1. Constant Ratio ($k$):

• If $y$ is directly proportional to $\ x$, it can be expressed as $\ y = kx$, where $\ k$ is the constant of proportionality.
• The ratio $\ \frac{y}{x} = k$ remains constant for all pairs of values of $\ x$ and $\ y$.

2. Graphical Representation:

• When graphed, a proportional relationship will always produce a straight line that passes through the origin ($0,0$).

• The slope of this line represents the constant of proportionality $\ k .$

3. Examples:

• Direct Proportion: If $4$ apples cost $£2$, then $8$ apples would cost $£4$, maintaining the same ratio of apples to cost.
• Real-Life Application: Speed is proportional to distance if the time is constant. For example, if a car travels $60$ miles in $1$ hour, it will travel $120$ miles in $2$ hours at the same speed.

🤔Exam Tip:

In an Edexcel A Level Maths exam, you might be asked to identify or work with proportional relationships. To do this:

• Recognize or calculate the constant of proportionality $\ k .$
• Use the equation $\ y = kx$ to solve problems.

• Check if the relationship graph is a straight line through the origin to confirm proportionality.

📑Example: Direct Proportion

Question:

The amount of paint required to cover a wall is directly proportional to the area of the wall. If $4$ litres of paint are needed to cover a $20 m²$ wall, how much paint is required to cover a $35$ m² wall?

Solution:

4. Set up the proportion:

• Let $p$ be the amount of paint required and $A$ the area of the wall.
• Since $p$ is directly proportional to $A$, we can write $p=kA$, where $k$ is the constant of proportionality.

5. Find the constant of proportionality:

• From the information given: $4=k×20$.

• Solve for $k$: $k= \frac{4}{20} = 0.2$

6. Calculate the paint needed for a $35$ m² wall:

• Substitute $A=35$ into the equation $p=kA$: $p=0.2×35=7 litres$ Answer: $7$ litres of paint are required to cover the $35$ m² wall.

📑Example: Inverse Proportion

If $y$ is inversely proportional to $x$ and $y=8$ when $x=2$, find the value of $y$ when $x=4$

Step-by-Step Solution:

1. Express the relationship: Since $y$ is inversely proportional to $x$ we have: $y = \frac{k}{x}$

where $k$ is the constant of proportionality.

2. Find the constant $k$: Use the given values $y=8$ and $x=2$ to find $k$: $8 = \frac{k}{2}$

$k=8×2=16$

3. Formulate the equation: Substitute $k=16$ into the equation $y= \frac{k}{x}$ $y = \frac{16}{x}$

4. Find $y$ when $x=4$: $y = \frac{16}{4} = 4$

Final Answer: When $x=4$, $y=4$

📝Practice Question:

If $\ y$ is directly proportional to$\ x ,\ and \ y = 10$ when $\ x = 2$, find the value of $\ y$ when $\ x = 7 .$

Solution:

5. Find the constant of proportionality $\ k$: $k = \frac{y}{x} = \frac{10}{2} = 5$
6. Use $\ k\ to\ find \ y \ when \ x = 7 :$ $y = kx = 5 \times 7 = 35$ Thus, $\ y = 35$ when $\ x = 7$.