The Product Of Two Rational Numbers Is -16/9. If One Of The Numbers Is -4/3 Find The Other

The Product Of Two Rational Numbers Is -16/9. If One Of The Numbers Is -4/3 Find The Other

Understanding the relationship between rational numbers and their products is fundamental in algebra. When given the product of two rational numbers and one of the numbers, the problem typically involves finding the other number. This type of problem emphasizes the importance of inverse operations, specifically division, to isolate the unknown. In this article, we will explore how to approach and solve the problem: "The product of two rational numbers is -16/9. If one of the numbers is -4/3, find the other." We will delve into the concepts of rational numbers, multiplication, and division, and walk through the step-by-step solution process, complemented by illustrative examples and explanations.

Understanding Rational Numbers and Their Properties

What Are Rational Numbers?

• Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
• Examples include: 1/2, -3/4, 7, -5, 0, and -4/3.
• Rational numbers are dense on the number line, meaning between any two rational numbers, there exists another rational number.

Properties of Rational Numbers in Multiplication and Division

• Closure Property: The product or quotient of two rational numbers is always a rational number.
• Commutative Property: a × b = b × a.
• Associative Property: (a × b) × c = a × (b × c).
• Multiplicative Inverse: For any non-zero rational number a/b, its inverse is b/a such that a/b × b/a = 1.

Understanding these properties helps in manipulating and solving equations involving rational numbers.

Formulating the Problem Mathematically

Given Data

• Product of two rational numbers: \( P = -\frac{16}{9} \)
• One of the numbers: \( a = -\frac{4}{3} \)

Objective

• Find the other rational number, \( b \).

Mathematical Approach to the Solution

Using the Multiplication Formula

The key relationship is: \[ a \times b = P \] Substituting the known values: \[ -\frac{4}{3} \times b = -\frac{16}{9} \] To find \( b \), we need to isolate it: \[ b = \frac{P}{a} \]

Performing the Calculation

• Since both \( a \) and \( P \) are rational numbers, dividing \( P \) by \( a \) involves multiplying \( P \) by the reciprocal of \( a \):

\[ b = P \times \frac{1}{a} \]

• Find the reciprocal of \( a = -\frac{4}{3} \):

\[ \frac{1}{a} = -\frac{3}{4} \]

• Now multiply \( P = -\frac{16}{9} \) by \( \frac{1}{a} = -\frac{3}{4} \):

\[ b = -\frac{16}{9} \times -\frac{3}{4} \]

• Multiply numerator and denominator:

\[ b = \left(-16 \times -3\right) / \left(9 \times 4\right) \] \[ b = (48) / (36) \]

Simplifying the Result

• Simplify the fraction:

\[ b = \frac{48}{36} = \frac{4 \times 12}{3 \times 12} = \frac{4}{3} \]

• Since both numerator and denominator are divisible by 12, the simplified form is:

\[ b = \frac{4}{3} \]

Final Answer and Interpretation

Conclusion

• The other rational number is \( \frac{4}{3} \).
• This makes sense because:

\[ a \times b = -\frac{4}{3} \times \frac{4}{3} = -\frac{16}{9} \] which matches the original product.

Summary of the Solution Steps

• Identify the known quantities: \( a = -\frac{4}{3} \), \( P = -\frac{16}{9} \).
• Express the unknown \( b \) as \( b = \frac{P}{a} \).
• Replace \( P \) and \( a \) with their fractions and find the reciprocal of \( a \).
• Multiply \( P \) by the reciprocal of \( a \) to obtain \( b \).
• Simplify the resulting fraction to find the value of \( b \).

Additional Examples and Practice Problems

Example 1: Find the other number if the product is 10/7 and one of the numbers is 2/3.

• Solution:

\[ b = \frac{\frac{10}{7}}{\frac{2}{3}} = \frac{10}{7} \times \frac{3}{2} = \frac{10 \times 3}{7 \times 2} = \frac{30}{14} = \frac{15}{7} = \frac{15}{7} \]

• Final answer: \( b = \frac{15}{7} \).

Practice Problems for Readers

• Find the other number if the product is \( \frac{25}{16} \) and one of the numbers is \( -\frac{5}{4} \).
• Determine the missing rational number if the product is \( -3/8 \) and one number is \( 3/2 \).
• Given the product \( -9/4 \) and one number \( -3/2 \), find the other number.

Key Takeaways

• To find the unknown rational number given the product and one rational factor, divide the product by the known number.
• Remember that dividing by a fraction is equivalent to multiplying by its reciprocal.
• Simplify the resulting fraction to its lowest terms for clarity.
• Always verify the solution by multiplying the two numbers to ensure the product matches the original value.

Conclusion

In conclusion, solving for an unknown rational number when the product and one number are known involves straightforward algebraic manipulation rooted in the properties of rational numbers. The main steps include expressing the unknown as a division problem, finding the reciprocal, and simplifying the result. In the specific problem discussed, the other rational number is \( \frac{4}{3} \), which, when multiplied by \( -\frac{4}{3} \), yields the original product of \( -\frac{16}{9} \). Mastery of these concepts enhances problem-solving skills in algebra and provides a strong foundation for more complex mathematical topics involving rational expressions and equations.