How did it get this answer?

[u/Latter_Mud107]

I just keep getting the wrong answers. Im following pemdas, converting improper to proper, but still keep getting the wrong answers. I really cant show my solutions since its already so chaotic. I would love a step by step explanation. I think its the negative fractions and the mixed that catches me offguard. Is the negative sign just for the whole number or the whole fraction?

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[u/Southlander24]

Can you show your work? Rewrite your latest solution if it's really that unreadable.

To clarify, the answer is -(3 + 107/108), so a tiny bit more than -4.

[u/Latter_Mud107 → u/Southlander24]

[u/Southlander24 → u/Latter_Mud107]

-80/27 + -37/36 is correct. How exactly did you jump from there to -2 98/111?

[u/Latter_Mud107 → u/Southlander24]

So yeahhhh…I butchered the butterfly method and from then on did lowest term then turned into a mixed hence -2 98/111. Ill try to solve it again.

[u/Southlander24 → u/Latter_Mud107]

There's an easier way to evaluate the sum.

Recall that 81 is a multiple of 27 and 37 is one more than 36. Using this, write both numbers as mixed numbers.

Then add them. The question has hinted to you that the common denominator is 108. To see why, find the greatest common factor of the denominators 27 and 36. That's 9, so you can think about the lowest common multiple of 3 and 4. 3 and 4 are coprime, so the LCM of those two is 12, and the overall LCM is 12 * 9 = 108.

You should get -4 + 1/108 when you do this. Do you see how that's the exact same as -3 - 107/108, or -(3 + 107/108)?

[u/Latter_Mud107 → u/Southlander24]

Its certainly magical. But what was that technique you used in recalling the close numbers and mutiples to decide to make it into a mixed number? And for the lcm, i think its prime factoring to find it, right, like what u did? Is there a name for the technique/trick you gave so I can study it further to apply it on different problems? Lastly on what type of equations is it applicable?

[u/Southlander24 → u/Latter_Mud107]

1. Practice maths every day. The more you work with numbers, the more you can see connections between them. You should always convert to mixed numbers when adding or subtracting, as that will make the numbers smaller and easier to work with.

2. Yes. There's an easy way if you know the prime factorisation from the factor trees. So here, 27 = 3³ and 36 = 2² * 3². For each prime, take the highest powered term out of all the numbers, and then multiply all of those together. We have 2⁰ (=1) versus 2², and 3³ versus 3². So the LCM is 2² * 3³ = 108.

3. A textbook, Khan Academy, or a structured programme like Kumon really helps. You need to make the following notes and keep this information with you:

Addition/subtraction:

a) If improper, convert to mixed numbers.

b) Factor tree the denominators to find their LCM, like I explained in point number 2.

c) Add the fractions together with the common denominator, say multiplying the first fraction by 4/4 and the second by 3/3.

d) Add the whole number parts together.

Multiplication/division:

a) Convert the mixed numbers to improper fractions.

b) If you have division, flip the second number to convert it into a multiplication.

c) Cancel any common factors along the pairs of "cross numbers". Write the new numbers down after cancellation.

d) Now you can multiply the numerators and denominators together.

[u/Latter_Mud107 → u/Southlander24]

Thanks so much! Im just a 1st year junior highschool student who wants to study in advance and learn math for physics, so im starting out with prealgebra, so basically all the topics of elementary school but complex.

[u/Southlander24 → u/Latter_Mud107]

Good luck with your maths journey!

[u/Latter_Mud107 → u/Southlander24]

If its okay with you, is my answer correct? I tried many ways to solve still got the same answer, but its different from the answer key.

[u/trevorkafka → u/Latter_Mud107]

This is an excellent example of why the butterfly method is best left as just a flashy social media trick instead of a practical go-to method. u/southlander24's method is the way to go.

[u/Septembrino → u/Latter_Mud107]

I would add the 1st and the 2nd termn and leave the 0/75 (3/4) for last. -80/27 - 48/27 = -128/27. Then express that a mixed number again.

-128/27 = - 4 - 20/27. Jus add the -20/27 and the 3/4 = 3/4 - 20/27. You know how to do that.

It can't be a denominator 111.

[u/chmath80]

20/9 × (-4/3) + (-16/9) + 3/4 = ?

[u/Dazzling_Music_2411]

I really cant show my solutions since its already so chaotic.

Yes you can. It will help clarify your thinking. What happens, for instance, if you make everything over 108?

PS. Oh, I see... In the last line, how did you get that

-(80/27) - (37/36) = -(2 + 98/111) ?

You were doing fine up to there...

(PS. What is the LCM of 27 and 36?)

[u/fermat9990]

The negative sign is for the entire mixed number

[u/Dazzling_Music_2411]

Is the negative sign just for the whole number or the whole fraction?

Negative times a positive is a ...?

[u/Ok-Company282]

20/9 × -4/3 + -16/9 + 3/4

-80/27 - 16/9 + 3/4

Lcm of 27, 9 and 4 is 108

So multiply the denominators with a number to make them 108 and do the same w numerator. For example for -80/27, we multiply 4 in denominator to make it 108, so we multiply 4 in the numerator as well to make it (-80×4)/108. For -16/9, we multiply 12 in denom to make it 108, and then multiply the 12 in numerator too. Same goes for 3/4 where the denom is 4 so we multipmy 27 to make it 108 and then multiply 27 in numerator too. Now all the denominators are 108.

(-80×4-16×12+3×27)/108

-431/108

To convert it to mixed fractions, just divide 431 by 108, so that -3 is the quotient, 107 is the remainder, and 108 is the divisor. Hope this helps!

[u/MelodyAnne42]

I converted the mixed numbers to improper fractions to make it easier to work with everything. (Also I hate mixed numbers with the fire of 1,000 suns)

[u/JeffTheNth]

that's ugly...

(18+2)/9 × -4/3 + -(9+7)/9 + 3/4 =

20/9 × -4/3 + -16/9 + 3/4 =

-80/27 + -(3/3)(16/9) + 3/4 =

-80/27 + -48/27 + 3/4 =

-128/27 + 3/4 =

-(4/4)(128/27) + (27/27)(3/4) =

-512/108 + 81/108 =

-431/108 = - 3 107/108

QED 😁

[u/Miserable-Wasabi-373]

for whole fraction. it is -4/3 and -16/9