Question

If a varies directly with b, then does b vary directly with a? Explain your reasoning.

Short answer

Yes, if a varies directly with b, then b also varies directly with a, as the relationship in direct variation is symmetrical.

Step-by-step solution

Understanding of Direct Variation

In direct variation, when one variable increases, the other variable also increases proportionally. The same is true when one variable decreases; the other also decreases proportionally. In mathematical terms, this is represented as \( y = kx \), where \( y \) and \( x \) are the variables and \( k \) is a constant which is a factor of proportionality.

Applying the Concept to the Problem

In the given problem, it is stated that \( a \) varies directly with \( b \). That means if \( b \) increases or decreases, \( a \) will also increase or decrease proportionally. In mathematical terms, this can be represented as \( a = kb \), where \( k \) is a constant. Since in direct proportion, the relationship is symmetrical, the same rule applies in reverse, i.e., \( b \) also varies directly with \( a \). Thus, the equation can also be rearranged to give \( b = ka \).

Conclusion

Therefore, it can be concluded that if \( a \) varies directly with \( b \), then \( b \) also varies directly with \( a \) because the relationship in direct variation is symmetrical and reciprocal. So, any change in \( a \) will result in a proportional change in \( b \) and vice versa.