Ratios and Rates

Ratios and rates is the most practical topic in the course, and one of the most heavily examined in Standard 1. Questions come from maps, house plans, shopping, power and water bills, and cars.

• Ratios: their link with fractions, simplest form, the ratio of two quantities, and dividing a quantity in a given ratio.
• Scale: map scales and scale drawings; reading measurements from building plans (with their standard symbols); estimating materials and costs.
• Rates: the difference between ratios and rates, unit rates, converting rates, and comparing rates (best buys).
• Energy: power in watts and kilowatts, energy in kilowatt hours, and electricity bills; water use and water bills.
• Fuel consumption in L/100 km, to compare the efficiency of vehicles.

A ratio compares two or more quantities of the same kind, in the same units. It has no units itself.

A ratio is linked to fractions: in a class where boys : girls = 2 : 3, there are 5 parts altogether, so $\frac{2}{5}$ of the class are boys and $\frac{3}{5}$ are girls. Note that the ratio 2 : 3 is not the fraction $\frac{2}{3}$ of the class.

Simplest form

Write both quantities in the same units, then divide by the highest common factor.

Example. 1.2 kg : 450 g = 1200 g : 450 g = 120 : 45 = 8 : 3 (dividing by 10, then by 15).

Dividing a quantity in a ratio

1. Add the parts.
2. Divide the total by the number of parts to find one part.
3. Multiply by each number in the ratio.

Example. A $3600 prize is shared in the ratio 2 : 3 : 7.

Check: $600+900+2100=3600$.

A scale is a ratio of the length on the drawing to the real length. A scale of 1 : 200 000 means 1 cm on the map is 200 000 cm (2 km) in real life.

• Drawing → real: multiply by the scale factor, then convert units.
• Real → drawing: convert to the same units, then divide by the scale factor.
• Length conversions: 1 m = 100 cm, 1 km = 1000 m = 100 000 cm.

Example. On a 1 : 200 000 map, two towns are 6.3 cm apart. Real distance $=6.3\times 200000=1260000$ cm $=12.6$ km.

Scales can also be shown as a scale bar (a line marked 0, 1, 2 km), or in words ("1 cm represents 50 m"). Measure on the drawing with a ruler and compare with the scale bar.

Building plans

House plans are drawn to scale (often 1 : 100) and use standard symbols: a door is drawn as a gap in the wall with an arc showing which way it swings; a window is a thin double or triple line in the wall; stairs are parallel lines with an arrow. A plan usually has a scale, a north arrow and room labels. Measurements on plans are often written in millimetres: 3600 means 3600 mm = 3.6 m.

Estimating materials and costs

1. Use the scale to find the real lengths.
2. Calculate the area (or perimeter, or volume) you need.
3. Allow for waste if the question says so.
4. Round up when materials come in whole units (boxes, square metres, litres of paint).
5. Multiply by the price.

Worked example. A deck is drawn on a 1 : 50 plan as a rectangle 9 cm by 7 cm. Decking costs $65 per square metre and is sold in whole square metres.

Round up to 16 m², so the cost is $16\times 65=1040$ dollars.

Convert lengths before finding the area. Areas scale by the square of the scale factor, so multiplying an area on the plan by 50 would be wrong; it would need $50^2=2500$.

A rate compares quantities of different kinds, so it has units: km/h, $/kg, L/min, beats per minute. That is the main difference from a ratio.

A unit rate is the amount for one unit: $4.50 per kg, 75 km per hour.

Converting rates

Convert the top and the bottom units separately.

Example. A hose delivers 1.5 L per minute. In litres per hour: $1.5\times 60=90$ L/h.

Example. 90 km/h in m/s:

Shortcut: km/h ÷ 3.6 = m/s, and m/s × 3.6 = km/h.

Best buys

Compare unit prices. Example. Yoghurt: 600 g for $4.20 is $7.00 per kg; 1 kg for $6.30 is $6.30 per kg; 1.25 kg for $7.80 is $6.24 per kg. The 1.25 kg tub is the best value, just ahead of the 1 kg tub.

Speed, distance and time

Convert times to hours before using km/h: 2 hours 24 minutes is $2+\frac{24}{60}=2.4$ hours.

• Power is the rate of using energy, measured in watts (W) or kilowatts (kW). 1 kW = 1000 W.
• Energy used is measured in kilowatt hours (kWh): $\text{energy (kWh)}=\text{power (kW)}\times \text{time (h)}$
• Electricity is charged per kWh, often plus a daily supply charge.

Example. An 1800 W kettle is used for 6 minutes a day for 30 days, at 30 cents per kWh.

Example: a power bill. A household uses 640 kWh in a 92-day quarter. The usage rate is 31.5 c/kWh and the supply charge is $1.10 per day.

Water use

Water is billed per kilolitre (1 kL = 1000 L), often with a fixed service charge. Example. A shower uses 9 L per minute. An 8-minute shower each day for a year uses $9\times 8\times 365=26280$ L, which is 26.28 kL. At $3.10 per kL, that costs about $81.47.

In Australia, fuel consumption is given in litres per 100 km (L/100 km). A smaller number means a more fuel-efficient vehicle.

Worked example. A car used 41.6 L of petrol to travel 640 km.

For a 380 km trip it would use $6.5\times \frac{380}{100}=24.7$ L. At $2.04 per litre, the fuel would cost $24.7\times 2.04\approx \$50.39$

To compare cars, find the yearly fuel cost of each for the same distance, then compare the difference with any difference in purchase price.

• Writing a ratio without first making the units the same.
• Dividing by one number in the ratio instead of the total number of parts.
• Multiplying an area on a plan by the scale factor instead of converting the lengths first.
• Rounding materials down: you can't buy 15.75 m² of decking.
• Mixing W and kW, or cents and dollars, in energy calculations.
• Converting minutes to hours by writing 1 h 45 min as 1.45 h (it's 1.75 h).
• Calculating km per litre when the question wants L/100 km.

• Write the units at every step; they show your method and catch errors.
• For scale questions, say which direction you're going (plan to real, or real to plan).
• When a question asks which option is better value or more efficient, compare the same kind of rate for each option and finish with a sentence.
• Read bills and tables carefully: supply charges are per day, usage is per kWh or kL.

• Applications of measurement (Year 11): area, volume, capacity and unit conversions are used in every plan and water question.
• Formulas and equations (Year 11): speed, distance and time.
• Right-angled triangles (Year 12): scale drawings and maps are often combined with bearings and Pythagoras' theorem.
• Managing money (Year 11): household bills and the running costs of a car.