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you should have an identity that says 2sinAcosA = sin2A. Use this.

So, 8sinAcosA becomes... ?

you now have an equation with one thing you do not know so you can solve it now. Have a go :-)

Jon

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you need to convert the 1 into quarters so that both terms have the same denominator.

So, how many quarters is one whole one? Answer 4/4

So we have 4/4 - 3/4 which is 1/4. That is your answer

Jon

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true

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negative number = -2 and -10
positive number = +6

so 2 examples - one with a digit value closer to zero than 6 and one larger than 6:

+6 - -2 --> +6 + 2 = +8
+6 - -10 --> +6 + 10 = +16

subtracting a minus number is like adding. So adding a number to an already positive number will always give you a positive answer.

cheers
jon.

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speed=distance/time.

let distance=d
let t = time going
let T = time returning

going: 30 = d/t --> t = d/30
returning: 50 = d/T --> T = d/50

so t+T = d/30 + d/50

but we are told that the total time of the trip (ie t+T) is 8hours, so we have 8 = d/30 + d/50

so we have a fractional equation to solve. If we multiply every term by 150, we get
and now we can cancel the fractions, leaving

1200 = 8d
d = 1200/8
d = 150miles

cheers
Jon.

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2% of 761,000

2% is 2 percent --> "per-cent" means "out of 100" so 2% is 2/100.

"of" means multiply, like "what is three lots of 4": answer is 3x4 = 12

so we get (2/100) x 761,000

if you want to write it simpler, then 2/100 is 0.02 so we get 0.02 x 761,000.

Use a calculator to get the value of this commission.
=============================================================

So, a 5% commission on 86,000 would be 5/100 x 86,000 or 0.05x86,000

cheers
Jon.

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L:M ratio is 7:4 which is a total of 11. So scale this up by a factor of 10 and we get a total of 110 from a ratio of 70:40.

cheers
Jon

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2 fundamental rules of indices are:

x and

using the second one, what happens if the powers m and n were the same?


would become


since, the left hand side has something divided by itself --> this is always 1.

So ANY number a to the power zero is always 1. And a can be zero, so

As to what this physically means, well that is a difficult question to answer and is not something i could answer but i know what the maths rules tell me and i believe them. That is the best i can give you.

cheers
jon.

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let the number be x, then:





So first find where . The solution to this is not straight forward and i don't have time to write it all down here, sorry. You can re-submit or ask your teacher, sorry.

cheers
Jon

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giving
giving and hence

After this, the next powers just return through the same 4 values, so etc.

So, is the same as which is -1.

Anything as a negative power, like means , so 1/(-1) is -1. That is your answer.

cheers
jon

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let the number be a, so:
Twelve less than 2 times a number is equal to 15 minus 7 times the number
12 less than 2a = 15 - 7a

so, 12 less than 2a:
what is 12 less than 20? answer is found by 20-12
what is 12 less than 30? answer is found by 30-12
so, what is 12 less than 2a? answer is found by 2a-12

so, 2a-12 = 15-7a is the equation.

2a-12 = 15-7a --> add 7a to both sides
9a-12 = 15 --> add 12 to both sides
9a = 27
a = 27/9
a = 3

cheers
jon

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These are all the 20 different versions of 5 Boys and 3 girls.
However, you can go one step further depending upon what your question implies...
in each version, the 5 boys can be arranged as b1...b2...b3...b4...b5 or other ways. These are a total of 5! = 5x4x3x2x1 = 120 ways
Also, the 3 girls can be sat 3! = 6 ways.
In addition, each version of the boys seating plan allows for all 6 of the girls seating plans so in total there, there are 120x6 = 720 versions.
so in total there are 20x720 = 14400 different permutations.
jon.

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this is just a fraction problem. We treat it just like numerical fractions:



if we had we would have to find the LCM of both 3 and 4 --> 3*4 which we can evaluate --> 12.
We do the same with (3x-2) and 2x. Their LCM is (3x-2)*2x which we cant evaluate so we leave it like that.

Next step is to multiply both fractions by 1 (so they remain unchanged) but write the 1 as a fraction:





Algebraically we cannot do all these steps. We can do the first one:

We can multiply the numerators together and the denominators too:

And now add them together since they have the same denominator:


So that is the fractions done. The rest is just manipulation of the algebra to try to simplify it:




or perhaps multiply out the denominator:


but we tend not to do this - we like things factorised. The answer is best quoted as

hope this helps
cheers
Jon.

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First, the Natural numbers are the counting numbers 1,2,3,4,5,... etc. Note no negatives and usually no zero although some definitions have zero in the Natural numbers.

A larger set of numbers are the Integers - all whole numbers. Integers include the Natural numbers plus zero and all negative whole numbers.

Rational numbers are those numbers that can be written as a fraction (a/b) where a and b are integers eg 6 is rational since it can be written as 6/1 or 12/2 etc. 12.3 is rational since 12.3 is 123/10. Even repeating decimals are rational like 0.232323 recurring is 23/99.

At first glance then you would assume that every number is rational. This is not the case. There are some special numbers that never repeat themselves but go on for ever (at least we have never found the end). The classic example is 3.1415.... which is given a special name since we cannot write it: as soon as we stop writing it, that becomes an approximation. We call that irrational number, pi and its symbol is

Similarly is a never ending, never repeating decimal so that too is not rational --> it is irrational. Same for . However, is 2 which is rational. But in general, the majority of square roots are irrational.

To complete this level of maths understanding, if we have the set of rational number and all the irrationals, then this is called the Real set of numbers - basically the real set covers any eventuality for the majority of maths. Beyond Real numbers are Complex numbers but that is another story entirely.

Hope this helps

cheers
Jon

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by re-arranging you mean?


by taking the denominator to the other side to remove the fraction part.
multiplying out the bracket
collecting all terms with "b" in on one side and all others on the other side
factorise to get the "b" by itself
removing the unwanted factor by dividing.

jon

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Let the number be written as xy where x is the number of tens and y is the number of units.

Mathematically, the number is 10x+y

So, --1
and --2

Simplifying eq2:



or

--1 is which then becomes


so using and knowing that y=2, we get x=4

--> number is 42


cheers
Jon.

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you need to start using brackets to group things together as your expression is one long list and very difficult to understand.

I am assuming you meant the following (after deciphering it for about 5 minutes):

which, in effect, is just:

Now we need to start factorising these 6 expressions where possible.




and now we can cancel out anything that appears on the top and bottom:





so the final version is
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Jon

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v=(-5 6) and w=(-2 -3).. both should be column vectors (ie one number on top of the other but i cannot write this here.

v.w = (-5x-2) + (6x-3)
v.w = +10 - 18
v.w = -8

so, yes you are correct :-)

cheers
Jon

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it doesn't look like much like this: . However, let , which means that . We now can re-write this as: which is a quadratic you can solve by factorisation. Once you have your answers for y, put them into to find your values for x.

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Jon

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and

fg(0) says wherevere x is, put zero as its value --> in the g(x) function. So:

becomes





Now put this value into f(x):
becomes




Now lets work out gf(1). This says to put 1 into f(x) for x. Work out the answer and then put that answer into g(x).

becomes




then
becomes



Does this help. Do you understand any better? Read my tutorial on functions - it hopefully explains things in more depth.

cheers
jon

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i have never used this rational zeros theorem - i have had a quick look on the web and i think i understand, so here goes.
q is the factors of 1 --> the coefficient of the x^4 term
p is the factors of 12

q = +1, -1
p = 1,2,3,4,6,12,-1,-2,-3,-4,-6,-12

Now, N=p/q gives all RATIONAL values that when put into the polynomial and give zero are solutions. For our values of p and q, we get N=1,2,3,4,6,12,-1,-2,-3,-4,-6,-12 as possible solutions to the polynomial.

However, putting each of these as x into the polynomial - none gives us zero when we raise it to the power 4 and then subtract 12.

So, the conclusion is that if there is a real solution, it has to be irrational. Thinking about this particular case, we have and so there is a real solution...there is a number that multiplies by itself 4 times to give 12 and it is between 1 and 2 since and .

cheers
Jon.

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you have not given enough info in the question to say for certain, but the espression (x^2+5x+6)(x+5) is an area times a length so the quadratic part could be the area of the bottom.

However, conversely it could be the area of the front and the x+5 is the "width".

cheers
Jon

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now - i know that factorises to , so we can write


and now we can cancel some terms down

or


this is as far as we can go with just real numbers - the quadratic does simplify but you need to know some Complex Numbers: its final version is


where the denominator has "i" and not a 1.

cheers
jon

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1.



2. think of as where . You can factorise this quadratic in y by looking and then re-write y as x^2 for each solution and find out what x is for each.

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Jon

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first, define the number to be, say, x

right then...
"seven less than a number" is written as: x-7: just think of 7 less than 10... we would say 3 because 10-7 is 3.

right then...
"square root of seven less than a number" is...

and finally...
"one eighth of the square root of seven less than a number" is...

and this equals 2. So,


Opposite (or reverse) of a square root is to square something, so square both sides:




Hope this helps?

cheers
Jon.

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Starting with i shall re-write it as .
I am going to prove this is correct instead by looking at the Left hand side and showing that it equals the right hand side .

So,

--> one of laws of Logs
--> multiply out brackets

now we are told that so sub that into what we have and we get:
re-arrange order so you can see the squared terms together
--> sub in the other equality
--> add together like terms
--> cancel out the 9's
use one of the Laws of Logs

which is what i set out to prove it equalled.

cheers
jon

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2 numbers that multiply to give 4 --> either 1x4 or 2x2

Added complication: the x^2 term has a 6 so either we have 6x1 or 3x2 there. We now need to mess around with these 2 pairings until we get 5.

3x2 and 1x4 could give us a 5 --> 3x1 and 2x4 could by having -3+8
no othere pairings give a 5

So (3x 4)(2x 1) is the order we need. We just need to figure out the signs. The 4 and 1 multiply to give -4 so we need 1 +ve and 1 -ve. So either:
(3x + 4)(2x - 1) or (3x - 4)(2x + 1)
Looking at the first version: -3x+8x --> +5x which is what we need, so this is the answer. The second version would give us -5x because the only difference is the order of the signs.

answer: (3x + 4)(2x - 1)

the more you do, the quicker/better you get.

cheers
Jon

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right angled triangle --> use Pythagoras. Find the height of the ladder in the first instance then subtract 3 from that answer and recalculate the floor distance using that new height remembering that the hypotenuse is fixed at 10.

The first part is a really quick calculation because you have the classic 3,4,5 Pythagorean sides scaled up--> 6,8,10

cheers
Jon

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time to be a little sneaky then...

or...

so think about the 2 sets of terms, bracketed:



and you notice that both factorised terms have a common term which can be factorised out, giving us:


The first term is a difference of 2 squares so:


there you go :-)

cheers
Jon

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2 approaches:
1. trial and error seeing as how the number is not so large:
23+1
21+3
19+5
17+7
15+9
13+11 ... there you go.

2. using algebra.

let x be the smaller of the 2 numbers. The larger number is 2 more than this (as you can see from the answers...but consecutive odd numbers will always have a difference of 2) so larger number is x+2
so x+x+2=24
2x+2=24
2x=22
x=11

so numbers are 11 and 13.

cheers
Jon

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solve means find the answer for the variable (here x)... You need an equals sign hence the word "equation"

The first one... you cannot find the value of x because you are not told what all this equals... no equals sign. All you can do is mess around with the expression and simplify it, as follows:





then perhaps:



but still, this does not tell you what value x is.

cheers
Jon