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U2, L2: Sums & Products of Rational Numbers

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Sure! Sums and products of rational numbers are fundamental operations in arithmetic that involve numbers that can be expressed as fractions. Here’s a brief overview of how to work with sums and products of rational numbers.

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) is an integer and \( b \) is a non-zero integer.

Sums of Rational Numbers

To add two rational numbers:

  1. Identify the Rational Numbers: Let’s say we have two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \).

  2. Find a Common Denominator: The common denominator is usually the least common multiple (LCM) of the denominators \( b \) and \( d \).

  3. Convert to an Equivalent Fraction:

    • Convert each fraction to an equivalent fraction with the common denominator: \[ \frac{a}{b} = \frac{a \cdot (d/d)}{b \cdot (d/d)} = \frac{ad}{bd} \] \[ \frac{c}{d} = \frac{c \cdot (b/b)}{d \cdot (b/b)} = \frac{cb}{bd} \]

  4. Add the Numerators: \[ \frac{ad + cb}{bd} \]

  5. Simplify: If possible, simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

Example of Sums

Let’s add \( \frac{2}{3} \) and \( \frac{1}{4} \):

  1. The common denominator of \( 3 \) and \( 4 \) is \( 12 \).
  2. Convert: \[ \frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12} \] \[ \frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12} \]
  3. Add the numerators: \[ \frac{8}{12} + \frac{3}{12} = \frac{8 + 3}{12} = \frac{11}{12} \]

Products of Rational Numbers

To multiply two rational numbers:

  1. Identify the Rational Numbers: Let’s say we are multiplying \( \frac{a}{b} \) and \( \frac{c}{d} \).

  2. Multiply the Numerators and Denominators: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \]

  3. Simplify: If possible, simplify the resulting fraction.

Example of Products

Let’s multiply \( \frac{2}{3} \) and \( \frac{3}{4} \):

  1. Multiply the numerators: \[ 2 \times 3 = 6 \]
  2. Multiply the denominators: \[ 3 \times 4 = 12 \]
  3. Combine: \[ \frac{6}{12} \]
  4. Simplify: \[ \frac{6}{12} = \frac{1}{2} \]

Summary

  • To add rational numbers, find a common denominator, convert, and add the numerators.
  • To multiply rational numbers, multiply the numerators together and the denominators together, then simplify if necessary.

If you have specific examples or problems in mind, feel free to share, and I can walk you through them!