08Arates

= Identifying Rates =


 * We often want to study how things change
 * grow, move, shrink, fill, empty, rotate, etc


 * The ** rate ** at which something is changing is important


 * The ** rate ** describes how much one quantity changes with respect to another quantity


 * For example, if a bucket is being filled at a rate of 0.8 litres per minute
 * the quantity that is changing is the volume of water in the bucket
 * the rate at which the bucket is filling is with respect to the time (in minutes)
 * a rate of 0.8 litres per minute means that in one minute, 0.8 litres will have flowed into the bucket


 * For example, the price of petrol is $1.48 per litre
 * the quantity that is changing is the price you pay for a tank of petrol
 * the price of petrol is with respect to the volume of petrol (in litres)
 * a rate of $1.48 per litre means that if you buy 1 litre of petrol, you will pay $1.48


 * For most rates, you can recognise them by
 * the word "per" ... (or)
 * the symbol "/"


 * For example,
 * A speed of 75 km/hour (75 km per hour) is a rate
 * A rental cost of $20/week ($20 per week) is a rate
 * A test score of 83% (83 per cent = 83 points per 100 points) is a rate


 * The exception to the above (per or /) is when the rate is given as a gradient
 * For example
 * the roof of the house shown here has a gradient of 3 in 5
 * this means the height of the roof increases by 3 units for every 5 units across
 * this ratio will be true regardless of what units you measure it in
 * up 3m for every 5m across
 * up 3 feet for every 5 feet across
 * up 3 cubits for every 5 cubits across
 * the units are the same for both directions,
 * so a gradient of 3 in 5 is the same as
 * a change of 3 units up __per__ 5 units across

math . \qquad \text{A gradient of 3 in 5 could also be written as a fraction or a percentage:} \\ .\\ . \qquad \qquad \text{Gradient } = \dfrac{3}{5} \qquad (or) \\. \\ . \qquad \qquad \text{Gradient } = 60\% math


 * For example,
 * a straight line with the equation y = 3x + 2, has a gradient of 3
 * this means the y value increases by 3 units
 * for every time the x value increases by 1 unit

math . \qquad \text{A gradient of } 3 \text{ could also be written as the fraction } \dfrac{3}{1} \\. \\ . \qquad \qquad \text{ie a change of 3 units up per 1 unit to the right} math .