More intuitive solution to simplifying complex fraction?

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My problem is this:

$$frac3 - frac1xfrac13x - 1$$

This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is there a more intuitive way to solve this problem? Currently, there are three major steps.

1. Multiply each term in the expression to get a common denominator of $3x$ in each term and simplify


2. Then, we can rearrange the denominator:


3. Factor out $(3x-1)$ and simplify.

algebra-precalculus fractions

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edited Jul 28 at 22:16

Xander Henderson

13.1k83150

asked Jul 28 at 19:29

user9995331

1114

• 
  
  
  

  
  

  
  
  
  
  
  

  
  see what happens if you just multiply the numerator and denominator by $3$.
  – John Joy
  Jul 29 at 0:34
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

My problem is this:

$$frac3 - frac1xfrac13x - 1$$

This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is there a more intuitive way to solve this problem? Currently, there are three major steps.

1. Multiply each term in the expression to get a common denominator of $3x$ in each term and simplify


2. Then, we can rearrange the denominator:


3. Factor out $(3x-1)$ and simplify.

algebra-precalculus fractions

share|cite|improve this question

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

asked Jul 28 at 19:29

user9995331

1114

• 
  
  
  

  
  

  
  
  
  
  
  

  
  see what happens if you just multiply the numerator and denominator by $3$.
  – John Joy
  Jul 29 at 0:34
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

My problem is this:

$$frac3 - frac1xfrac13x - 1$$

This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is there a more intuitive way to solve this problem? Currently, there are three major steps.

1. Multiply each term in the expression to get a common denominator of $3x$ in each term and simplify


2. Then, we can rearrange the denominator:


3. Factor out $(3x-1)$ and simplify.

algebra-precalculus fractions

share|cite|improve this question

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

asked Jul 28 at 19:29

user9995331

1114

My problem is this:

$$frac3 - frac1xfrac13x - 1$$

This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is there a more intuitive way to solve this problem? Currently, there are three major steps.

1. Multiply each term in the expression to get a common denominator of $3x$ in each term and simplify


2. Then, we can rearrange the denominator:


3. Factor out $(3x-1)$ and simplify.

algebra-precalculus fractions

share|cite|improve this question

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

asked Jul 28 at 19:29

user9995331

1114

share|cite|improve this question

share|cite|improve this question

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

edited Jul 28 at 22:16

Xander Henderson

13.1k83150

13.1k83150

asked Jul 28 at 19:29

user9995331

1114

asked Jul 28 at 19:29

user9995331

1114

asked Jul 28 at 19:29

user9995331

1114

1114

• 
  
  
  

  
  

  
  
  
  
  
  

  
  see what happens if you just multiply the numerator and denominator by $3$.
  – John Joy
  Jul 29 at 0:34
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  see what happens if you just multiply the numerator and denominator by $3$.
  – John Joy
  Jul 29 at 0:34
  

  
  
  
  

see what happens if you just multiply the numerator and denominator by $3$.
– John Joy
Jul 29 at 0:34

see what happens if you just multiply the numerator and denominator by $3$.
– John Joy
Jul 29 at 0:34

add a comment | 

5 Answers
5

active

oldest

votes

up vote
2
down vote

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $frac 3x3x$
$$frac 3x3xcdot dfrac3 - frac1xfrac13x - 1=frac 9x-31-3x=-3$$

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

add a comment | 

up vote
2
down vote

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

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up vote
1
down vote

Note that $$frac3-frac1xfrac13x-1=frac3x-1frac13-x=frac9x-31-3x$$

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

up vote
1
down vote

I would set

$$y=frac13ximplies dfrac3 - frac1xfrac13x - 1=dfrac3 - 3yy - 1=-3cdot dfracy-1y - 1=-3$$

share|cite|improve this answer

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

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up vote
0
down vote

I would try to make the numerator look similar to the denominator as follows:
$$frac 3 - frac 1 x frac 1 3x - 1 = - frac 3 - frac 1 x 1 - frac 1 3 x = - 3 cdot frac 1 - frac 1 3x 1 - frac 1 3x = -3.$$
By the way, this is quite similar to both gcbound's and gimusi's answers.

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

add a comment | 

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5 Answers
5

active

oldest

votes

5 Answers
5

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $frac 3x3x$
$$frac 3x3xcdot dfrac3 - frac1xfrac13x - 1=frac 9x-31-3x=-3$$

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

add a comment | 

up vote
2
down vote

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $frac 3x3x$
$$frac 3x3xcdot dfrac3 - frac1xfrac13x - 1=frac 9x-31-3x=-3$$

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

add a comment | 

up vote
2
down vote

up vote
2
down vote

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $frac 3x3x$
$$frac 3x3xcdot dfrac3 - frac1xfrac13x - 1=frac 9x-31-3x=-3$$

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $frac 3x3x$
$$frac 3x3xcdot dfrac3 - frac1xfrac13x - 1=frac 9x-31-3x=-3$$

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 19:49

Ross Millikan

275k21185351

answered Jul 28 at 19:49

Ross Millikan

275k21185351

answered Jul 28 at 19:49

Ross Millikan

275k21185351

275k21185351

add a comment | 

add a comment | 

up vote
2
down vote

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

add a comment | 

up vote
2
down vote

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

add a comment | 

up vote
2
down vote

up vote
2
down vote

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 20:33

gcbound

1212

answered Jul 28 at 20:33

gcbound

1212

answered Jul 28 at 20:33

gcbound

1212

1212

add a comment | 

add a comment | 

up vote
1
down vote

Note that $$frac3-frac1xfrac13x-1=frac3x-1frac13-x=frac9x-31-3x$$

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

up vote
1
down vote

Note that $$frac3-frac1xfrac13x-1=frac3x-1frac13-x=frac9x-31-3x$$

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

up vote
1
down vote

up vote
1
down vote

Note that $$frac3-frac1xfrac13x-1=frac3x-1frac13-x=frac9x-31-3x$$

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

Note that $$frac3-frac1xfrac13x-1=frac3x-1frac13-x=frac9x-31-3x$$

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

answered Jul 28 at 19:36

Dr. Sonnhard Graubner

66.7k32659

66.7k32659

add a comment | 

add a comment | 

up vote
1
down vote

I would set

$$y=frac13ximplies dfrac3 - frac1xfrac13x - 1=dfrac3 - 3yy - 1=-3cdot dfracy-1y - 1=-3$$

share|cite|improve this answer

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

add a comment | 

up vote
1
down vote

I would set

$$y=frac13ximplies dfrac3 - frac1xfrac13x - 1=dfrac3 - 3yy - 1=-3cdot dfracy-1y - 1=-3$$

share|cite|improve this answer

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

add a comment | 

up vote
1
down vote

up vote
1
down vote

I would set

$$y=frac13ximplies dfrac3 - frac1xfrac13x - 1=dfrac3 - 3yy - 1=-3cdot dfracy-1y - 1=-3$$

share|cite|improve this answer

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

I would set

$$y=frac13ximplies dfrac3 - frac1xfrac13x - 1=dfrac3 - 3yy - 1=-3cdot dfracy-1y - 1=-3$$

share|cite|improve this answer

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

share|cite|improve this answer

share|cite|improve this answer

edited Jul 28 at 20:07

edited Jul 28 at 20:07

edited Jul 28 at 20:07

answered Jul 28 at 20:02

gimusi

64.7k73482

answered Jul 28 at 20:02

gimusi

64.7k73482

answered Jul 28 at 20:02

gimusi

64.7k73482

64.7k73482

add a comment | 

add a comment | 

up vote
0
down vote

I would try to make the numerator look similar to the denominator as follows:
$$frac 3 - frac 1 x frac 1 3x - 1 = - frac 3 - frac 1 x 1 - frac 1 3 x = - 3 cdot frac 1 - frac 1 3x 1 - frac 1 3x = -3.$$
By the way, this is quite similar to both gcbound's and gimusi's answers.

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

add a comment | 

up vote
0
down vote

I would try to make the numerator look similar to the denominator as follows:
$$frac 3 - frac 1 x frac 1 3x - 1 = - frac 3 - frac 1 x 1 - frac 1 3 x = - 3 cdot frac 1 - frac 1 3x 1 - frac 1 3x = -3.$$
By the way, this is quite similar to both gcbound's and gimusi's answers.

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

add a comment | 

up vote
0
down vote

up vote
0
down vote

I would try to make the numerator look similar to the denominator as follows:
$$frac 3 - frac 1 x frac 1 3x - 1 = - frac 3 - frac 1 x 1 - frac 1 3 x = - 3 cdot frac 1 - frac 1 3x 1 - frac 1 3x = -3.$$
By the way, this is quite similar to both gcbound's and gimusi's answers.

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

I would try to make the numerator look similar to the denominator as follows:
$$frac 3 - frac 1 x frac 1 3x - 1 = - frac 3 - frac 1 x 1 - frac 1 3 x = - 3 cdot frac 1 - frac 1 3x 1 - frac 1 3x = -3.$$
By the way, this is quite similar to both gcbound's and gimusi's answers.

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

share|cite|improve this answer

share|cite|improve this answer

answered Jul 29 at 12:08

Luca Bressan

3,8322935

answered Jul 29 at 12:08

Luca Bressan

3,8322935

answered Jul 29 at 12:08

Luca Bressan

3,8322935

3,8322935

add a comment | 

add a comment | 

 

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