PowerPoint Presentation: Putting everything in proportion Proportional relationships tell us that two variables scale with each other in a predictable way: If you drive twice as fast, you'll go twice as far in a given time: Speed and distance are directly proportional to each other. If you drive twice as fast, it takes half the time to go a set distance: Speed and time are inversely proportional to each other. To show proportionality we use a Greek alpha symbol in place of an =. Sometimes we use a constant ” k ” to show that need to either multiply or divide something by our other variable. The most common and best example of this is of ‘Builders and Time Taken’ . More the Builders , lesser time taken for building a house ; Less builders , more time will be taken to build the house. The equation for any inverse proportion is as follows: a α K y ( The symbol ‘ α ’ is not the letter a but a Greek letter used to represent proportions in maths ). K is a constant which is always present in any type of proportion and has a fixed number , in the builder and time taken example the constant would be the time it takes one builder to make a house.
Direct Proportionality: Direct Proportionality If two things are directly proportional to each other, they will increase and decrease in the same proportion to each other - double one, double the other; increase one by 12%, the other increases by 12%. We can find the exact link by looking at the numbers: If we know that the cost of a taxi journey is directly proportional to the distance travelled we can use any pair of values to find the relationship between cost and distance: If a 300 mile journey cost £ 1200, we could work out that a 30 mile taxi ride would cost £ 120, and a 3 mile journey would be £ 12 - meaning that the trip costs £ 4 a mile. We could then find the cost of any distance travelled, or how far we could get for a given price. In this example, our constant k = 4, as the cost = 4 x the distance in miles.
Graph of Direct Proportion: Graph of Direct Proportion As we know that in Direct Proportion if one variable increases the other variable will also increase, and both of them increase at the constant rate which keeps the line of the graph straight.
Inverse Proportionality : Inverse Proportionality Inverse proportionality is a similar idea, but one of our variables decreases as the other increases . The changes are still proportional, but inversely - double one the other variable gets halved, one gets four times smaller, the other gets four times bigger.
Graph of Inverse Proportion: Graph of Inverse Proportion In inverse proportion One variable increases Where-as the other one Decreases or vice versa. So this does not allow the graph to go on in a straight line from the bottom to the top, the graph of inverse proportion will always show the line from top to Bottom.
PowerPoint Presentation: Just a fact on proportionality: m (number of men) w (total weight of men, in pounds) 0 0 1 160 2 330 3 475 4 655 Note that, in real life, these relationships are not always exact! For instance, suppose m is the number of men in the room, and w is the weight of all the men in the room. The data might appear something like this: Not all men weigh the same. So this is not exactly a direct proportion. However, looking at these numbers, you would have a very good reason to suspect that the relationship is more or less direct proportion. How can you confirm this? Recall that if this is direct proportion, then it follows the equation w = km , or w / m = k . So for direct variation, we would expect the ratio w/m to be approximately the same in every case. If you compute this ratio for every pair of numbers in the above table, you will see that it does indeed come out approximately the same in each case.