Question on Subtracting Mixed Numbers With Variables

[littlelimabean]

Hello!

Okay, *claps hands* let's get down to business.

I am trying to solve this problem 2 7/8 - 4 1/3 and simplify per textbook instructions, but I am not understanding the simplifying part at the end. The way I did it initially was to line it up in a horizontal-stacked type of way. So, I did 2 7/8 and 4 1/3 on top of each other and made a horizontal line with the subtraction sign. I made the equivalent fractions for the two by using the common denominator 24. That got me 2 21/24 - 4 8/24. From here, I did the subtraction from the whole numbers and the fractions, ending up with -2 13/24. This is where I get stuck with this problem and all others like it. I looked at the answer key and my textbook told me the answer was -1 11/12.

At first, I tried to fix it by redoing the question, but this time in its improper fraction form. However, I got the same answer. Then, I was like... I must be tripping. Let me whip out a calculator and see how it shows its work (the calculator I used). However, it says I did everything right. At the end, however, it says -2 13/24 = -1 11/12 .

How is -2 13/24 equal to -1 11/12 ???

I thought that maybe it was simplification, but 13/24 is the simplest and it isn't an improper fraction. Every time I found a person, textbook, calculator, it skips over the last bit. I also have been similar questions wrong.

Could we be taking a one form the two? But why would we do that??? It isn't needed...

help. I feel like its something obvious lol.

Thank you in advance. I appreciate you taking the time to help

 

[Dr.Peterson]

Hello!

Okay, *claps hands* let's get down to business.

I am trying to solve this problem 2 7/8 - 4 1/3 and simplify per textbook instructions, but I am not understanding the simplifying part at the end. The way I did it initially was to line it up in a horizontal-stacked type of way. So, I did 2 7/8 and 4 1/3 on top of each other and made a horizontal line with the subtraction sign. I made the equivalent fractions for the two by using the common denominator 24. That got me 2 21/24 - 4 8/24. From here, I did the subtraction from the whole numbers and the fractions, ending up with -2 13/24. This is where I get stuck with this problem and all others like it. I looked at the answer key and my textbook told me the answer was -1 11/12.

At first, I tried to fix it by redoing the question, but this time in its improper fraction form. However, I got the same answer. Then, I was like... I must be tripping. Let me whip out a calculator and see how it shows its work (the calculator I used). However, it says I did everything right. At the end, however, it says -2 13/24 = -1 11/12 .

How is -2 13/24 equal to -1 11/12 ???

I thought that maybe it was simplification, but 13/24 is the simplest and it isn't an improper fraction. Every time I found a person, textbook, calculator, it skips over the last bit. I also have been similar questions wrong.

Could we be taking a one form the two? But why would we do that??? It isn't needed...

help. I feel like its something obvious lol.

Thank you in advance. I appreciate you taking the time to help

The problem is that you can't subtract a larger number from a smaller one in this way. What you actually calculated was this:

(2 + 7/8) - (4 + 1/3) = (2 - 4) + (7/8 - 1/3) = -2 + 13/24​

Do you see that this is not the same as -2 13/24, which means -(2 + 13/24)?

But you could rescue your answer this way:

-2 + 13/24 = -1 + (-1 + 13/24) = -1 - (1 - 13/24) = -1 - 11/24 = -(1 + 11/24) = -1 11/24​

The more efficient way to do this is to first see that the answer will be negative, and then subtract the smaller number from the larger to get the absolute value of the answer:

2 7/8 - 4 1/3 = -(4 1/3 - 2 7/8) = -(4 8/24 - 2 21/24) = -(3 32/24 - 2 21/24) = -(1 11/24) = -1 11/24​

Now I have a question for you: where are the variables you mentioned in the title?

 

[littlelimabean]

The problem is that you can't subtract a larger number from a smaller one in this way. What you actually calculated was this:

(2 + 7/8) - (4 + 1/3) = (2 - 4) + (7/8 - 1/3) = -2 + 13/24​

Do you see that this is not the same as -2 13/24, which means -(2 + 13/24)?

But you could rescue your answer this way:

-2 + 13/24 = -1 + (-1 + 13/24) = -1 - (1 - 13/24) = -1 - 11/24 = -(1 + 11/24) = -1 11/24​

The more efficient way to do this is to first see that the answer will be negative, and then subtract the smaller number from the larger to get the absolute value of the answer:

2 7/8 - 4 1/3 = -(4 1/3 - 2 7/8) = -(4 8/24 - 2 21/24) = -(3 32/24 - 2 21/24) = -(1 11/24) = -1 11/24​

Now I have a question for you: where are the variables you mentioned in the title?

Sorry about the mention of variables in the title. I meant to say unlike denominators not variables. That was a mistake on my end. I will definitely double check the title just like I did the body of text.

To continue, I want to make sure I understand. Looking at the more efficient method, you factored out a -1 first. This allowed you to reorder this in a way where you could subtract a bigger number from the smaller one. You take away a 1 from the 4 at this part -(4 8/24 - 2 21/24) to distribute it to its fraction. Now, you were able to subtract properly making the new arrangement -(3 32/24 - 2 21/24). Then, you factor the -1 back in. This gives you the answer -1 11/24.

Am I correct in my analysis of your answer?

This helps and I want to practice more of this! Thank you for taking the time to answer.

 

[JeffM]

What does [imath]2 \dfrac{7}{8}[/imath] mean?

It is a not very useful way to say [imath]2 + \dfrac{7}{8}.[/imath]

Similarly, [imath]4 \dfrac{1}{3} = 4 + \dfrac{1}{3}.[/imath]

Thus,

[math]2\dfrac{7}{8} - 4\dfrac{1}{3} = \left (2 + \dfrac{7}{8} \right ) - \left (4 + \dfrac{1}{3} \right ) =\\ (2 - 4) + \left ( \dfrac{7}{8} - \dfrac{1}{3} \right ) = - 2 + \dfrac{21}{24} - \dfrac{8}{24} = \dfrac{13}{24} - 2 =\\ \dfrac{13}{24} - 1 + 1 - 2 = \dfrac{13}{24} - \dfrac{24}{24} - 1 = \\ - \dfrac{11}{24} - 1 = - \left ( 1 + \dfrac{11}{24} \right ) = -1\dfrac{11}{24}. [/math]
The fact is that mixed numbers are a poor notation that is virtually never used in anything but grade school math.
Change things into improper fractions and forget about mixed numbers, which will cause all kinds of errors.

 

[Dr.Peterson]

To continue, I want to make sure I understand. Looking at the more efficient method, you factored out a -1 first. This allowed you to reorder this in a way where you could subtract a bigger number from the smaller one. You take away a 1 from the 4 at this part -(4 8/24 - 2 21/24) to distribute it to its fraction. Now, you were able to subtract properly making the new arrangement -(3 32/24 - 2 21/24). Then, you factor the -1 back in. This gives you the answer -1 11/24.

Am I correct in my analysis of your answer?

Yes, though I was at first confused by "You take away a 1 from the 4 ... to distribute it to its fraction"; I think you're referring to what is commonly called "borrowing", not "distributing". And you said this backward, probably a typo: "you could subtract a bigger number from the smaller one".

But the way one would normally carry this out would be less formal. Just as we subtract, say, 78-94 by subtracting 94-78, and then tacking on a negative sign, we would just subtract 4 8/24 - 2 21/24 = 1 11/24 and tack on the negative. We wouldn't write it all out in factored form, keeping the negative sign. I was showing the reason for the method.

Note that the same difficulty you encountered arises in my whole number example. If you write 94 below 78 and subtract 8-4=4, you'll end up with nonsense.

The fact is that mixed numbers are a poor notation that is virtually never used in anything but grade school math.
Change things into improper fractions and forget about mixed numbers, which will cause all kinds of errors.

I would say this differently. Mixed numbers are used heavily in real life areas where "grade school math" is used (at least in non-metric regions). For example, I would much rather have a recipe call for 2 1/3 cups of water than 7/3, because the former fits how I measure.

But we typically don't do the calculations using mixed numbers, especially for more complicated problems (such as algebra). We would do the work using improper fractions, and convert to mixed numbers in stating the final answer for use.

(We don't say that base-ten notation is a poor notation because subtraction of 78-94 is confusing; we just learn how to do it right.)

 

[pka]

I am trying to solve this problem 2 7/8 - 4 1/3 and simplify per textbook instructions,

I say no to so called mixed fractions period. So here is my method.
[imath]2\tfrac{7}{8}-4\tfrac{1}{3}=\dfrac{23}{8}-\dfrac{13}{3}=\dfrac{69-104}{24}=\dfrac{-35}{24}=-1\tfrac{11}{24}[/imath]
SEE HERE

[imath][/imath]

 

[Deleted member 4993]

I say no to so called mixed fractions period

I am surprised.

When I calculate \(\displaystyle 7\frac{2}{3} - 5\frac{1}{3}\) - without pencil/paper - I do it in mixed number mode.

\(\displaystyle 7\frac{2}{3} - 5\frac{1}{3}\) = \(\displaystyle (7 - 5) + (\frac{2}{3} - \frac{1}{3})\)

 

[littlelimabean]

Yes, though I was at first confused by "You take away a 1 from the 4 ... to distribute it to its fraction"; I think you're referring to what is commonly called "borrowing", not "distributing". And you said this backward, probably a typo: "you could subtract a bigger number from the smaller one".

But the way one would normally carry this out would be less formal. Just as we subtract, say, 78-94 by subtracting 94-78, and then tacking on a negative sign, we would just subtract 4 8/24 - 2 21/24 = 1 11/24 and tack on the negative. We wouldn't write it all out in factored form, keeping the negative sign. I was showing the reason for the method.

Note that the same difficulty you encountered arises in my whole number example. If you write 94 below 78 and subtract 8-4=4, you'll end up with nonsense.


I would say this differently. Mixed numbers are used heavily in real life areas where "grade school math" is used (at least in non-metric regions). For example, I would much rather have a recipe call for 2 1/3 cups of water than 7/3, because the former fits how I measure.

But we typically don't do the calculations using mixed numbers, especially for more complicated problems (such as algebra). We would do the work using improper fractions, and convert to mixed numbers in stating the final answer for use.

(We don't say that base-ten notation is a poor notation because subtraction of 78-94 is confusing; we just learn how to do it right.)

Okayyyyyyyy... I think my problem and my confusion came from this part:

Note that the same difficulty you encountered arises in my whole number example. If you write 94 below 78 and subtract 8-4=4, you'll end up with nonsense.

I actually had a bit of a problem with dealing with questions like this back in grade school. I dealt with it, but it comes back to haunt my math problems sometimes.

I will pay more attention to this in the future. The mixed number notation is only confusing to me in this instance. I will practice it and try to check my answer by turning it into an improper fractions to make sure I am not neglecting anything.

And yes, I meant to say borrow/distribute, but I had trouble articulating it in the way I desired to.

Thank you so much for helping!

 

[Dr.Peterson]

Okayyyyyyyy... I think my problem and my confusion came from this part:

Note that the same difficulty you encountered arises in my whole number example. If you write 94 below 78 and subtract 8-4=4, you'll end up with nonsense.

I actually had a bit of a problem with dealing with questions like this back in grade school. I dealt with it, but it comes back to haunt my math problems sometimes.

Just to finish that 78-94 example, you would (wrongly) take 8 - 4 to get 4 in the ones place; then 7 - 9 to get -2 in the tens place; but the correct answer is not -24. What you would really have found is -20 + 4, which is -16. That's what we get when we do it the proper way: 94 - 78 = 16, and we make it negative because we reversed it.

So the first way is not entirely nonsense, but it's extremely easy to misinterpret the answer.

 

[Deleted member 4993]

You had said:

I am trying to solve this problem 2 7/8 - 4 1/3 and simplify per textbook instructions, but I am not understanding the simplifying part at the end. The way I did it initially was to line it up in a horizontal-stacked type of way. So, I did 2 7/8 and 4 1/3 on top of each other and made a horizontal line with the subtraction sign. I made the equivalent fractions for the two by using the common denominator 24. That got me 2 21/24 - 4 8/24. From here, I did the subtraction from the whole numbers and the fractions, ending up with -2 13/24.

Many moons ago, we used log & anti-log the same way.

In "log-table" -2 13/24.would be written as \(\displaystyle \overline{2}.5417\) ******************* added later

 

[Steven G]

To continue, I want to make sure I understand. Looking at the more efficient method, you factored out a -1 first. This allowed you to reorder this in a way where you could subtract a bigger number from the smaller one.

No, you subtracted a smaller number from a larger number.

In 7-4, you are subtracting 4 from 7, NOT 7 from 4.

 

[littlelimabean]

No, you subtracted a smaller number from a larger number.

In 7-4, you are subtracting 4 from 7, NOT 7 from 4.

I see... Alright, I think I have got it now. There was a lot of little holes before this question. Most of them surrounded similar subtraction problems. Went back to make a more solid foundation on it.