Ratios and Rates

Ratios and rates is a very practical focus area: maps, building plans, rainfall, fuel and power bills. It was worth 5 marks in NESA's sample HSC paper and has averaged about 9 marks a year in recent HSC papers, so it is worth getting fast and accurate at the conversions.

• Ratios: simplest form, the ratio of two quantities, dividing a quantity in a ratio, and the capture–recapture method.
• Scales: map scales, scale drawings and building plans; estimating materials and costs from a plan.
• Land and water: perimeter and area of land from a plan, aerial photo, radial survey or map, and rainfall volume $V=Ah$.
• Rates: unit rates, converting rates, comparing rates (best buys), and distance–time graphs.
• Fuel consumption in L/100 km, and energy in watts, kilowatts and kilowatt hours.

A ratio compares quantities of the same kind in the same units, such as 2 : 3. It is like a fraction: in a 2 : 3 split, the first share is $\frac{2}{5}$ of the total and the second is $\frac{3}{5}$.

• Simplest form: convert to the same units, then divide by the highest common factor. 750 mL : 2 L becomes 750 : 2000, then 3 : 8.
• Dividing in a ratio: add the parts, find one part, then multiply.

Example. Concrete is mixed from cement, sand and gravel in the ratio 1 : 2 : 3. How much of each is needed for 0.9 m³ of mix?

So 0.15 m³ cement, 0.3 m³ sand and 0.45 m³ gravel. Check: they add to 0.9 m³.

Capture–recapture

To estimate the size of an animal population: catch and tag some animals, release them, later catch a second sample and count how many are tagged. Assume the tagged proportion in the sample matches the whole population.

Example. 80 frogs are tagged and released. Later, 90 frogs are caught and 12 are tagged. Then $\frac{12}{90}=\frac{80}{N}$ so $N=\frac{80\times 90}{12}=600$ frogs (an estimate). This assumes the tagged animals mix evenly and none die or leave between samples.

A scale of 1 : 500 means 1 unit on the plan is 500 of the same units in real life: 1 cm represents 500 cm, which is 5 m.

• Plan to real: multiply by the scale factor, then convert units.
• Real to plan: convert to the same units, then divide.
• Areas scale by the square: a 1 : 500 plan shrinks lengths 500 times but areas 250 000 times. Convert each length to real size first, then find the area.

Building plans use standard symbols for doors, windows and so on, and are drawn to scale. Use measurements from the plan to estimate materials (tiles, paint, fencing) and their cost; round up when you must buy whole items.

Example. On a map with scale 1 : 25 000, a farm is a rectangle 3.2 cm by 1.5 cm. Real lengths: $3.2\times 25000=80000$ cm, which is 800 m, and $1.5\times 25000$ cm is 375 m. The area is $800\times 375=300000$ m², which is 30 hectares (1 ha = 10 000 m²).

Land area and rainfall

For land with an irregular boundary, use the trapezoidal rule from Year 11, or split the block into triangles; with a radial survey, use $\frac{1}{2}ab\sin C$ for each triangle (see Trigonometry). Then rainfall collected is:

$V$ is the volume, $A$ the area and $h$ the depth of rain. Put everything in metres first: 1 m³ = 1000 L.

Worked example: rain on a block

A site plan has scale 1 : 500. A rectangular block measures 6 cm by 4.4 cm on the plan. How many litres of rain fall on it during 12 mm of rain?

Step 1: real lengths.

Step 2: area, and rain depth in metres.

Step 3: volume, then litres.

About 7920 litres of rain fall on the block.

A rate compares quantities of different kinds, so it has units: km/h, $/kg, L/min, beats per minute. A ratio has no units. A unit rate is "per one" of something: $7 per kg, 72 L per hour.

Converting rates

Convert the top and bottom units separately.

A shortcut for speed: km/h ÷ 3.6 = m/s, and m/s × 3.6 = km/h. For a tap flowing at 650 mL per minute: $0.65\times 60=39$ L per hour.

Best buys

Compare the price for the same amount. A 600 g pack for $4.20 costs $4.20÷6=0.70$ per 100 g. A 1.25 kg pack for $7.65 costs $7.65÷12.5=0.612$ per 100 g. The 1.25 kg pack is the better buy (about 61 cents per 100 g, compared with 70 cents).

Distance–time graphs

On a distance–time graph, the gradient is the speed. A steeper line means faster travel, and a horizontal line means the object is stopped. Speed between two points = change in distance ÷ change in time.

Fuel consumption

Fuel consumption is given in litres per 100 km (L/100 km). Smaller is more efficient.

Example. A car uses 7.8 L/100 km. For a 380 km trip it uses $\frac{7.8\times 380}{100}=29.64$ L. At $1.95 per litre, that costs $29.64\times 1.95=57.80$ dollars, so $57.80. To compare a second car that used 31 L for 450 km, find its rate: $\frac{31}{450}\times 100=6.9$ L/100 km (1 d.p.), so the second car is more efficient.

Energy

Power is measured in watts (W) or kilowatts (kW): 1000 W = 1 kW. Energy used is measured in kilowatt hours (kWh): a 1 kW appliance running for 1 hour uses 1 kWh.

Worked example: running a TV

A 150 W television is used for 5 hours a day for a year (365 days). Electricity costs 32 c/kWh. Find the cost.

It costs $87.60 a year. The two conversions that earn the marks are watts to kilowatts and cents to dollars.

• Writing a ratio with different units, such as 750 mL : 2 L as 750 : 2.
• Dividing in a ratio by one of the parts instead of the total number of parts.
• Squaring the scale for a length, or not squaring it for an area found on the plan.
• Using rain depth in millimetres with area in square metres. Convert to metres.
• Multiplying watts by hours and calling it kWh. Divide by 1000 first.
• Calculating fuel consumption as km per litre when the question wants L/100 km.

• Write units on every line; many marks in this topic are lost in unit conversions, not the maths.
• For a best-buy question, show the unit price of each option before you choose.
• In a multi-step problem (plan → area → rain → tanks needed), set out each step on its own line with a label.
• Round up for things you must buy whole (bags, tanks, tins of paint), and say so.
• Read scales carefully: 1 : 200 and 1 cm : 2 m mean the same thing.

• Applications of measurement (Year 11): unit conversions, area, volume and the trapezoidal rule.
• Trigonometry: radial surveys and areas of triangular blocks.
• Linear relationships (Year 11): the gradient of a line is a rate; direct variation.
• Managing money (Year 11): household bills use energy and water rates.