If \(a\) is proportional to \(b\) state which of the following are true and which are false: (a) \(a\) multiplied by \(b\) is a constant (b) \(a\) divided by \(b\) is a constant (c) \(\sqrt{a}\) is proportional to \(\sqrt{b}\)

Condition (a) says that \(a\) multiplied by \(b\) is a constant. Let's check if this is true by multiplying both sides of the proportionality equation by \(b\): \[ a \cdot b = k \cdot b \cdot b \] \[ a \cdot b = k \cdot b^2 \] Here, we see that \(a \cdot b\) is equal to the constant \(k\) multiplied by \(b^2\). Since \(b\) is not a constant, this condition is false.

Condition (b) says that \(a\) divided by \(b\) is a constant. Let's check if this is true by dividing both sides of the proportionality equation by \(b\): \[ \frac{a}{b} = \frac{k \cdot b}{b} \] \[ \frac{a}{b} = k \] Here, we see that \(\frac{a}{b}\) is equal to the constant \(k\). Thus, this condition is true.

Condition (c) says that \(\sqrt{a}\) is proportional to \(\sqrt{b}\). To check this, let's first rewrite \(a = k\cdot b\) in terms of \(\sqrt{a}\) and \(\sqrt{b}\): \[ \sqrt{a} = \sqrt{k\cdot b} \] Now, let's see if there exists another constant of proportionality, say \(l\), such that \(\sqrt{a} = l\cdot \sqrt{b}\). If we can find such an \(l\), then Condition (c) would be true. To find \(l\), we can divide both sides of the equation \(\sqrt{a} = \sqrt{k\cdot b}\) by \(\sqrt{b}\): \[ \frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{k\cdot b}}{\sqrt{b}} \] \[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{k\cdot b}{b}} \] \[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{k} \] Here, we see that \(\frac{\sqrt{a}}{\sqrt{b}}\) is equal to the constant \(\sqrt{k}\). So, we can rewrite the equation as: \[ \sqrt{a} = \sqrt{k} \cdot \sqrt{b} \] This shows that \(\sqrt{a}\) is proportional to \(\sqrt{b}\) with the constant of proportionality as \(\sqrt{k}\). Therefore, condition (c) is true. In conclusion, Condition (a) is false, while Conditions (b) and (c) are true.