Answer to Question #129591 in Algebra for Sphesihle Zuma

Question

QUESTION 1
1.1. Define the following concepts and give an example in each case.
1.1.1. Ratio (3)
1.1.2. Direct proportion (3)
1.1.3. Indirect proportion

Answer 1

1.1.1. A ratio, which is a comparison of two numbers by division, is the quotient obtained when the first number is divided by the second, nonzero number.

In general, the ratio of $a$ to $b$ can be expressed as

$${a \over b}\ \ \text{or}\ \ a \div b \ \ \text{or}\ \ a : b$$

To find the ratio of two quantities, both quantities must be expressed in the same unit of measure before their quotient is determined.

The ratio has no unit of measure.

Example: sine ratio is the ratio of the length of the opposite side divided by the length of the hypotenuse. 

1.1.2. Two values $x$ and $y$ are directly proportional to each other when the ratio $x: y$ or $\dfrac{x}{y}$  is a constant (i.e. always remains the same). This would mean that $x$ and $y$ will either increase together or decrease together by an amount that would not change the ratio.

Example: If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality. 

1.1.3. Two values $x$ and $y$ are inversely proportional (indirect proportion)to each other when their product $xy$ is a constant (always remains the same). This means that when $x$ increases $y$ will decrease, and vice versa, by an amount such that $xy$ remains the same.

Example: Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

$$F_{gr}=G\dfrac{m_1m_2}{r^2}$$

where $F_{gr}$ is the gravitational force acting between two objects, $m_1$ and $m_2$ are the masses of the objects, $r$ is the distance between the centers of their masses, and $G$ is the gravitational constant.