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adkor95
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 14:31
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Hi,

I'm confused by this foundation theory and was hoping you could help.

Why is it that, in some cases of complex fractions, we multiply both the numerator and denominator - yet in others we multiply only the numerator?

Example: \(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

My answer is either:

\(\frac{15}{8}\) - when all fractions are multiplied by \(\frac{60}{1}\)

\(\frac{ 3}{1}\) - when the top fraction numerators are multiplied by 10 and bottom fraction denominator are multiplied by 12

Thanks
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anniaustin
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 15:12
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adkor95
Hi,

I'm confused by this foundation theory and was hoping you could help.

Why is it that, in some cases of complex fractions, we multiply both the numerator and denominator - yet in others we multiply only the numerator?

Example: \(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

My answer is either:

\(\frac{15}{8}\) - when all fractions are multiplied by \(\frac{60}{1}\)

\(\frac{ 3}{1}\) - when the top fraction numerators are multiplied by 10 and bottom fraction denominator are multiplied by 12

Thanks
Show more
There is a specific answer to this. If you multiply by 60, you need to multiply both the numerator and denominator by 60.

Btw, both your answers are wrong. For this particular one, it is \(\frac{18}{5}\)
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AndrewN
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 15:22
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adkor95
Hi,

I'm confused by this foundation theory and was hoping you could help.

Why is it that, in some cases of complex fractions, we multiply both the numerator and denominator - yet in others we multiply only the numerator?

Example: \(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

My answer is either:

\(\frac{15}{8}\) - when all fractions are multiplied by \(\frac{60}{1}\)

\(\frac{ 3}{1}\) - when the top fraction numerators are multiplied by 10 and bottom fraction denominator are multiplied by 12

Thanks
Show more
Hello, adkor95. I think you may be getting your theory mixed up and looking to take shortcuts instead. Neither answer you calculated above is correct. You need a common denominator to add or subtract fractions, but either approach--finding a common denominator for all fractions in the expression or finding one for each of the top and bottom fractions--will lead you to the same answer if you make your conversions correctly. Consider:

a) Choose a common denominator for all fractions:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{30}{30}-\frac{1}{5}*\frac{12}{12}}{\frac{1}{3}*\frac{20}{20}-\frac{1}{4}*\frac{15}{15}}\)

\(\frac{\frac{30}{60}-\frac{12}{60}}{\frac{20}{60}-\frac{15}{60}}\)

\(\frac{\frac{18}{60}}{\frac{5}{60}}\)

\(\frac{18}{60}*\frac{60}{5}\)

\(\frac{18}{5}\)

b) Choose a common denominator for the numerator and the denominator separately:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{5}{5}-\frac{1}{5}*\frac{2}{2}}{\frac{1}{3}*\frac{4}{4}-\frac{1}{4}*\frac{3}{3}}\)

\(\frac{\frac{5}{10}-\frac{2}{10}}{\frac{4}{12}-\frac{3}{12}}\)

\(\frac{\frac{3}{10}}{\frac{1}{12}}\)

\(\frac{3}{10}*\frac{12}{1}\)

\(\frac{36}{10}\)

\(\frac{18}{5}\)

I would refer you to any post by Quant Expert Bunuel for further information about theory. (Just look in the signature for a link to the Ultimate GMAT Quant Megathread.) Good luck with your studies.

- Andrew
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 15:38
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adkor95
Hi,

I'm confused by this foundation theory and was hoping you could help.

Why is it that, in some cases of complex fractions, we multiply both the numerator and denominator - yet in others we multiply only the numerator?

Example: \(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

My answer is either:

\(\frac{15}{8}\) - when all fractions are multiplied by \(\frac{60}{1}\)

\(\frac{ 3}{1}\) - when the top fraction numerators are multiplied by 10 and bottom fraction denominator are multiplied by 12

Thanks
Hello, adkor95. I think you may be getting your theory mixed up and looking to take shortcuts instead. Neither answer you calculated above is correct. You need a common denominator to add or subtract fractions, but either approach--finding a common denominator for all fractions in the expression or finding one for each of the top and bottom fractions--will lead you to the same answer if you make your conversions correctly. Consider:

a) Choose a common denominator for all fractions:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{30}{30}-\frac{1}{5}*\frac{12}{12}}{\frac{1}{3}*\frac{20}{20}-\frac{1}{4}*\frac{15}{15}}\)

\(\frac{\frac{30}{60}-\frac{12}{60}}{\frac{20}{60}-\frac{15}{60}}\)

\(\frac{\frac{18}{60}}{\frac{5}{60}}\)

\(\frac{18}{60}*\frac{60}{5}\)

\(\frac{18}{5}\)

b) Choose a common denominator for the numerator and the denominator separately:

\(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

\(\frac{\frac{1}{2}*\frac{5}{5}-\frac{1}{5}*\frac{2}{2}}{\frac{1}{3}*\frac{4}{4}-\frac{1}{4}*\frac{3}{3}}\)

\(\frac{\frac{5}{10}-\frac{2}{10}}{\frac{4}{12}-\frac{3}{12}}\)

\(\frac{\frac{3}{10}}{\frac{1}{12}}\)

\(\frac{3}{10}*\frac{12}{1}\)

\(\frac{36}{10}\)

\(\frac{18}{5}\)

I would refer you to any post by Quant Expert Bunuel for further information about theory. (Just look in the signature for a link to the Ultimate GMAT Quant Megathread.) Good luck with your studies.

- Andrew
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Hi Andrew,

Thanks very much for this post and pointing out that I posted the wrong answer (I will change if forum rules permit?). I think I understand where I'm going wrong - it's just my bad habits from my pre-GMAT Quant days (before learning the underlying theory) catching up with me.

If I understand correctly: since we need a common denominator to add/subtract fractions, we must multiply both the numerator and denominator by the fraction that aligns the denominators.

Therefore in this expression, we aren't actually multiplying each numerator by 10 or 12 or 60 - we're multiplying both numerators and denominators by the fractions that would result in each fraction having a common denominator and then simply just finding the difference. We can choose to do that either for the numerator and denominator separately, or find the common denominator of all fractions and follow the same process.
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 15:43
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anniaustin
adkor95
Hi,

I'm confused by this foundation theory and was hoping you could help.

Why is it that, in some cases of complex fractions, we multiply both the numerator and denominator - yet in others we multiply only the numerator?

Example: \(\frac{\frac{1}{2}-\frac{1}{5}}{\frac{1}{3}-\frac{1}{4}}\)

My answer is either:

\(\frac{15}{8}\) - when all fractions are multiplied by \(\frac{60}{1}\)

\(\frac{ 3}{1}\) - when the top fraction numerators are multiplied by 10 and bottom fraction denominator are multiplied by 12

Thanks
There is a specific answer to this. If you multiply by 60, you need to multiply both the numerator and denominator by 60.

Btw, both your answers are wrong. For this particular one, it is \(\frac{18}{5}\)
Show more


Thanks anniaustin - I've realised what I've done wrong here, and the wrong answer I've posted.

Based on the comment by Andrew below, it seems as though you also take a shortcut? Rather than multiplying through each fraction by the fraction (e.g. \(\frac{5}{5}\) needed to reach a common denominator in the top fraction, instead you multiply each fraction by the common multiple as a whole number (i.e. 60). Would be great to hear your thinking here.
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It is not exactly a shortcut, but what anniaustin was pointing out is that you have to multiply both the numerator and denominator through by the same number, effectively multiplying by 1:

Numerator:

\(60*\frac{1}{2}=30\)

\(60*\frac{1}{5}=12\)

\(30-12=18\)

Denominator:

\(60*\frac{1}{3}=20\)

\(60*\frac{1}{4}=15\)

\(20-15=5\)

Hence, the answer is

\(\frac{18}{5}\)

I hope that helps. (It is quite similar to the first approach I outlined above.)

- Andrew
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 16:04
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It is not exactly a shortcut, but what anniaustin was pointing out is that you have to multiply both the numerator and denominator through by the same number, effectively multiplying by 1:

Numerator:

\(60*\frac{1}{2}=30\)

\(60*\frac{1}{5}=12\)

\(30-12=18\)

Denominator:

\(60*\frac{1}{3}=20\)

\(60*\frac{1}{4}=15\)

\(20-15=5\)

Hence, the answer is

\(\frac{18}{5}\)

I hope that helps. (It is quite similar to the first approach I outlined above.)

- Andrew
Show more

Thanks Andrew but I think that's where I'm getting confused.

How is it that \(60*\frac{1}{2}=30\) can be multiplied through the whole fraction, but \(10*\frac{1}{2}=5\) can't be used for the numerator alone?
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When do you multiply both numerator and denominator vs numerator only? 06 Apr 2020, 17:22
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adkor95
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It is not exactly a shortcut, but what anniaustin was pointing out is that you have to multiply both the numerator and denominator through by the same number, effectively multiplying by 1:

Numerator:

\(60*\frac{1}{2}=30\)

\(60*\frac{1}{5}=12\)

\(30-12=18\)

Denominator:

\(60*\frac{1}{3}=20\)

\(60*\frac{1}{4}=15\)

\(20-15=5\)

Hence, the answer is

\(\frac{18}{5}\)

I hope that helps. (It is quite similar to the first approach I outlined above.)

- Andrew

Thanks Andrew but I think that's where I'm getting confused.

How is it that \(60*\frac{1}{2}=30\) can be multiplied through the whole fraction, but \(10*\frac{1}{2}=5\) can't be used for the numerator alone?
Show more
60 is a common multiple of each number in the denominator of the individual fractions, but 10 is not. You could multiply both the top and bottom of the larger fraction through by 10, but that would only wipe out the fractions in the numerator. Run it through to check:

Numerator:

\(10*\frac{1}{2}=5\)

\(10*\frac{1}{5}=2\)

\(5-2=3\)

Denominator:

\(10*\frac{1}{3}=\frac{10}{3}\)

\(10*\frac{1}{4}=\frac{10}{4}\)

\(\frac{10}{3}-\frac{10}{4}\)

Now you will run into a little problem. You need to find a common denominator to combine the fractions.

\(\frac{10}{3}*\frac{4}{4}-\frac{10}{4}*\frac{3}{3}\)

\(\frac{40}{12}-\frac{30}{12}\)

\(\frac{10}{12}\)

\(\frac{5}{6}\)

We can now combine the numerator and denominator:

\(\frac{\frac{3}{1}}{\frac{5}{6}}\)

\(\frac{3}{1}*\frac{6}{5}\)

\(\frac{18}{5}\)

The answer is the same, but it just took a little more work to get there. This illustrates why, if you were going to use the multiply-through-and-eliminate-the-denominators approach, you would probably want to choose a common multiple of all the denominators. In any case, whatever number you multiply through on top in this method must be the same as the one you multiply through on the bottom. In this manner, you are effectively multiplying by 1, but if you change the top and the bottom in different ways, then the original expression and the one you have manipulated are no longer equivalent.

- Andrew
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When do you multiply both numerator and denominator vs numerator only? 07 Apr 2020, 02:41
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adkor95
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It is not exactly a shortcut, but what anniaustin was pointing out is that you have to multiply both the numerator and denominator through by the same number, effectively multiplying by 1:

Numerator:

\(60*\frac{1}{2}=30\)

\(60*\frac{1}{5}=12\)

\(30-12=18\)

Denominator:

\(60*\frac{1}{3}=20\)

\(60*\frac{1}{4}=15\)

\(20-15=5\)

Hence, the answer is

\(\frac{18}{5}\)

I hope that helps. (It is quite similar to the first approach I outlined above.)

- Andrew

Thanks Andrew but I think that's where I'm getting confused.

How is it that \(60*\frac{1}{2}=30\) can be multiplied through the whole fraction, but \(10*\frac{1}{2}=5\) can't be used for the numerator alone?
60 is a common multiple of each number in the denominator of the individual fractions, but 10 is not. You could multiply both the top and bottom of the larger fraction through by 10, but that would only wipe out the fractions in the numerator. Run it through to check:

Numerator:

\(10*\frac{1}{2}=5\)

\(10*\frac{1}{5}=2\)

\(5-2=3\)

Denominator:

\(10*\frac{1}{3}=\frac{10}{3}\)

\(10*\frac{1}{4}=\frac{10}{4}\)

\(\frac{10}{3}-\frac{10}{4}\)

Now you will run into a little problem. You need to find a common denominator to combine the fractions.

\(\frac{10}{3}*\frac{4}{4}-\frac{10}{4}*\frac{3}{3}\)

\(\frac{40}{12}-\frac{30}{12}\)

\(\frac{10}{12}\)

\(\frac{5}{6}\)

We can now combine the numerator and denominator:

\(\frac{\frac{3}{1}}{\frac{5}{6}}\)

\(\frac{3}{1}*\frac{6}{5}\)

\(\frac{18}{5}\)

The answer is the same, but it just took a little more work to get there. This illustrates why, if you were going to use the multiply-through-and-eliminate-the-denominators approach, you would probably want to choose a common multiple of all the denominators. In any case, whatever number you multiply through on top in this method must be the same as the one you multiply through on the bottom. In this manner, you are effectively multiplying by 1, but if you change the top and the bottom in different ways, then the original expression and the one you have manipulated are no longer equivalent.

- Andrew
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Perfect - this makes sense Andrew.

Essentially, I need to keep in mind that I must always follow the basic rule of maintaining the fraction by multiplying both the numerator and the denominator through by 1.

In this case I can either do that based on the common denominator for both fractions on the top (10) and both fractions on the bottom (12) separately, or I can do it for all fractions in the top and bottom (60) at once
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Perfect - this makes sense Andrew.

Essentially, I need to keep in mind that I must always follow the basic rule of maintaining the fraction by multiplying both the numerator and the denominator through by 1.

In this case I can either do that based on the common denominator for both fractions on the top (10) and both fractions on the bottom (12) separately, or I can do it for all fractions in the top and bottom (60) at once
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Yes, now you have it. Just do not forget that you cannot change the larger or overall numerator and denominator in different ways. Multiplying by 1 effectively changes nothing; multiplying by something different on the top and bottom is not multiplying by 1. Any approach you choose from the above will require a certain facility with fractions, so if that issues lies at the core of your misunderstanding, then at least you know what you need to attack.

Good luck.

- Andrew
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When do you multiply both numerator and denominator vs numerator only? 08 Apr 2020, 16:18
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I think for you it would be hard to do LCM of 4 numbers in your head. For this reason I would advise to follow these steps attached.
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I would advise to brush up on your fundamentals. 1 to 1 coaching for a few classes might help.
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