Problem 63 - Prealgebra

Question

Choose two rational numbers whose product is a number between 0 and 1.

Step-by-Step Solution

Answer

The two numbers are \( \frac{1}{2} \) and \( \frac{2}{3} \) with a product of \( \frac{1}{3} \).

Step 1: Understanding Rational Numbers

A rational number is any number that can be expressed as the fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). For this exercise, we need two rational numbers whose product is between 0 and 1.

Step 2: Example of Rational Numbers

Let's consider two simple rational numbers: \( \frac{1}{2} \) and \( \frac{2}{3} \). Both of these numbers are rational because they can be expressed as fractions.

Step 3: Calculating the Product

To find their product, multiply the numerators and denominators separately: \( \frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} \).

Step 4: Simplifying the Product

Simplify \( \frac{2}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{2}{6} = \frac{1}{3} \).

Step 5: Checking the Condition

The simplified product is \( \frac{1}{3} \), which is indeed between 0 and 1, satisfying the condition.

Key Concepts

fraction multiplication

Fraction multiplication is a fundamental operation when dealing with rational numbers. It involves multiplying the numerators together and the denominators together. If you have two fractions, say \( \frac{a}{b} \) and \( \frac{c}{d} \), the product of these fractions is calculated by multiplying the numerators and the denominators, respectively. The product is given by:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)
• Resulting Fraction: \( \frac{a \times c}{b \times d} \)

For example, let's multiply \( \frac{1}{2} \) and \( \frac{2}{3} \). Their product is:\[\frac{1 \times 2}{2 \times 3} = \frac{2}{6}\]This process creates a new fraction, which may need further simplification to make it easier to understand.

simplifying fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common divisors other than 1. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator.Let's look at \( \frac{2}{6} \) from our previous calculation:

• Find the GCD: The GCD of 2 and 6 is 2.
• Divide the numerator and denominator by the GCD: \[ \frac{2 \div 2}{6 \div 2} = \frac{1}{3} \]

Simplified fractions are often easier to interpret and work with, especially when you need to compare the size of the fractions or convert them to decimals. Once you simplify a fraction, it remains equivalent to the original fraction, retaining the same value.

product of fractions

The product of fractions is the result obtained after multiplying two or more fractions together. It often results in a value that needs checking or simplifying. In the context of rational numbers, we frequently seek products that meet specific criteria, such as being between specific values.For the exercise, the goal was to find two rational fractions whose product lies between 0 and 1. The reasoning is that since both fractions \( \frac{1}{2} \) and \( \frac{2}{3} \) are less than 1, their product will also be less than 1. This is a basic property of any two positive fractions less than one:

• The multiplication keeps them less than 1.
• It's a method to ensure the product stays within desired limits.

The application of these concepts helps in comparing, analyzing, and making predictions based on the properties of fractions, which is invaluable in solving problems that involve rational numbers.