Problem 63 - Prealgebra

Question

Explain why each number is a rational number. $$1 \frac{1}{2}$$

Step-by-Step Solution

Answer

The number \(1 \frac{1}{2}\) is rational because it can be expressed as the fraction \(\frac{3}{2}\).

Step 1: Understand Rational Numbers

A rational number is a number that can be expressed as the quotient of two integers, where the numerator is an integer and the denominator is a non-zero integer.

Step 2: Convert Mixed Number to Improper Fraction

Convert the mixed number \(1 \frac{1}{2}\) into an improper fraction. A mixed number such as \(1 \frac{1}{2}\) can be converted by multiplying the whole number (1) by the denominator (2) and adding the numerator (1). So, \(1 \times 2 + 1 = 3\). This gives us the improper fraction \(\frac{3}{2}\).

Step 3: Verify as a Rational Number

Since \(\frac{3}{2}\) has an integer numerator and a non-zero integer denominator, it is considered a rational number by definition.

Key Concepts

Mixed Numbers

A mixed number is a combination of two parts: a whole number and a fraction. Think of it as having a whole pizza and a few extra slices. For instance, when you have 1 and a half pizzas, it means you have one full pizza and half of another, which can be written as \(1 \frac{1}{2}\).

Understanding mixed numbers is important because they show us a straightforward way to visualize amounts that are not whole. It allows you to easily see how much more than an integer, like 1 or 2, you have. Mixed numbers are often used in everyday life. For example, if a recipe calls for \(2 \frac{1}{4}\) cups of flour, it means you need two full cups plus a quarter cup more.

Reading mixed numbers is quite simple. The whole number is the amount you have, while the fraction shows how much extra you have. Learning how to work with mixed numbers can make math operations a lot more intuitive.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is larger than the denominator (bottom number), like \(\frac{7}{4}\). It might seem a little strange at first because we usually think of fractions as less than a whole number. However, improper fractions are vital when dealing with mathematical operations like addition or subtraction because they provide a simpler form.

Think of the improper fraction \(\frac{7}{4}\) as dividing 7 pizzas equally into 4 groups. Instead of saying that you have 'one pizza and three quarters of another,' an improper fraction gives you a clear amount: 7 quarters altogether.

Improper fractions can also be easily converted back into mixed numbers for everyday use. This helps because, in most situations, it's simpler to visualize and understand quantities in mixed numbers.

Converting Fractions

Converting between mixed numbers and improper fractions is a helpful skill. It allows you to switch between an easy-to-read form (mixed number) and a mathematically efficient form (improper fraction).

To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number by the fraction's denominator.
• Add the numerator to the result.
• Place that sum over the original denominator.

For example, take \(1 \frac{1}{2}\):

- Multiply 1 (whole number) by 2 (denominator): 1 × 2 = 2.
- Add the numerator (1): 2 + 1 = 3.
- Place over the denominator: \(\frac{3}{2}\).

Conversely, to convert an improper fraction to a mixed number, divide the numerator by the denominator to get the whole number. The remainder becomes the numerator of the fractional part, over the original denominator.