8.3 Proportions

**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?

**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.

**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?

**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?

**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: $C = 0.30n$.

**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?

**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.

**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 $t = k/v$.

**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.

**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.

**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.

**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?