Question

Given \(a\) is proportional to \(b\), state which of the following are true and which are false: (a) when \(a\) doubles, then \(b\) also doubles (b) when \(a\) is halved, then \(b\) is doubled (c) a graph of \(a\) against \(b\) is a straight line graph (d) \(a\) divided by \(b\) is a constant

Short answer

(a) If \(a\) doubles, then \(b\) also doubles. (b) If \(a\) is halved, then \(b\) also doubles. (c) A graph of \(a\) against \(b\) is a straight line. (d) The ratio of \(a\) to \(b\) is constant. Answer: (a) True, (b) False, (c) True, (d) True

Step-by-step solution

Evaluate statement (a)

Since \(a\) is proportional to \(b\), we know that \(a = k b\). If \(a\) doubles, then the new value of \(a\) would be \(2a\). So, we have: $$2a = k(2b),$$ which is equivalent to: $$a = k b.$$ Since \(a\) and \(2a\) are both proportional to their corresponding values of \(b\) with the same constant, statement (a) is true.

Evaluate statement (b)

If \(a\) is halved, then the new value of \(a\) would be \(\frac{a}{2}\). So, we have: $$\frac{a}{2} = k b',$$ where \(b'\) is the new value of \(b\). To determine if this new equation is equivalent to the original, we can multiply both sides of the equation by 2: $$2 \times \frac{a}{2} = 2 \times k b'.$$ This simplifies to: $$a = 2kb'.$$ If \(b'\) were twice the original value of \(b\), we would have: $$a = k (2b).$$ However, the above derivation shows that halving \(a\) does not lead to an equation in this form. Thus, statement (b) is false.

Evaluate statement (c)

In order to determine whether a graph of \(a\) against \(b\) is a straight line, we can examine the equation \(a = k b\). This equation is in the form of a linear equation with \(a\) as the dependent variable and \(b\) as the independent variable. The slope of the line is \(k\), and the line passes through the origin, since when \(b = 0\), \(a = 0\). Therefore, statement (c) is true.

Evaluate statement (d)

Given the equation \(a = k b\), we can divide both sides of the equation by \(b\) to obtain: $$\frac{a}{b} = k.$$ Since \(k\) is a constant, it shows that the ratio of \(a\) to \(b\) is constant, and this holds true for any valid values of \(a\) and \(b\). Therefore, statement (d) is true. In conclusion: - Statement (a) is true. - Statement (b) is false. - Statement (c) is true. - Statement (d) is true.

Key concepts

Direct Proportionality

Direct proportionality is a fundamental concept in mathematics that describes a relationship between two variables where they increase or decrease together at the same rate. If you are given that one variable, let's call it a, is directly proportional to another variable, b, it means that as a increases, b also increases, and the same is true for decreases.

This relationship can be expressed by the equation a = k * b, where k is known as the constant of proportionality. This constant remains the same regardless of the values of a and b, representing the consistent rate of change between them. As such, we can understand why, in our original problem, statement (a) is true, and statement (b) is false. When a doubles, b must also double to maintain the same ratio dictated by the constant k.

Linear Equations

Linear equations form the backbone of understanding relationships between variables in algebra. They have the general format y = mx + c, where m represents the slope, and c the y-intercept. However, in a directly proportional relationship, where the variables pass through the origin, the equation simplifies to y = mx, or in our textbook problem, to a = kb.

What's crucial about linear equations, as indicated in statement (c), is that when graphed, they always produce a straight line. This visual representation makes it easier to predict values and understand the relationship. Since the value of a depends on the value of b in a predictable, constant way, we obtain a straight line through the origin, confirming the truth of statement (c) in our exercise. This feature helps in identifying linear relationships quickly - if the graph isn't a straight line, the relationship isn't linear.

Constants in Proportionality

Discussing constants in proportionality brings us to the notion that in a directly proportional relationship, the quotient of the two variables remains unchanged. This quotient is the constant of proportionality, k. It is a measure of how many units of b correspond to one unit of a.

When dealing with these constants, statement (d) from our practice problem demonstrates the importance of understanding that the ratio, \( \frac{a}{b} = k \), is consistent for all values of a and b within a direct proportion. Recognizing the constant nature of k provides a powerful tool to solve numerous problems involving proportional relationships and underpins many real-world applications such as calculating rates, densities, and conversion factors.