Question
In the proof that a/b × c/d
Answers
Answered by
GPT-4o mini
To prove that \((a/b) \times (c/d) = (a \times c) / (b \times d)\), we can rely on the properties of fractions and multiplication. Here's how the proof generally goes:
1. **Write the fractions**: We start with the two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).
2. **Multiply the fractions**: The rule for multiplying fractions states:
\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]
3. **Understanding the multiplication**:
- When you multiply two fractions, you multiply the numerators together and the denominators together. This is valid because of how fractions are constructed.
- The fraction \(\frac{a}{b}\) represents \(a \div b\) and \(\frac{c}{d}\) represents \(c \div d\). Therefore:
\[
\frac{a}{b} \times \frac{c}{d} = \left(a \div b\right) \times \left(c \div d\right)
\]
4. **Using the definition of division**:
- The expression \(a \div b\) can be rewritten as \(a \times \frac{1}{b}\) and \(c \div d\) can be rewritten as \(c \times \frac{1}{d}\).
- Hence,
\[
\left(a \div b\right) \times \left(c \div d\right) = \left(a \times \frac{1}{b}\right) \times \left(c \times \frac{1}{d}\right)
\]
5. **Associative property of multiplication**:
- Using the associative property of multiplication, we can regroup the terms:
\[
= a \times c \times \left(\frac{1}{b} \times \frac{1}{d}\right)
\]
- The product \(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\).
6. **Final expression**:
- Therefore, we have:
\[
a \times c \times \frac{1}{b \times d} = \frac{a \times c}{b \times d}
\]
Putting it all together, we can conclude that:
\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]
This completes the proof.
1. **Write the fractions**: We start with the two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).
2. **Multiply the fractions**: The rule for multiplying fractions states:
\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]
3. **Understanding the multiplication**:
- When you multiply two fractions, you multiply the numerators together and the denominators together. This is valid because of how fractions are constructed.
- The fraction \(\frac{a}{b}\) represents \(a \div b\) and \(\frac{c}{d}\) represents \(c \div d\). Therefore:
\[
\frac{a}{b} \times \frac{c}{d} = \left(a \div b\right) \times \left(c \div d\right)
\]
4. **Using the definition of division**:
- The expression \(a \div b\) can be rewritten as \(a \times \frac{1}{b}\) and \(c \div d\) can be rewritten as \(c \times \frac{1}{d}\).
- Hence,
\[
\left(a \div b\right) \times \left(c \div d\right) = \left(a \times \frac{1}{b}\right) \times \left(c \times \frac{1}{d}\right)
\]
5. **Associative property of multiplication**:
- Using the associative property of multiplication, we can regroup the terms:
\[
= a \times c \times \left(\frac{1}{b} \times \frac{1}{d}\right)
\]
- The product \(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\).
6. **Final expression**:
- Therefore, we have:
\[
a \times c \times \frac{1}{b \times d} = \frac{a \times c}{b \times d}
\]
Putting it all together, we can conclude that:
\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]
This completes the proof.
There are no human answers yet. A form for humans to post answers is coming very soon!