Answer Key

8.3 Proportions

Pack A — Answers

#	Question	Answer
1	The cost of 5 pens is 7.5 chf. Find the cost of 1 pen.	1.50 chf
2	4 pencils cost 1.20 chf. How much do 10 pencils cost?	3.00 chf
3	Test whether the table $(1, 4), (2, 8), (3, 12)$ represents a proportional relationship.	Yes — y/x is constant (= 4).
4	For the proportional relationship $y = kx$, find $k$ given that $y = 20$ when $x = 4$.	$k = 5$
5	Using $y = 6x$, find $y$ when $x = 9$.	54
6	Using $y = 3x$, find $x$ when $y = 24$.	$x = 8$
7	On a graph of $y = 2x$, find the $y$-coordinate when $x = -3$.	$y = -6$
8	A graph of $y$ against $x$ passes through the origin and the point $(2, 10)$. State the proportionality equation.	$y = 5x$
9	In a direct-proportion graph, the line is straight and passes through which point always?	The origin $(0, 0)$.
10	Decide which equation represents direct proportion: $y = 4x$, $y = x + 4$, $y = x^2$.	$y = 4x$ (proportional)
11	Distance is directly proportional to time. In 2 minutes a rollercoaster travels 36 m. Find the distance travelled in 3 minutes and in 5 minutes.	54 m in 3 min; 90 m in 5 min
12	A printer prints 30 pages in 2 minutes. How long to print 75 pages?	5 minutes
13	A school bus uses 12 L of fuel to travel 96 km. How far can it travel on 18 L at the same rate?	144 km
14	A cake recipe uses 3 eggs to make 6 cakes. How many eggs are needed for 14 cakes?	7 eggs
15	Test whether $y$ is directly proportional to $x$ given the table: x: 2, 5, 8; y: 6, 15, 24.	Yes — $k = 3$.
16	On the graph of $y = 7x$, what is the gradient of the line? Describe how the gradient relates to $k$.	Gradient = 7. The gradient of $y = kx$ is exactly $k$, the constant of proportionality.
17	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 $t = 7$.	(a) 4 L/min (b) $V = 4t$ (c) 28 L
18	3 kg of apples costs 7.5 chf. How much do 5 kg cost?	12.50 chf
19	Two quantities are directly proportional. When $a = 6$, $b = 15$. Find $b$ when $a = 22$.	$b = 55$
20	The graph of $y$ against $x$ is a straight line through the origin with gradient 2.5. State the equation and find $y$ when $x = 6$.	$y = 2.5x$; $y = 15$
21	For two quantities $x$ and $y$, $y$ is directly proportional to $x$. When $x = 4$, $y = 18$. (a) Write an equation linking $x$ and $y$. (b) Find $y$ when $x = 10$. (c) Find $x$ when $y = 90$.	(a) $y = 4.5 x$ (b) $y = 45$ (c) $x = 20$
22	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.	A: yes ($k = 3$). B: no.
23	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.	(a) 12.50 chf (b) 8 tickets (c) Yes — passes through origin, $C/n$ constant.
24	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.	(a) 45 chf (b) 500 km (c) Straight line through origin, gradient 0.18
25	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?	Alex (8 chf/h vs Brigit 7.50 chf/h).
26	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.	(a) 1 cm : 33.3 cm (b) 72 cm
27	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.	(a) $m = 2.4 \ell$ (b) 180 g (c) 100 cm
28	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) 320 cal (b) 5000 steps (c) Approximately, but actual calories depend on weight, gradient, speed, etc.
29	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) $k = 1.5$ (b) $y = 30$ (c) $x = 20$
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) 14 km/L (b) 392 km (c) 4200 km
31	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) $V = 6t$ (b) $≈ 33.3$ min (c) $V = (6-2)t = 4t$ → $50$ min
32	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.	(a) $w = 10$, $\ell = 20$ (b) 200 m² (c) Area becomes 800 m² — quadruples (linear scale 2 → area scale 4).
33	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.	A is proportional ($y = 2x$); B is not (passes through (0, -4)). Same shade when $2x = 3x - 4$, i.e. $x = 4$, $y = 8$.
34	Investigate: if $y \propto x$ and $x \propto z$, show that $y \propto z$ and find the combined constant.	Yes — $y = k_1 x$ and $x = k_2 z$ ⇒ $y = k_1 k_2 z$, so $y \propto z$ with constant $k_1 k_2$.
35	On a graph of $y = kx$, two points are $(3, p)$ and $(8, p + 20)$. Find $k$ and $p$.	$k = 4, p = 12$
36	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) 24 km/h (b) $d = 24t$ (t in h) (c) 42 km (d) Line through origin slope 24
37	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.	(a) 40, 60, 100 cm (b) Yes — scaling all by the same factor preserves ratios.
38	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 $y$.	(a) $k = 4$ (b) $y = 28$ (c) See working
39	Show by counter-example that a relationship may have constant ratio at two points but not be proportional overall.	E.g. data (1, 3) and (2, 6) gives ratio 3, but adding (3, 8) breaks proportionality.
40	Investigate: is the perimeter of a square proportional to its side length? Is the area? Justify.	Perimeter: yes ($P = 4s$). Area: no ($A = s^2$, a power relationship).

Pack B — Answers

#	Question	Answer
1	The cost of 8 pens is 12 chf. Find the cost of 1 pen.	1.50 chf
2	4 pencils cost 1.20 chf. How much do 15 pencils cost?	4.50 chf
3	Test whether the table $(1, 3), (2, 9), (3, 27)$ represents a proportional relationship.	No — y/x is not constant (3, 4.5, 9).
4	For the proportional relationship $y = kx$, find $k$ given that $y = 35$ when $x = 5$.	$k = 7$
5	Using $y = 4x$, find $y$ when $x = 12$.	48
6	Using $y = 5x$, find $x$ when $y = 40$.	$x = 8$
7	On a graph of $y = 2x$, find the $y$-coordinate when $x = -3$.	$y = -6$
8	A graph of $y$ against $x$ passes through the origin and the point $(2, 10)$. State the proportionality equation.	$y = 4x$
9	In a direct-proportion graph, the line is straight and passes through which point always?	The origin $(0, 0)$.
10	Decide which equation represents direct proportion: $y = 4x$, $y = x + 4$, $y = x^2$.	$y = 4x$
11	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.	75 m in 3 min; 175 m in 7 min
12	A printer prints 24 pages in 3 minutes. How long to print 60 pages?	7.5 minutes
13	A school bus uses 15 L of fuel to travel 180 km. How far can it travel on 25 L at the same rate?	300 km
14	A cake recipe uses 2 eggs to make 6 cakes. How many eggs are needed for 15 cakes?	5 eggs
15	Test whether $y$ is directly proportional to $x$ given the table: x: 2, 5, 8; y: 6, 15, 24.	Yes — $k = 4$.
16	On the graph of $y = 3x$, what is the gradient of the line? Describe how the gradient relates to $k$.	Gradient = 3. Same general rule.
17	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 $t = 7$.	(a) 2.5 L/min (b) $V = 2.5t$ (c) 22.5 L
18	4 kg of apples costs 10 chf. How much do 7 kg cost?	17.50 chf
19	Two quantities are directly proportional. When $a = 6$, $b = 15$. Find $b$ when $a = 22$.	$b = 37.5$
20	The graph of $y$ against $x$ is a straight line through the origin with gradient 0.4. State the equation and find $y$ when $x = 15$.	$y = 0.4x$; $y = 6$
21	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 $y = 91$.	(a) $y = 7 x$ (b) $y = 84$ (c) $x = 13$
22	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.	A: yes. B: no.
23	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.	(a) 8.50 chf (b) 11 (with 6 chf left) (c) Yes.
24	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.	(a) 55 chf (b) ≈ 409 km (c) Gradient 0.22
25	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?	Alex (9 chf/h vs Brigit ≈ 8.33 chf/h).
26	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.	(a) 1 cm : 40 cm (b) 50 cm
27	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.	(a) $m = 2.5 \ell$ (b) 187.5 g (c) 96 cm
28	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.	See Pack A.
29	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) $k = 4/3$ (b) $y = 80/3 ≈ 26.67$ (c) $x = 22.5$
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) 12 km/L (b) 336 km (c) 2880 km
31	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.	See Pack A.
32	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.	(a) $w = 15$, $\ell = 30$ (b) 450 m² (c) Quadruples to 1800 m².
33	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.	A proportional. Same at $3x = 4x - 5$, $x = 5$, $y = 15$.
34	Investigate: if $y \propto x$ and $x \propto z$, show that $y \propto z$ and find the combined constant.	Same.
35	On a graph of $y = kx$, two points are $(3, p)$ and $(8, p + 20)$. Find $k$ and $p$.	$k = 5, p = 10$
36	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) 27 km/h (b) $d = 27t$ (c) 60.75 km (d) slope 27
37	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.	See Pack A.
38	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 $y$.	(a) $k = 3$ (b) $y = 21$ (c) See working
39	Show by counter-example that a relationship may have constant ratio at two points but not be proportional overall.	Similar — use a parabola or piecewise example.
40	Investigate: is the perimeter of a square proportional to its side length? Is the area? Justify.	Same.

Problems — Worked Solutions

**Direct proportion + a graph.** Distance is directly proportional to time for a rollercoaster: in 2 minutes it travels 36 m. (a) Find the unit rate in metres per minute. (b) Write an equation $d = \ldots$ linking distance $d$ (m) and time $t$ (minutes). (c) Use your equation to find how far the rollercoaster travels in 12 minutes. (d) The rollercoaster's track is 540 m long. How long does it take to traverse the full track at constant speed?

Answer

(a) 18 m/min (b) $d = 18t$ (c) 216 m (d) 30 minutes

(a) Rate $= 36 \div 2 = 18$ m/min. (b) $d = kt$ with $k = 18$, so $d = 18t$. (c) $d = 18 \times 12 = 216$ m. (d) $t = 540 \div 18 = 30$ minutes.

**Tour de Suisse training.** A cyclist trains over a 4-week plan, increasing her weekly distance by 20% each week. In week 1 she covers 80 km. (a) How far does she cycle in week 4? (b) Find her total distance over the four weeks (to the nearest km). (c) Express her week-4 distance as a percentage of her week-1 distance. (d) Is the relationship between distance and week number proportional? Justify.

Answer

(a) 138.24 km (b) 429 km (c) 172.8% (d) No — geometric (multiplicative), not proportional.

Weekly distances: $80, 96, 115.2, 138.24$. (a) Week 4 $= 80 \times 1.2^3 = 138.24$ km. (b) Total $\approx 429$ km. (c) $138.24 / 80 = 1.728 = 172.8\%$. (d) Not proportional — week-vs-distance is a geometric (exponential) sequence, not a constant-ratio (linear) one. Plotting gives a curve, not a straight line through the origin.

**Testing proportionality.** A student records distances driven by a car against fuel used: | Fuel (L) | 5 | 10 | 15 | 20 | |----------|---|----|----|----| | Distance (km) | 60 | 120 | 180 | 240 | (a) Show that distance is proportional to fuel and find $k$. (b) Write the equation. (c) Predict the distance for 28 L of fuel. (d) How much fuel is needed for 300 km?

Answer

(a) Ratios all equal 12 km/L (b) $d = 12f$ (c) 336 km (d) 25 L

(a) $60/5 = 120/10 = 180/15 = 240/20 = 12$. Constant ratio → proportional. (b) $d = 12f$ (km/L). (c) $d(28) = 336$ km. (d) $f = 300/12 = 25$ L.

**Recipe per person.** A recipe to feed 8 people uses: 480 g flour, 320 g sugar, 6 eggs. (a) Find the amount of each ingredient per person. (b) How much of each is needed for 12 people? (c) A chef has 1 kg flour, 800 g sugar, 10 eggs. What is the limiting ingredient if she wants to scale the recipe up exactly?

Answer

(a) 60 g flour, 40 g sugar, 0.75 eggs per person (b) 720 g flour, 480 g sugar, 9 eggs (c) Eggs

(a) Per person: $480/8 = 60$ g flour; $320/8 = 40$ g sugar; $6/8 = 0.75$ eggs. (b) For 12 people: flour $= 720$ g; sugar $= 480$ g; eggs $= 9$. (c) Max people from each: flour $1000/60 \approx 16.7$; sugar $800/40 = 20$; eggs $10/0.75 \approx 13.3$. **Eggs are the limiting ingredient** — max 13 people.

**Investigation.** Decide which of these is a proportional relationship and justify by sketching the graph. (a) Earnings $E$ at a flat hourly rate (no fee): $E = 12h$. (b) Taxi fare: $F = 4 + 1.5d$ (flat fee + per km). (c) Area of a square: $A = s^2$. (d) Cost of pencils at 30p each: $C = 0.30n$.

Answer

(a) Yes (b) No (has $y$-intercept 4) (c) No (curve, not linear) (d) Yes

Direct proportion has the form $y = kx$ (line through origin, no offset, power 1). (a) $E = 12h$ ✓ — proportional. (b) $F = 4 + 1.5d$ — line with $y$-intercept 4, not through origin → **not** proportional. (c) $A = s^2$ — curve (parabola), not linear → not proportional. (d) $C = 0.30n$ ✓ — proportional.

**Volume vs surface area.** A cube of side $s$ has volume $V = s^3$ and surface area $S = 6s^2$. (a) Are $V$ and $s$ directly proportional? Justify. (b) Are $S$ and $s$ directly proportional? Justify. (c) Find the ratio $V/S$ in terms of $s$. What happens to this ratio as $s$ grows?

Answer

(a) No (b) No (c) $V/S = s/6$ — grows linearly with $s$.

(a) $V = s^3$ is a cube relationship; doubling $s$ gives $V \times 8$, not $V \times 2$. Not proportional. (b) $S = 6s^2$ — quadratic. Doubling $s$ gives $S \times 4$. Not proportional. (c) $V/S = s^3 / (6s^2) = s/6$. As $s \to \infty$, this grows without bound. **Bigger cubes have proportionally more volume per surface area** — this is the "square-cube law" of biology and engineering.

**Currency conversion.** On a particular day, 1 USD = 0.92 CHF. (a) Write the conversion equation $C = kU$. (b) Convert 250 USD to CHF. (c) Convert 460 CHF to USD. (d) Is this a proportional relationship? Justify.

Answer

(a) $C = 0.92U$ (b) 230 CHF (c) 500 USD (d) Yes

(a) $C = 0.92U$ where $C$ is CHF and $U$ is USD. (b) $C = 0.92 \times 250 = 230$ CHF. (c) $U = 460 / 0.92 = 500$ USD. (d) Yes — passes through origin (0 USD = 0 CHF) and is linear with constant rate $k = 0.92$.

**Investigating inverse relationships.** Distance $d$, speed $v$ and time $t$ are linked by $d = vt$. (a) For a fixed distance of 60 km, complete the table: | $v$ (km/h) | 60 | 30 | 20 | 15 | |------------|----|----|----|----| | $t$ (h) | ? | ? | ? | ? | (b) Is $t$ directly proportional to $v$? Justify. (c) Define inverse proportion and write $t$ in the form $t = k/v$.

Answer

(a) 1, 2, 3, 4 (b) No (c) $t = 60/v$

(a) $t = d/v = 60/v$: $t = 1, 2, 3, 4$ h for $v = 60, 30, 20, 15$. (b) Not directly proportional — as $v$ doubles, $t$ halves. Direct proportion would mean both double together. (c) **Inverse proportion**: $t = k/v$ with $k = 60$. The product $vt = k = 60$ is constant.

**Population growth.** A town's population grows from 12 000 to 15 000 over 5 years. Assume linear (proportional) growth. (a) Find the average annual growth rate (in absolute numbers). (b) Write an equation $P = P_0 + kt$ and identify $P_0$ and $k$. (c) Estimate the population in 10 years. (d) Is **percentage** growth proportional? Justify.

Answer

(a) 600 people/year (b) $P = 12000 + 600t$ (c) 18 000 (d) No — exponential, not proportional.

(a) Growth $= 15000 - 12000 = 3000$ over 5 years → 600 per year. (b) $P = 12000 + 600t$. $P_0 = 12000$, $k = 600$. (c) $P(10) = 12000 + 6000 = 18000$. (d) Percentage growth (e.g. 5% per year) makes the population follow $P = P_0(1.05)^t$ — exponential, not proportional to $t$.

**Best buy investigation.** A market sells potatoes: - 1 kg sack: 2.20 chf - 2.5 kg sack: 5.25 chf - 5 kg sack: 9.50 chf (a) Find the price per kg for each pack. (b) Plot 'price' against 'mass'. Does the relationship form a straight line through the origin? Discuss. (c) Which sack is best value per kg? (d) Suggest a reason why larger sacks are cheaper per kg.

Answer

(a) 2.20, 2.10, 1.90 chf/kg (b) Not a straight line through the origin — bulk discount (c) 5 kg sack (d) Bulk discounts / lower per-unit packaging cost.

(a) 1 kg: 2.20 chf/kg. 2.5 kg: $5.25/2.5 = 2.10$. 5 kg: $9.50/5 = 1.90$ chf/kg. (b) Plotting (1, 2.20), (2.5, 5.25), (5, 9.50) gives points that are roughly on a straight line, but not perfectly through origin — the per-kg rate decreases as size grows, so the points actually bend slightly. (c) 5 kg sack at 1.90 chf/kg is the best value. (d) Larger sacks have lower per-unit packaging cost and incentivise bulk buying — common retail strategy. Strict direct proportion would imply equal per-kg cost.

**Petrol cost.** Petrol costs 1.80 chf per litre. Your car uses 7 litres per 100 km. (a) Write a formula for petrol cost $C$ (chf) as a function of distance $d$ (km). (b) Find $C$ for $d = 320$ km. (c) If petrol rises to 2.00 chf/L, find the new cost for 320 km. (d) For a 600-km trip, find the change in cost between the old and new petrol prices.

Answer

(a) $C = 0.126d$ (b) ≈ 40.32 chf (c) ≈ 44.80 chf (d) ≈ 8.40 chf

(a) Petrol per km: $7/100 = 0.07$ L/km. Cost per km: $0.07 \times 1.80 = 0.126$ chf/km. So $C = 0.126d$. (b) $C(320) = 0.126 \times 320 = 40.32$ chf. (c) New per-km cost: $0.07 \times 2.00 = 0.14$ chf/km. For 320 km: $44.80$ chf. (d) Old (1.80): $0.126 \times 600 = 75.60$ chf. New (2.00): $0.14 \times 600 = 84.00$ chf. Difference: $8.40$ chf extra.

**Hours and earnings.** A part-time worker is paid 18 chf/h for the first 20 hours per week and 25 chf/h for any extra hours. (a) Is total weekly pay directly proportional to hours worked? Justify. (b) Plot weekly pay versus hours from 0 to 30 hours. (c) The worker earns 510 chf one week. How many hours did she work?

Answer

(a) No — piecewise linear with a kink at 20 h (b) Two segments (c) 26 hours

(a) Not proportional — for $h \leq 20$: pay $= 18h$ (proportional in this range). For $h > 20$: pay $= 360 + 25(h - 20)$, which has a different gradient and a non-zero constant. (b) Two straight-line segments: from (0, 0) to (20, 360) with slope 18, then from (20, 360) to (30, 610) with slope 25. There is a "kink" at $h = 20$. (c) 510 chf is above the 20-hour threshold (360 chf). Extra: $510 - 360 = 150$ at 25 chf/h → $150/25 = 6$ extra hours. Total: $20 + 6 = 26$ hours.