8.3 Proportions

The cost of 8 pens is 12 chf. Find the cost of 1 pen.

4 pencils cost 1.20 chf. How much do 15 pencils cost?

Test whether the table $(1, 3), (2, 9), (3, 27)$ represents a proportional relationship.

For the proportional relationship $y = kx$, find $k$ given that $y = 35$ when $x = 5$.

Using $y = 4x$, find $y$ when $x = 12$.

Using $y = 5x$, find $x$ when $y = 40$.

On a graph of $y = 2x$, find the $y$-coordinate when $x = -3$.

A graph of $y$ against $x$ passes through the origin and the point $(2, 10)$. State the proportionality equation.

In a direct-proportion graph, the line is straight and passes through which point always?

Decide which equation represents direct proportion: $y = 4x$, $y = x + 4$, $y = x^2$.

Distance is directly proportional to time. In 2 minutes a rollercoaster travels 50 m. Find the distance travelled in 3 minutes and in 7 minutes.

A printer prints 24 pages in 3 minutes. How long to print 60 pages?

A school bus uses 15 L of fuel to travel 180 km. How far can it travel on 25 L at the same rate?

A cake recipe uses 2 eggs to make 6 cakes. How many eggs are needed for 15 cakes?

Test whether $y$ is directly proportional to $x$ given the table: x: 2, 5, 8; y: 6, 15, 24.

On the graph of $y = 3x$, what is the gradient of the line? Describe how the gradient relates to $k$.

A tap fills a bucket at a constant rate. After 3 minutes there are 12 L in the bucket. (a) Find the rate. (b) Write $V = kt$. (c) Find $V$ at $t = 7$.

4 kg of apples costs 10 chf. How much do 7 kg cost?

Two quantities are directly proportional. When $a = 6$, $b = 15$. Find $b$ when $a = 22$.

The graph of $y$ against $x$ is a straight line through the origin with gradient 0.4. State the equation and find $y$ when $x = 15$.

For two quantities $x$ and $y$, $y$ is directly proportional to $x$. When $x = 5$, $y = 35$. (a) Write an equation linking $x$ and $y$. (b) Find $y$ when $x = 12$. (c) Find $x$ when $y = 91$.

Decide whether each table represents direct proportion. Justify. Table A: x = 1, 2, 4, y = 3, 6, 12. Table B: x = 1, 2, 4, y = 3, 6, 8.

In a sale, the cost $C$ of buying $n$ tickets is $C = 12.50n$. (a) State the unit cost. (b) How many tickets can you buy with 100 chf? (c) Is this a proportional relationship? Justify.

The cost of running a car for $d$ km is $C = 0.18d$ chf. (a) Find the cost for 250 km. (b) Find $d$ for a budget of 90 chf. (c) Sketch the graph and label the gradient.

Two friends share a rate. Alex earns 40 chf for 5 hours of work. Brigit earns 60 chf for 8 hours. Whose hourly rate is higher?

A scale model is a proportional copy of a real building. The model door is 6 cm tall and the real door is 2 m tall. (a) Find the scale factor. (b) The real building is 24 m tall. Find the height of the model.

The mass of a uniform metal bar is directly proportional to its length. A 50 cm bar has mass 120 g. (a) Write an equation. (b) Find the mass of a 75 cm bar. (c) Find the length of a 240 g bar.

On a fitness tracker, calories burned $C$ is roughly proportional to steps $s$: $C = 0.04s$. (a) How many calories for 8000 steps? (b) How many steps to burn 200 calories? (c) Is the relationship truly proportional in real life? Justify briefly.

A graph of $y$ versus $x$ shows a straight line through the origin. From the graph, $y = 12$ when $x = 8$. (a) Find $k$. (b) Find $y$ when $x = 20$. (c) Find $x$ when $y = 30$.

A car travels $d$ km on $\ell$ litres of fuel: $d = 14\ell$. (a) State the unit rate (km/L). (b) Find $d$ when $\ell = 28$. (c) The driver only has 600 chf and fuel costs 2 chf/L. What is the maximum distance she can drive?

A tap fills a bath proportionally to time. After 5 min the bath has 30 L. (a) Write a model $V = kt$. (b) The bath holds 200 L. How long to fill from empty? (c) The drain leaks at 2 L/min (constant). Modify the model to account for the leak, and find the time to fill.

A rectangular field has length proportional to width: $\ell = 2w$. The perimeter is 60 m. (a) Find $\ell$ and $w$. (b) Find the area. (c) If the perimeter doubles, what happens to the area? Justify.

Compare two paint-mixing schemes. Scheme A: $y = 2x$ (yellow per blue). Scheme B: $y = 3x - 4$. Identify which is proportional and find values that give the same shade in both.

Investigate: if $y \propto x$ and $x \propto z$, show that $y \propto z$ and find the combined constant.

On a graph of $y = kx$, two points are $(3, p)$ and $(8, p + 20)$. Find $k$ and $p$.

Distance is directly proportional to time. A cyclist covers 12 km in 30 min. (a) Find the speed. (b) Write a model with units. (c) Determine how far in 1 h 45 min. (d) Sketch the d–t graph for the first 2 hours.

A rope is divided into 3 lengths in the ratio $2:3:5$. The longest is 60 cm longer than the shortest. (a) Find each length. (b) If the lengths are doubled, are the ratios still $2:3:5$? Justify.

Two quantities are proportional: $y$ doubles every time $x$ doubles. The point $(2, 8)$ lies on the graph. (a) Find the constant of proportionality. (b) Find $y$ when $x = 7$. (c) Sketch the graph and explain why doubling $x$ doubles $y$.

Show by counter-example that a relationship may have constant ratio at two points but not be proportional overall.

Investigate: is the perimeter of a square proportional to its side length? Is the area? Justify.