Proportion (OCR GCSE Maths): Revision Notes

Example:

• If $x$ is the number of KitKat Chunkys you buy, and $y$ is the total cost, then $y$is directly proportional to $x.$
• Graph: A straight line through the origin $(0,0)$ shows a direct linear relationship.

Example:

• If $x$ is the amount of money spent advertising a gig, and $y$ is the number of people who attend, then $y$ is directly proportional to $x²$.
• Graph: A curve that gets steeper as $x$ increases.

Example:

• If $x$ is the time you spend revising, and $y$is your exam mark, then $y$ is directly proportional to $x³$.
• Graph: A curve that becomes very steep very quickly as $x$ increases.

Example 1: Direct Linear Proportion Problem: The cost of buying KitKat Chunkys is directly proportional to the number of bars you buy. If $3$ bars cost $£6$, how much will $7$ bars cost?

Solution:

1. Set Up the Proportion:

• $y∝x$ means $y=kx$, where $k$ is the constant of proportionality.

2. Find the Constant $k$:

• $6=k×36=k×3$
• $k=\frac{6}{3}=2$

3. Calculate the Cost for $7$ Bars:

• $y=2×7=14$ Final Answer: $7$ bars will cost $£14$.

Example 2: Quadratic Proportion Problem: The number of people who hear about a concert is directly proportional to the square of the amount of money spent on advertising. If spending $£4,000$ results in $1,600$ people hearing about the concert, how many people will hear about it if $£9,000$ is spent?

Solution:

Step 1: Set Up the Proportion

The number of people, $y$, is directly proportional to the square of the amount spent, $x²$. This can be written as:

[$$ y=kx²

y = 1600 \ \text{people when } x = 4000 \ \text{pounds}.

1600=k×4000²

4000²=16,000,000

k=\frac {1600}{16,000,000}

k=\frac {1600}{16×10^6}= \frac{1}{10,000}

y=0.0001×9000^2

9000^2=81,000,000

y=0.0001×81,000,000=8100

Example:

• $x$ could be the number of people sharing a car ride, and $y$ could be the amount each person has to pay for petrol. As more people join the ride, the amount each person pays decreases.
• Graph: The graph of $y$ against $x$ is a hyperbola, which approaches both axes but never touches them.

Worked Example 1: Simple Inverse Proportion Problem: The time taken to complete a journey is inversely proportional to the speed. If it takes $4$ hours to complete a journey at $30$ mph, how long will it take at $60$ mph?

Step-by-Step Solution:

1. Set Up the Proportion:

• Since $y∝\frac{1}{x}$, we have $y=\frac{k}{x}$

2. Find the Constant kk:

• Given: $y=4$ hours, $x=30$ mph.
• $4=\frac{k}{30}$
• $k=4×30=120$

3. Use $k$ to Find the New Time:

• At $60$ mph: $y=\frac{120}{60}=2$ hours.

Final Answer: It will take $2$ hours to complete the journey at $60$ mph.

• Example:
• $x$could be the number of hours spent watching TV, and $y$ could be the number of brain cells remaining. As the number of hours increases, the number of brain cells decreases at an accelerating rate.
• Graph: The graph of $y$against $x$is a steeper curve than in simple inverse proportion, showing a faster rate of decrease.

Worked Example 2: Quadratic Inverse Proportion Problem: The intensity of light $I$ is inversely proportional to the square of the distance $d$ from the light source. If the intensity is $100$ units at a distance of $2$ metres, what is the intensity at $5$ metres?

Step-by-Step Solution:

4. Set Up the Proportion:

• Since $I∝\frac{1}{d^2}$, we have $I=\frac{k}{d^2}.$

5. Find the Constant $k$:

• Given: $I=100$ units, $d=2$ metres.
• $100=\frac{k}{2^2}= \frac{k}{4}$
• $k=100×4=400$

6. Use $k$ to Find the New Intensity:

• At $5$ metres: $I=\frac{400}{5^2}=\frac{400}{25}=16$ units.

Final Answer: The intensity at $5$ metres is $16$ units.

• Example: $y∝x$ or $y∝\frac{1}{x}$.

• Example: $y=kx$ or $y=\frac{k}{x}$.

📑Example 1: Problem: $y$ is directly proportional to $x$. Given that $y=12$ when $x=4$, calculate the value of:

• (a) $y$ when $x=6$
• (b) $x$ when $y=66$

Step-by-Step Solution:

7. Identify the Proportion Type:

• The problem states that $y$is directly proportional to $x$, so we write:

8. Convert to an Equation:

• Replace the proportionality sign with an equation using $k$:

9. Find the Constant $k$:

• Use the given values $y=12$ and $x=4$ to find $k$:

10. Write the Complete Formula:

• Now that $k=3$, the relationship between $y$and $x$is:

11. Solve Part (a): Find y when $x=6$.

• Substitute $x=6$ into the formula:

Answer: $y=18$ when $x=6$.

12. Solve Part (b): Find $x$ when $y=66$

• Substitute $y=66$ into the formula and solve for $x$:

Answer: $x=22$ when $y=66$.

📑Example 2:

📑Example 3: Problem:

$z$ is inversely proportional to $t$. Given that when $t=0.3$, the value of $z=16$, find the value of $z$ when $t=0.5$.

Step-by-Step Solution:

13. Identify the Proportionality:

• The problem states that $z$ is inversely proportional to $t$. We express this relationship as:

14. Write the Equation:

• Replace the proportionality sign ($∝$) with an equals sign and introduce a constant $k$:

15. Substitute Known Values:

• We know that when $t=0.3$, $z=16$. Substitute these values into the equation to find $k$:

16. Solve for $k$:

• Rearrange the equation to solve for $k$:

17. Write the Full Equation:

• Now that we have $k$, the full equation becomes:

18. Substitute the New Value of $t$:

• To find $z$ when $t=0.5$, substitute $t=0.5$ into the equation:

19. Calculate $z$:

• Perform the division:

Final Answer:

When $t=0.5$, $z=9.6$.

Example 4: Problem:

(a) Describe the variation using the proportionality symbol ($∝$).

(b) Find the equation connecting the two variables.

Given:

• $x=5, y=5$
• $x=25, y=125$

Step-by-Step Solution:

20. Identify the Type of Proportion:

• The problem states that as $x$ increases from $5$ to $25$, $y$ also increases from $5$ to $125$.
• To determine the type of direct proportion, observe the changes:
• $x$ increases by a factor of $5$ (since $25=5×5$).
• $y$ increases by a factor of $25$ (since $125=5×25$).
• Since $25$ is $5^2$, this suggests a quadratic proportion.

21. Express the Proportion:

• The relationship between $y$ and $x$ is expressed as:

• This means that $y$ is directly proportional to the square of $x$.

22. Write the Equation:

• Replace the proportionality sign ($∝$) with an equals sign and introduce a constant $k$:

23. Determine the Constant k:

• Use the given values to find $k$. Substitute $x=25$ and $y=125$ into the equation:

• Simplify the equation to solve for $k$:

24. Write the Final Equation:

• Substitute the value of $k$ back into the equation:

25. Verify the Equation:

• Check the equation with the other set of values ($x=5 and y=5$) to ensure it works correctly:

• The equation $y=0.2x^2$ works with both sets of values, confirming its accuracy.

Final Answer:

The equation connecting $y$and $x$ is $y=0.2x^2$