Hey guys.

I'm just totally drawing a blank here...

$${\frac{{\mathtt{2}}}{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{X}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\mathtt{5}}}} = {\mathtt{\,-\,}}{\frac{{\mathtt{17}}}{{\mathtt{30}}}}$$

When I solve these equations I like to turn everything to decimals and solve them, but here they want the answer in fraction form. I can figure out that X = -2.05 then turn that into X = -2 & 1/20 but can you show me to work this out keeping everything in fractions?

Thanks,

~Ninja

NinjaDevo
Jun 24, 2014

#3**+10 **

It is always best to leave things in terms of fractions for a most exact answer. The first thing that is tough about this is that you dont have a common denominator, so let's fix that!

We want out common denominator to be 30 here because it is easy to turn 3 and 5 into thirty with multiplication. Multiply $${\frac{{\mathtt{2}}}{{\mathtt{3}}}}$$2/3 by 10/10 (which you should notice is legal because we are just multiplying by one) and multiply 4/5 by 6/6. This gives us a new expression

(20/30)x+(24/30)=-17/30

Let's subtract (24/30) from both sides to get (20/30)x=-41/30

Now we can multiply both sides by 30 so that there are no longer any fractions

20x=-41 And now we divide by 20 to get x=-41/20 which is exactly the same as -2.05

jboy314
Jun 24, 2014

#1**+10 **

Subtract 4/5 from each side ..... we have...

(2/3)x = -(17/30) - (4/5) and 4/5 = 24/30 ....so we have

(2/3)x = -(17/30) - (24/30)

(2/3)x = -41/30 multiply both sides by (3/2)

x = (-41/30) * ( 3/2) (cancel the 30 and the 3 )

= (-41/10) * (1/2) = (-41/20)

And that's it, ND.....

CPhill
Jun 24, 2014

#2**+5 **

Thanks CPhill.

I'm not sure why I was getting tripped up on that one, but you explained it well.

*"Thumbs up and points!"*

NinjaDevo
Jun 24, 2014

#3**+10 **

Best Answer

It is always best to leave things in terms of fractions for a most exact answer. The first thing that is tough about this is that you dont have a common denominator, so let's fix that!

We want out common denominator to be 30 here because it is easy to turn 3 and 5 into thirty with multiplication. Multiply $${\frac{{\mathtt{2}}}{{\mathtt{3}}}}$$2/3 by 10/10 (which you should notice is legal because we are just multiplying by one) and multiply 4/5 by 6/6. This gives us a new expression

(20/30)x+(24/30)=-17/30

Let's subtract (24/30) from both sides to get (20/30)x=-41/30

Now we can multiply both sides by 30 so that there are no longer any fractions

20x=-41 And now we divide by 20 to get x=-41/20 which is exactly the same as -2.05

jboy314
Jun 24, 2014

#4**+5 **

Thanks jboy.

I get what your saying with the whole fractions being the best answer thing. When I just turn everything into decimals on equations like this, sometimes I get answers like X = .454545 and I have no idea how to turn that into a fraction...but it's easy to turn a fraction into a decimal!

Thanks for the feedback guys, I appreciate it!

NinjaDevo
Jun 24, 2014

#5**0 **

Exactly! It is always easy to turn a fraction into a decimal but sometimes it is very difficult and cumbersone to turn a decimal into a fraction.

jboy314
Jun 24, 2014

#6**+5 **

This is a little off- topic, but let me show you how to turn a "repeating" decimal into a fraction........

Suppose we have the decimal .127 where the '27" part repeats

Combine the non-repeating part (1) with the repeating part (27)...so we have... 127

Now from this, subtract the non-repeating part, so we have 127 - 1 = 126

Divide this by a number comprised of 9s and 0s, where the number of 9s = the number of repeating digits (2) and the number of 0s = the number of non-repeating digits (1). So the number we divide by = 990....so we have

126/990 = .127(2727....)

Using your example, ND, of .454545

Combining the repeating part with the non-repeating part = 45 (The non-repeating part = 0, in this case)

Subtracting the non-repeating part from the repeating part gives 45 - 0 = 45

Write this over 99 (the number of 9s = the number of repeated digits, we have no non-repeating ones, thus, there are no 0s)

So 45/99 = .4545(4545.....)

In effect, if there is no non-repeating part, we just write our repeating part over the number of 9s = the number of digits in the repeating part !!!

Note also that you may be able to reduce the resulting fraction to a more simple form.....

And that's it !!

(Try writing .817(1717.....) as a fraction using this procedure....)

CPhill
Jun 24, 2014

#7**0 **

That's an interesting trick, CPhill.

Let's try the example you gave: .817(1717.....)

Add the repeating part to the non-repeating part.

817

Subtract the non-repeating part

817-8 = 809

Put this over 9's and 0's

809/990 = .817(1717...)

This seems to work pretty well, but I think I'll stick to solving it in the fraction form if possible!

NinjaDevo
Jun 24, 2014

#8**+5 **

2/3X +4/5 = -17/30

30 is the lowest common denominator so I would just multiply both sides by 30 right from the beginning.

$$\begin{array}{rll}

30\left(\frac{2x}{3}+\frac{4}{5}\right)&=&30\times\frac{-17}{30}\\\\

20x+24&=&-17\\\\

20x&=&-41\\\\

x&=&\frac{-41}{20}\\\\

x&=&-2.05

\end{array}$$

Melody
Jun 25, 2014