+0  
 
+5
1131
8
avatar+3451 

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

 Jun 24, 2014

Best Answer 

 #3
avatar+576 
+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

 Jun 24, 2014
 #1
avatar+111326 
+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.....

 

 Jun 24, 2014
 #2
avatar+3451 
+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!"

 Jun 24, 2014
 #3
avatar+576 
+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
avatar+3451 
+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!

 Jun 24, 2014
 #5
avatar+576 
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.

 Jun 24, 2014
 #6
avatar+111326 
+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....)

 

 Jun 24, 2014
 #7
avatar+3451 
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!

 Jun 24, 2014
 #8
avatar+109520 
+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.

$$\begin{array}{rll}
30\left(\frac{2x}{3}+\frac{4}{5}\right)&=&30\times\frac{-17}{30}\\\\
20x+24&=&-17\\\\
20x&=&-41\\\\
x&=&\frac{-41}{20}\\\\
x&=&-2.05
\end{array}$$

.
 Jun 25, 2014

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