How To Solve Hard Word Problems In Algebra
The professor explains that 'plus' in mathematical notation can show up in word problems as
Hi, I'm Doctor Shah. I was the National Lecture competition winner in 1989 and I'm the Maths master at Mathscool. Now, ready for a new way of doing maths? Word problems in algebra. So we're going to learn to convert the English in a word problem into maths. So if you see, that would be an equals sign in mathematical notation. And, some, greater than, longer, heavier, those kind of things are a plus in mathematical notation. Minus, less, difference, less than, below, those sort of things would be a minus sign. Multiply by, product and of are all multiply by signs. 'Divide', 'shared', 'per' would be a divide by sign. 'Times itself' is another way of saying squared so that would be square. And sometimes you'll see consecutive numbers. If we call the first of our consecutive numbers n, then the second of our consecutive numbers would be n plus one. So n would be an unknown number, n plus one would be the next number on from that. Similarly, if it said, instead of saying consecutive numbers, if it said an even number and we did didn't know what the even number was, we would call it two n. The two in front of it making sure that it's even, because whatever n is, it will be multiplied by two to give us an even number and if we wanted an odd number, we would call it 2n plus one. Make it even, add one and that will ensure it's odd. And if we wanted consecutive odd numbers, then if our first consecutive number was 2n + 1, the next odd number would have to add two to that, so that would be 2n + 3 would be our consecutive odd numbers. You don't have to do it as 2n + 1 and 2n + 3, you can also do it as 2n - 1 and 2n + 1. So in other words, subtracting two from that one to get your other odd number. Okay, so let's move on to some examples where the question is written in English and we have to convert it into mathematical equations before we can solve it. So, a slightly more involved example this time. Fred is more popular than Pete, together they have 66 friends, Fred has 6 fewer than the square of the number Pete has and we need to find out how many friends Fred has. So this is slightly more complicated, so I'm going to start off by defining some variables. I'm going to say the number of friends Fred has is called F, and the number of friends Pete has is called P. Okay, so the first thing I see here is that it says together they have 66 friends. So together would mean adding, so F plus P equals 66. Now there's my first equation, but I can't solve that equation because it has two unknowns in it. So I'm going to have to look for another equation. And the next thing they tell me is Fred has 6 fewer than the square of the number that Pete has. So Fred, F, has 6 fewer (has is equals) 6 fewer than the square of the number that Pete has. Now, when you're doing fewer, which is subtract, you must be careful to get it the right way round. So the square of the number that Pete has is P squared and then I must minus the 6 afterwards, so 6 fewer doesn't mean 6 subtract P squared, it means P squared subtract 6. Okay, so we have two equations, simultaneous equations, which means we have got to substitute one of the equations into the other. This equation already has F as the subject, so what I'm going to do is replace this F with this equation here, so I'm going to put that equation into there. So that P squared minus 6 is going to replace that F there. And then rub that out, and so there's my equation, now with only one variable in it, and we're going to solve this equation. First of all, take everything over to one side, so minus 66. This is a quadratic equation, so I want to write them in the correct order. I'm going to write my P squared first, plus P second and my minus 72 last. And then, now I'm going to factorize this, this is a quadratic, so we know we need to factorise it. Two sets of brackets, to get a P squared, I need a P at the front of both of those. 72, minus 72, could be 9 times 8.