To understand the relationship between the equations [tex]\( \frac{a}{b} = \frac{c}{d} \)[/tex] and [tex]\( a \cdot d = b \cdot c \)[/tex], we need to consider the properties of proportions and how the terms can be rearranged. Here is a step-by-step explanation:
1. Start with the given proportion:
[tex]\[
\frac{a}{b} = \frac{c}{d}
\][/tex]
This means that the ratio of [tex]\(a\)[/tex] to [tex]\(b\)[/tex] is equal to the ratio of [tex]\(c\)[/tex] to [tex]\(d\)[/tex].
2. Cross-multiplication:
To remove the fractions and simplify the comparison, we can use the method of cross-multiplication. Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other. This gives us:
[tex]\[
a \cdot d = b \cdot c
\][/tex]
Here’s the step-by-step breakdown:
- Multiply [tex]\(a\)[/tex] by [tex]\(d\)[/tex], which gives [tex]\( a \cdot d \)[/tex].
- Multiply [tex]\(b\)[/tex] by [tex]\(c\)[/tex], which gives [tex]\( b \cdot c \)[/tex].
- Set the two products equal to each other.
3. Interpreting the result:
The resulting equation [tex]\( a \cdot d = b \cdot c \)[/tex] means that the product of [tex]\(a\)[/tex] and [tex]\(d\)[/tex] is equal to the product of [tex]\(b\)[/tex] and [tex]\(c\)[/tex]. This result confirms that the proportion [tex]\( \frac{a}{b} = \frac{c}{d} \)[/tex] implies that these products are equal.
4. Conclusion:
Therefore, for real numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] (where [tex]\(b\)[/tex] and [tex]\(d\)[/tex] cannot be zero to avoid division by zero), the equation [tex]\( \frac{a}{b} = \frac{c}{d} \)[/tex] is indeed equivalent to [tex]\( a \cdot d = b \cdot c \)[/tex].
In summary, by cross-multiplying the terms of the given proportion, we see that the statement [tex]\( \frac{a}{b} = \frac{c}{d} \)[/tex] can be rearranged to [tex]\( a \cdot d = b \cdot c \)[/tex]. This transformation shows that these two equations are mathematically equivalent.
