2 13. ProportionWhat does proportion mean, and what’s that funny fish symbol?If two variables are proportional to each other, it just means that they are related to each other in a specific way.The funny fish symbol just means “is proportional to”1. Two types of ProportionAgain, how many of these you need to worry about depends on your maths set, and your exam board, and stuff like that, but here are the two main types of proportion:(a) Direct ProportionBoth variables increase or decrease together(i) LinearGraphFancy LingoExampleyx could be the number of KitKat Chunkys that you buyy could be the total cost of those KitKat ChunkysAs the number you buy increases, so too does the total costy is proportional to xy is directly proportional to xy varies directly as x variesx
3 (ii) Quadratic (iii) Cubic Graph Fancy Lingo Example x could be the amount of money you spend advertising a gigy could be the number of people who turn up to the gigAs the amount of advertising increases, word of mouth quickly spreads, and the number of people who go to the gig goes up by a lot.yy is proportional to x2y is directly proportional to x2y varies directly as x2 variesx(iii) CubicGraphFancy LingoExampleyx could be the amount of time you spend on mymaths.co.uky could be your maths exam markAs the amount of time you spend revising on the site increases, everything begins to fall into place, and your marks just get higher and higher with each extra minute!y is proportional to x3y is directly proportional to x3y varies directly as x3 variesx
4 (b) Inverse Proportion As one variable goes up, the other goes down (i) InverseGraphFancy LingoExampleNote: Not a straight line!x could be the number of people you convince to join you on a road tripy could be the amount each person must pay for petrolAs the number of people in the car increases, the amount everyone has to pay fallsyy is inversely proportional to xx(ii) Quadratic InverseGraphFancy LingoExamplex could be the number of hours you spend watching Eastendersy could be your number of brain cellsAs the hours increase, your brain cells disappear and an increasing rate!yy is inversely proportional to x2x
5 2. How to tackle proportion questions Whatever the question, whatever the type of proportion, this method will never let you down… hopefully!Method1. Decide on the type of Proportion- Direct or Indirect?- Linear, Quadratic or Cubic?2. Write the expression with the funny fish sign3. Make the expression into an equation by using = and k4. Use the numbers they give you to find out the value of k5. Write down the formula6. Answer the questions!And now I’ll take you through some pretty nasty examples, but each time we will use the same method, and everything will be fine… I promise!
6 (a) y when x = 6 (b) x when y = 66 Example 1y is directly proportional to x. Given that y = 12 when x = 4, calculate the value of:(a) y when x = 6 (b) x when y = 661. Okay, we’re in luck! The question has told us that we are dealing with direct proportion, and unless it says otherwise, we can also assume that it is linear.2. The expression to say that y is directly proportional to x is:3. Okay, this is the key bit. As I said at the start, proportional means related to in a specific way. Indeed, once you decide what kind of proportion you are dealing with, all you need to do to get from x to y is to multiply by a number, which we call k.Rule: Replace the sign with = and multiply the right hand side by k4. Now we use the numbers in the question and put them in our formula… when y = 12, x = 4And a little bit of rearranging gives us the value of k5. So now we have our formula:6. And now life is easy because we just need to substitute in numbers and maybe (for b) rearrange!!(a) Find y when x = 6(b) Find x when y = 66So…So…
7 Example 2The variables p and q are related so that p is directly proportional to the square of q. Complete the table of valuesp0.52q12271. Again, we’re in luck! The question has told us that we are dealing with direct proportion, and it also mentions the word “square” which means we are dealing with quadratic.2. The expression to say that p is directly proportional to the square of q is:3. Rule: Replace the sign with = and multiply the right hand side by k4. Now we look at the table and see what it tells us…. Well, it looks like when p = 2, q = 12, so let’s put that information into our formula!Note: I prefer a fraction to the horrible decimalAnd a little bit of rearranging gives us the value of k5. So now we have our formula:6. And now life is easy because we just need to substitute in numbers and maybe (for b) rearrange!!(a) Find p when q = 27(b) Find q when p = 0.5So…So…
8 Example 3z is inversely proportional to t. Given that when t = 0.3 the value of z = 16, find the value of z when t = 0.51. Happy days! The question has told us that we are dealing with inverse proportion, and unless it says otherwise, we can also assume that there are no nasty squares or cubes around.2. The expression to say that z is inversely proportional to t is:3. Rule: Replace the sign with = and multiply the right hand side by k, but be careful here!Note: We are multiplying by k, so it goes on the top!4. Now we use the numbers in the question and put them in our formula… when z = 16, t = 0.3And a little bit of rearranging gives us the value of k5. So now we have our formula:6. And now life is easy because we just need to substitute in numbers:(a) Find z when t = 0.5So…
9 (a) Describe the variation using Example 4 – Nightmare!!!!(a) Describe the variation using(b) Find the equation connecting the two variablesy5125x251. Okay, we have a problem! The question doesn’t tell us what type of proportion we are dealing with… but wait a minute, as x goes up (from 5 to 25), so does y (from 5 to 125), so it must be Direct.Now, look at this…x 25orx 52But what type of direct proportion?...Well, to get from the first x value (5) to the second (25), you must multiply by 5. But for the y values, you must multiply by 25. Now, 25 is 52, so we must be talking quadratic proportiony5125x25x 5(a) The expression to say that y is directly proportional to the square of x is:3. For the equation, we do what we always do… Rule: Replace the sign with = and multiply the right hand side by k4. Now we have two sets of values to choose from to find k! I am going to choose the second, but it doesn’t matter, the answer will be the same… when y = 125, x = 25(b) So… the equation is:Which works with both sets of values in the table!
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