8.2 Ratio and Rates

**World triathlon.** Out of 2000 competitors at the World Triathlon, 1650 finished the race. The ratio of female to male finishers was $20:13$. (a) How many men finished? (b) How many women finished? (c) What percentage of all competitors (including those who did not finish) was female finishers?

**Reverse percentage — sale price.** A bike is on sale for 590 chf. This is a 26% discount on the original price. (a) Find the original price to the nearest centime. (b) The shop later raises the sale price by 26%. Show that this does **not** restore the original price, and find the difference.

**Carbon footprint (a school trip to Nice).** Geneva to Nice round-trip distances and emissions data: - Plane (round-trip): 80 kg CO₂ per passenger. - High-speed rail: 0.014 kg CO₂ per passenger per km; one-way distance 547 km. - Small car: 0.134 kg CO₂ per km; one-way distance 617 km. For each mode, calculate the round-trip carbon footprint **for one person**. (Note: for the car, assume 1 person is in the vehicle.) Then express each as a percentage of the largest footprint.

**Mixing paint.** A shade of green is made by mixing blue and yellow paint in the ratio $3:5$. (a) How much yellow is needed to mix with 600 mL of blue? (b) How much blue is needed if a decorator wants to make exactly 4 litres of green paint? (c) The decorator only has 1.2 L of blue and 1.8 L of yellow. What is the largest amount of correctly-mixed green paint they can make? Which colour is the limiting one?

**Currency rates.** On a particular day, 1 chf = 1.04 EUR. (a) Convert 250 chf to euros. (b) Convert 312 EUR back to chf. (c) A currency exchange charges a 2% commission on euro purchases. How many euros do you actually receive when changing 250 chf?

**Recipe scaling — ratio.** A recipe for 6 brownies uses: - 120 g flour - 90 g sugar - 60 g butter - 2 eggs (a) Find the ratio flour : sugar : butter (by mass) in simplest form. (b) How much of each ingredient is needed to make 15 brownies? (c) A baker has 500 g of flour, plenty of everything else. What is the maximum number of brownies she can make?

**Combined ratio.** In a school the ratio of boys to girls is $4:5$. Of the boys, the ratio left-handed to right-handed is $1:9$. Of the girls, the ratio is $1:11$. (a) What fraction of the boys are left-handed? (b) What fraction of the **whole school** is left-handed? Give the answer as a fraction in simplest form.

**VAT and prices.** In Switzerland VAT is 7.7%. (a) A laptop has a pre-tax price of 1000 chf. What is the price with VAT? (b) A printer has a final (with VAT) price of 538.90 chf. What was its pre-tax price? (c) A customer asks for a 10% discount on the **pre-tax** price of a 1200-chf monitor; VAT is then added. Find the final price.

**Best value packs.** A shop sells cereal in three pack sizes: - 250 g pack: 3.20 chf - 500 g pack: 5.90 chf - 1 kg pack: 11.20 chf (a) Find the price per 100 g for each pack. (b) Which pack is best value? Justify. (c) The 500 g pack is offered with 'buy two, get one free'. Recompute its effective price per 100 g and compare with the others.

**Compound percentage growth.** A small forest has 800 trees in 2026. The population grows by 8% each year. (a) How many trees in 2027? (b) In 2030? (c) In which year does the forest first have more than 1500 trees?

**Unit-rate comparison.** Two cyclists train on Lake Geneva. - Alex covers 60 km in 2 h 30 min. - Brigit covers 45 km in 1 h 50 min. (a) Calculate each cyclist's average speed in km/h. (b) Express the ratio of Alex's speed to Brigit's speed in simplest form. (c) If they each ride at their average speed for 3 hours, how far apart are they when starting from the same point in opposite directions?

**Mixed-strategy savings.** A bank offers two savings plans. - Plan X: 5% interest per year for 3 years. - Plan Y: Year 1 8%, Year 2 4%, Year 3 3%. (a) If you deposit 1000 chf in each plan, find the value after 3 years. (b) Which plan gives the higher return? Justify by computing the difference.