subtract. Once they are the same, you subtract the numerators only and keep the denominator the same.

In our exercise, the fractions \( \frac{3}{4} - \frac{3}{4} \) already have the same denominator. So you simply subtract the top numbers to get:\[\frac{3-3}{4} = \frac{0}{4} = 0\]

• Ensure denominators are the same
• Subtract the numerators
• Simplify the result if needed

Subtraction may require simplifying the solution afterwards if it's not already in the simplest form.

Fraction Multiplication

Multiplying fractions is a bit like playing a matching game. You simply multiply the numbers across the top, and the numbers across the bottom. However, you can also simplify as you multiply, which can save you some work later.

Consider multiplying \( \frac{14}{15} \cdot \frac{15}{14} \). Notice that 14 appears once in the numerator and once in the denominator, as does 15. We can "cancel" these like terms beforehand. This makes the multiplication quite straightforward:

After canceling, you are left with \( \frac{1}{1} \), which equals 1.

Here’s a step-by-step approach to remember:

• Multiply the numerators together
• Multiply the denominators together
• Cancel terms that appear in both the numerator and the denominator before multiplying
• Simplify the result

Canceling common factors early helps eliminate bigger numbers and leaves the fraction already simplified.