Algebraic Fractions aren't making any sense to me. I'm not able to progress in math and I'm losing my mind.

Is it supposed to be ((c+2)/c)+((c+1)/2c)?
Can someone explain to me why my posts get downvoted? I'm looking for help and many people seem to hate that. Why is seeking help such a bad thing?

You can use distributive property to factor out a c in the numerator, and cancel that with one of the c in the denominator.

(3c^2 + 5c)/2c^2

= (c)(3c + 5)/(c)(2c) <-- this is distributive property in numerator

= c/c * (3c + 5)/(2c) <-- because pq/rs = p/r * q/s (fraction multiplication)

= 1 * (3c + 5)/(2c)

= (3c + 5)/(2c)

Your method of using the product of the denominators as your common denominator will work but the book's method is akin to using the least common multiple of the denominators as the common denominator. You used 2cc as your common denominator, but the book solution shows that you could simply use 2c as common denominator.

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It's like the difference between doing this with regular fractions:

your method: 1/8 + 1/6 = 6/48 + 8/48 = 14/48 = (7 * 2)/(24 * 2) = 7/24 * 2/2 = 7/24

book method: 1/8 + 1/6 = 3/24 + 4/24 = 7/24

With your method there is going to be a need to simplify your answer by dividing out a common factor in both numerator and denominator. It happens because the original denominators have a common factor of 2. The book method of using the simplest common multiple often avoids needing to simplify in the end.
Here's another example to show why finding simplest common multiple saves you time.

1/xyz + 2/yzw

If you simply multiply the denominators to get xyzyzw as common denominator, you would work it this way.

= (1/xyz * yzw/yzw) + (2/yzw * xyz/xyz)

= yzw/xyzyzw + 2xyz/xyzyzw

= (yzw + 2xyz)/xyyzzw

But then you would see that yz is a factor of each term in the numerator. Use the distributive property to express this as a multiplication.

= yz(w + 2x)/xyyzzw

Now the factor yz in numerator and factor yz in denominator cancel. You can cancel because y and z here are factors in numerator and factors in denominator, no longer part of a sum.

= ~~yz~~(w + 2x)/xy~~yz~~zw

= (w + 2x)/xyzw

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If you instead think about LCM of fractions, and extend that idea to finding simplest common multiple of the denominators:

The original denominators xyz and yzw have xyzw as their simplest common multiple. You multiply each fraction by a version of 1 that will create the common multiple as the new denominator. (So multiply first fraction by w/w, and multiply second fraction by x/x.)

1/xyz + 2/yzw

= (1/xyz * w/w) + (2/yzw * x/x)

= w/xyzw + 2x/xyzw

= (w + 2x)/xyzw

This takes less writing, but a little more thinking up front to find the common denominator that is best to use.
Too much work! The LCD is 2c, so multiply numerator and denominator of the 1st fraction by 2. Now add the the numerators and put the result over 2c. Next, simplify.

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