Ratio and Proportion for ESAT Mathematics 1

Understanding proportion is fundamental to modelling how one quantity changes in relation to another. The symbol used to denote proportionality is $\propto$.

Direct Proportion

Direct proportion occurs when two variables increase or decrease at the same rate. For example, if one chocolate bar costs £6, then 6 bars cost £36 and $x$ bars cost £6$x$. In this scenario, there is no bulk-buy discount, so the price per bar remains constant. The number of chocolate bars and the total price are in direct proportion: as one increases, the other increases by a proportional amount.

If we let $y$ represent the total cost and $x$ the number of bars, we write this relationship as $y \propto x$. This can be expressed as the equation $y = 6x$. In general, if $y$ is directly proportional to $x$, we write $y \propto x$ or $y = kx$, where $k$ is a constant. You can use any letter to represent this constant, provided it is not already used for a variable in the problem.

Graphs Illustrating Direct Proportion

When you plot a graph of $y$ against $x$ for variables in direct proportion, the resulting graph is always a straight line that passes through the origin $(0, 0)$. If a line is straight but does not pass through the origin, the variables are not in direct proportion.

Inverse Proportion

In an inverse proportion relationship, as one variable increases, the other decreases. For instance, consider the thickness of ice, $h$, on a pond relative to the air temperature, $t$. As the temperature decreases, the ice gets thicker. We say $h$ is inversely proportional to $t$, which is written as $h \propto \frac{1}{t}$.

Algebraically, this is expressed as $h = \frac{k}{t}$, where $k$ is a constant. It is important to understand that saying $x$ is inversely proportional to $y$ is mathematically equivalent to saying $x$ is proportional to $\frac{1}{y}$.

Graphs Showing Inverse Proportion

If you plot $y$ on the vertical axis against $x$ on the horizontal axis where $y$ and $x$ are inversely proportional, the graph takes the form of a reciprocal curve, as shown below:

Proportion Involving Integer and Fractional Indices

Proportionality is not limited to linear relationships. If $y \propto x^n$, then the equation is $y = kx^n$. The index $n$ can be any integer or fractional value. For example, $y$ might be proportional to the square of $x$ ($y = kx^2$) or the square root of $x$ ($y = kx^{1/2}$).

Worked Example: Direct Proportion

On days when the outside temperature is above $20^\circ\text{C}$, the number of ice creams sold in a shop, $n$, is proportional to $(t - 15)$, where $t^\circ\text{C}$ is the temperature. When the temperature is $20^\circ\text{C}$, the shop sells 30 ice creams. How many will be sold when the temperature is $28^\circ\text{C}$?

1. Set up the equation: $n \propto (t - 15)$, so $n = k(t - 15)$.
2. Find $k$: Substitute $t = 20$ and $n = 30$ into the equation. $30 = k(20 - 15)$, which simplifies to $30 = 5k$, so $k = 6$.
3. Solve for the new value: Substitute $t = 28$ and $k = 6$ into the equation. $n = 6(28 - 15) = 6 \times 13 = 78$. The shop will sell 78 ice creams.

Verifying Direct Proportion from a Graph

Consider the following graph of $x$ and $y$:

To determine if $y$ is directly proportional to $x$, we can use two methods:

Method 1: Check the origin. A direct proportion graph must be a straight line through the origin. While this graph is a straight line, it does not pass through $(0, 0)$, so $y$ is not directly proportional to $x$.

Method 2: Check the constant $k$. If $y \propto x$, then $y = kx$, meaning $\frac{y}{x}$ must be constant for all points. From the graph, when $x = 4$, $y = 18$, giving $k = 18/4 = 4.5$. However, when $x = 6$, $y = 24$, giving $k = 24/6 = 4$. Since the values of $k$ differ, it is not direct proportion.

Worked Example: Inverse Proportion

The number of pairs of gloves sold in a store, $n$, is inversely proportional to the outside temperature $t^\circ\text{C}$ for temperatures between 1 and 10. When $t = 5$, $n = 15$. How many are sold when $t = 3$?

1. Set up the equation: $n = \frac{k}{t}$.
2. Find $k$: Substitute $t = 5$ and $n = 15$. $15 = \frac{k}{5}$, so $k = 15 \times 5 = 75$.
3. Solve for the new value: When $t = 3$, $n = \frac{75}{3} = 25$. Therefore, 25 pairs of gloves are sold.

Verifying Inverse Proportion from a Graph

Look at the following graph and determine if $y$ could be inversely proportional to $x$:

If $y$ is inversely proportional to $x$, then $y = \frac{k}{x}$, or $xy = k$. We test multiple points from the line to see if $k$ remains constant:

• When $x = 1$, $y = 12$, so $k = 1 \times 12 = 12$.
• When $x = 2$, $y = 6$, so $k = 2 \times 6 = 12$.
• When $x = 3$, $y = 4$, so $k = 3 \times 4 = 12$.
• When $x = 4$, $y = 3$, so $k = 4 \times 3 = 12$.
• When $x = 6$, $y = 2$, so $k = 6 \times 2 = 12$.
• When $x = 12$, $y = 1$, so $k = 12 \times 1 = 12$.

Since $k$ is consistently 12 for these points, it is reasonable to assume $y$ is inversely proportional to $x$.

Worked Example: Proportion with Indices

The number of water bottles sold, $n$, is proportional to the square of the average daily temperature, $t$, for temperatures between 10 and 30. When $t = 12$, $n = 72$. Find $n$ when $t = 24$.

1. Set up the equation: $n \propto t^2$, so $n = kt^2$.
2. Find $k$: Substitute $n = 72$ and $t = 12$. $72 = k \times 12^2 = 144k$. This gives $k = 72/144 = 0.5$.
3. Solve for the new value: When $t = 24$, $n = 0.5 \times 24^2 = 0.5 \times 576 = 288$. The shop sells 288 bottles.