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# Some Fun With Maths

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I was having some fun with some people on facebook and thought about bringing the idea here.

Basically, I was posting some math problems, the idea is of course to solve it.

It went like this -

Let's see how good your maths is. I have created a problem and solved it myself so I know the answer.

simplify terms (involving fractions) and solve for $k$

$\frac{kr}{m} - S + (\frac{5}{ab} - \frac{3}{b^2})= T$

values of $b, r$ and $a$ are

$b = 2$

$r = 20$

$a = \sqrt{\frac{\frac{4 \cdot 154}{22}}{7}}$

I won't lie, some of it might be tricky. Some people particularly had trouble finding the value of $a$.

Enjoy!

Edited by Aethelwulf
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The value of [imath]a[/imath] isn't important.   If it's 14, a solution for k is: $k = \frac{m \left(7T + 7S + 4 \right)}{140}$   If it's 2, a solution is: $k = \frac{m \left(2T + No, addition is associative, so the equation [math] \frac{kr}{m} - S + (\frac{5}{ab} - \frac{3}{b^2})= T$ may validly be rewritten $\frac{kr}{m} - S + \frac{5}{ab} - \frac{3}{b^2}= T [/ma whatever you intend, [math]6 \frac{2}{\frac{4}{8}} = 6\frac{2}{\frac{8}{4}} = \frac{6 \cdot 2 \cdot 8}{4} = 24$ does not read well; it is ambiguous in the first form whether 6 is a multiplier or

Ok, thought since you gave numbers you wanted some numeric value for k...otherwise I agree completely with belovelife and think it is just basic algebra...

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kr/m - s + 1/2 = t

20k/m = s + t -1/2

20k=m( s+t-1/2)

k= m(s+t-1/2)

...... ----------

..........20

Nice try but it is wrong.

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Ok, thought since you gave numbers you wanted some numeric value for k...otherwise I agree completely with belovelife and think it is just basic algebra...

There is some basic algebra involved - but I also gave a numerical task. The previous poster was very close, the actual answer is

$k = Tm+Sm -1$

The numbers simplify, so I am not sure where the previous poster got 1/2 from... try it again.

Edited by Aethelwulf
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the sqrt is 2

then brackets give 5/4 -3/4=2/4=1/2

That's not right. the value of $a$ is what gave other people problems as well. Also, that is not how you simplfy what is inside the paranthesis.

The way you divide a number by a fraction is like this

$6 \frac{2}{\frac{4}{8}} = 6\frac{2}{\frac{8}{4}} = \frac{6 \cdot 2 \cdot 8}{4} = 24$

So when you divide a number by a fraction, you have to flip the fraction around. So let's go back to our example

$a = \sqrt{\frac{4 \cdot 154}{\frac{22}{7}}}$

swap the fraction round

$a = \sqrt{\frac{4 \cdot 154}{\frac{7}{22}}}$

calculate

$4 \cdot 154 \cdot 7 / 22 = 196$

Take the square root of this and it is 14.

Edited by Aethelwulf
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How do you simplify what is in the brackets?

That's easy

It is in paranthesis so you solve this part first $(\frac{5}{ab} - \frac{3}{b^2}) = \frac{5b - 3a}{ab^2}$ now plug in the values. Then simplify by plugging in the value for $r$ when you get the chance.

Edited by Aethelwulf
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i think the original problem could have been visually represented different, i thought the same thing as sanctus

the original representation of (a)

i should say

Edited by belovelife
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i think the original problem could have been visually represented different, i thought the same thing as sanctus

the original representation of (a)

i should say

You both wouldn't have needed to think anything if it had been calculated right. The rule for dividing whole numbers by fractions is well-known.

http://www.cimt.plymouth.ac.uk/resources/help/miscon5.pdf

An example from this site is

$3/1/4$

The answer they give is 12. Use the rule I gave, remember to swap the fraction about

$3/4/1$

is

$3 \cdot 4/1 = 12$

It is perfectly consistent and nothing wrong with any kind of representation. I did warn you it could be tricky.

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you should end up with

$kr + \frac{36}{56} = m(T + S)$

right? Plug in the value for $r$

you should end up with

$k + \frac{56}{56} = m(T + S)$

(I noticed in the answer I gave I forgot to plug in the value of 1 (I'll change it now), so what you have for you final answer is

$k = m(T + S) -1$

Edited by Aethelwulf
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I'll write up a calculus problem later but I need to dash for now.

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wulf: I do not agree. You say that $\frac{x}{\frac{y}{z}}\equiv \frac{x}{\frac{z}{y}}$. You should see right away that this is wrong.

What you want to say I guess is $\frac{x}{\frac{y}{z}}\equiv \frac{x}{1}\cdot{\frac{z}{y}}$, so you have the swapping around you mean.

In general you have the rule $\frac{\frac{x}{k}}{\frac{y}{z}}\equiv \frac{x}{k}\cdot \frac{z}{y}$.

Applying this to your a (squared): x=4*154=616, k=22, y=7, z=1 hence it follows: $\frac{616}{22}\cdot \frac{1}{7}=28\cdot\frac{1}{7}=4$.

Hence a=2.

But re-reading through your posts, you are not coherent on where you put the main fraction!! And this changes everything, as written in the opening post you have:

$a^2=\frac{\frac{4\cdot 154}{22}}{7}$ and then belovelife and I are right.

Where you say that we are wrong you actually develop:

$\frac{4\cdot 154}{\frac{22}{7}}$ which is different and leads to a =14 as you say.

So it is your typo in OP that started all this discussion ;-)

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How where you right? You are still doing it wrong.

Belovelife has calculated

4*154 then divides that by 22 and then by 7 - that does indeed give you 4 but that isn't the correct operation for dividing whole numbers by fractions. Let us for a moment, say we where doing it the other way around, say we where dividing a fraction by a whole number, that is still not what you get.

Dividing a fraction by a whole number, let's say we had

2/5/4 = 2/ 5 *4 = 2/20 = 1/10

So in my example

616/22/7 = 616/ 22*7 = 616/154 = 308/77

If you even tried both dividing a whole number by a fraction and a fraction by a whole number, one would be able to work out which one was probably intended.

Edited by Aethelwulf
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you should end up with

$kr + \frac{36}{56} = m(T + S)$

right?

I don't think so. Starting with post #1's

$\frac{kr}{m} - S + (\frac{5}{ab} - \frac{3}{b^2})= T$

I get

$k = \frac{m(T +S -\frac{5}{ab} +\frac{3}{b^2} )}{r}$

Substituting the givens [imath]a=14, b=2, r=20[/imath] and simplifying, that's

$k = \frac{m(7T +7S +4)}{40}$

Evaluating the original equation with sample values for the m, T, S, and k calculated with my result checks out. Using your doesn't. I think you've erred, Aetherwolf.

Edited by CraigD
Replaces incorrect given a=2 with correct a=14

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