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Which of the following sum is in the simplest form:
(a) \[\dfrac{4}{9}+\dfrac{-5}{9}\]
(b) \[\dfrac{-2}{5}+\dfrac{13}{20}\]
(c) \[\dfrac{-5}{12}+\dfrac{11}{-12}\]
(d) \[\dfrac{-7}{8}+\dfrac{1}{12}+\dfrac{2}{3}\]

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Last updated date: 20th Jun 2024
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Answer
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Hint: We will consider each option and do the necessary algebraic operation. We will be using the concept of addition and subtraction of fractions. So, in option (a), we have the same denominator, so we can directly add the numerators and divide it by the common denominator. In case of option (b), we will have to take the LCM to make the denominator the same and then continue. The sum which appears in the simplest form immediately after applying the algebraic operation is the sum in the simplest form.

Complete step-by-step answer:

Let us consider the four options that are given to us. Undergo the basic arithmetic operations like addition, subtraction, and multiplication to obtain the simplest form, where the fraction cannot be any smaller.
Consider the first option, \[\dfrac{4}{9}+\left( \dfrac{-5}{9} \right)\]. We can also write it as, \[\left( \dfrac{4}{9}-\dfrac{5}{9} \right)\]. Here the denominator is same for both the fractions. Hence we can do the arithmetic operation of subtraction directly.
\[\Rightarrow \]\[\dfrac{4}{9}-\dfrac{5}{9}=\dfrac{-1}{9}\], this fraction is in the simplest form.
Now let us consider the next option, \[\left( \dfrac{-2}{5}+\dfrac{13}{20} \right)\]. Here both the denominators of the fractions are not same. Thus let us simplify it.
 Now taking the LCM, we get,
\[\Rightarrow \dfrac{\left[ -2\times 4+13\times 1 \right]}{20}=\dfrac{-8+13}{20}=\dfrac{5}{20}\]
Not in the simplest form.
Here, \[\dfrac{5}{20}\] can be simplified further. But we want the fractions that cannot be simplified further. Hence we can say that \[\dfrac{5}{20}\] is not the simplest form.
Now let us take the next option, \[\left( \dfrac{-5}{12}+\dfrac{11}{-12} \right)\]. We can write them as \[\left( \dfrac{-5}{12}-\dfrac{11}{12} \right)\]. Both the fractions have the same denominator.
\[\Rightarrow \dfrac{-5}{12}-\dfrac{11}{12}=-\dfrac{16}{12}\]
Not in the simplest form.
Now in \[\left( \dfrac{-7}{8}+\dfrac{1}{12}+\dfrac{2}{3} \right)\], none of the fractions have the same denominator. Thus simplifying it,
Taking LCM we get,
\[\Rightarrow \dfrac{-7}{8}+\dfrac{1}{12}+\dfrac{2}{3}=\dfrac{\left[ -7\times 3+1\times 2+2\times 8 \right]}{24}=\dfrac{-21+2+16}{24}=\dfrac{-3}{24}\]
Thus out of all the four options we got only \[\left( \dfrac{4}{9}+\dfrac{-5}{9} \right)\] in the simplest form.
\[\therefore \] Option (a) is the correct answer.

Note: Option (b), (c), and (d) can be simplified further to obtain the simplest form. But for this particular question, we need the fraction that becomes form immediately after applying algebraic operations. We could have applied one more operation of canceling similar terms in the result obtained for all these options. We could have obtained \[\dfrac{5}{20}\] as \[\dfrac{1}{4}\], \[\dfrac{-16}{12}\] as \[\dfrac{-4}{3}\], \[\dfrac{-3}{24}\] as \[\dfrac{-1}{8}\]. This is the simplified form of all the options.