Question

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 \(\sqrt{b}\)

Short answer

(a) \(a\) multiplied by \(b\) is a constant. (b) \(a\) divided by \(b\) is a constant. (c) \(\sqrt{a}\) is proportional to \(\sqrt{b}\). Answer: (b) and (c) are true, while (a) is false.

Step-by-step solution

Checking Condition (a)

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.

Checking Condition (b)

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.

Checking Condition (c)

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.

Key concepts

Direct Proportion

Direct proportion is a fundamental concept in mathematics that describes a linear relationship between two quantities. When two variables, say \(a\) and \(b\), are in direct proportion, it means that as one variable increases, the other increases at a constant rate, known as the constant of proportionality. To put it simply, if \(a\) is directly proportional to \(b\), we can express this as: \[a = k \cdot b,\] where \(k\) is the constant of proportionality.

• This equation states that the ratio \(\frac{a}{b} = k\) should remain constant, irrespective of the values \(a\) and \(b\) might take, as long as they stay in direct proportion.
• Consequently, if \(a\) doubles, \(b\) must also double for the ratio to remain constant, showcasing a linear relationship.

This principle of direct proportion helps in making predictions and understanding the relationship between varying quantities in real-life problems, such as speed and time, where the speed is constant.

Inversely Proportional

Inversely proportional relationships are slightly different from direct ones. When you think of two quantities being inversely proportional, remember that as one quantity increases, the other decreases in such a manner that their product remains constant. Mathematically, if \(a\) is inversely proportional to \(b\), this relationship can be depicted as: \[a \cdot b = k,\] where \(k\) is a constant.

• If \(a\) increases, then \(b\) must decrease to maintain the product \(k\). Conversely, if \(b\) increases, \(a\) must decrease.
• This can be observed in real-life scenarios where, for example, as the speed of a car increases, the time it takes to travel a fixed distance decreases, given the distance remains constant.

Understanding inverse relationships is useful in determining behaviors of systems that have a fixed capacity or constraints.

Mathematical Proof

Mathematical proofs help establish the validity of concepts and relationships. In our example exercise, understanding whether different expressions and conditions for \(a\) and \(b\) are true or false requires proof. By using mathematical logic and formulas, we can verify our given conditions: Condition (a): - We tested if \(a \cdot b\) is a constant. - The proved equation \(a \cdot b = k \cdot b^2\) showed that this relationship does not hold as \(b\) alone cannot be a constant. Hence, this is false. Condition (b): - We confirmed if \(\frac{a}{b} = k\) holds true. - Proof demonstrated that dividing both sides of the equation by \(b\) does yield a constant ratio \(k\). Thus, this condition is true. By dissecting each condition step by step, a mathematical proof offers the rigor needed to definitively accept or reject assertions relating to proportionality.

Constant of Proportionality

A constant of proportionality is a unique value that defines the relationship between two proportionate quantities. In direct proportion, the constant \(k\) in the equation \(a = k \cdot b\) determines the rate at which \(a\) changes with respect to \(b\). Similarly, in an inverse proportion, it is represented in the equation \(a \cdot b = k\). This constant determines how one variable influences another.

• The value of \(k\) ensures that any proportional relationship provides consistent results.
• If \(k\) changes, the nature of the relationship also changes, which means recalibration is required for predictions or calculations.

In instances where square roots are involved, like proving \(\sqrt{a}\) is proportional to \(\sqrt{b}\), the constant becomes \(\sqrt{k}\), as shown in the exercise. This shows the versatility and applicability of the constant across different mathematical contexts, enabling a deeper understanding of how quantities relate to each other.