Question

U2, L2: Sums & Products of Rational Numbers

Answer 1

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!