Question

What is the value of \[3\dfrac{4}{9} + 3\dfrac{3}{4}\]?

Answer 1

Hint: In this question, we have to find out the sum of the given fractions.
For finding that first we have to find the improper fraction of these two then we need to find the lowest common multiple of the two denominators. Then increase the terms of each fraction so that the denominator of each equals to the L.C.M. Substituting these two new fractions for the original ones and adding them we will get the solution.

Complete step-by-step solution:
We need to find out \[3\dfrac{4}{9} + 3\dfrac{3}{4}\].
Simplifying we get,
\[\begin{gathered}
  3\dfrac{4}{9} + 3\dfrac{3}{4} \\
   = \dfrac{{31}}{9} + \dfrac{{15}}{4} \\
\end{gathered} \]
We have to find out the lowest common multiple of the two denominators.
First, we need to do prime factorization for the two numbers \[9\text{ and }4\].
We get,
\[9 = 3 \times 3\].
\[4 = 2 \times 2\].
Now there is no common prime number, for finding the L.C.M we will multiply the common term once and the other terms.
Thus, the L.C.M of \[9\text{ and }4\]=\[2 \times 2 \times 3 \times 3 = 36\]
Now we can convert the fractions like the following way:
\[\dfrac{{31}}{9} \times 1\] and \[\dfrac{{15}}{4} \times 1\]
Or,\[\dfrac{{31}}{9} \times \dfrac{4}{4}\] and \[\dfrac{{15}}{4} \times \dfrac{9}{9}\]
i.e., \[\dfrac{{124}}{{36}}\] and \[\dfrac{{135}}{{36}}\]
Hence,
 \[\begin{align}
  &\dfrac{{31}}{9} + \dfrac{{15}}{4} \\
   &= \dfrac{{124}}{{36}} + \dfrac{{135}}{{36}} \\
   &= \dfrac{{259}}{{36}} \\
  & = 7\dfrac{7}{{36}} \\
\end{align} \]
Therefore, \[3\dfrac{4}{9} + 3\dfrac{3}{4} = 7\dfrac{7}{{36}}\].

Note: Proper fraction:
A fraction where the numerator (the top number) is less than the denominator (the bottom number). For example,\[\dfrac{1}{4},\dfrac{3}{5}\] etc.
Improper fraction:
A fraction where the numerator (the top number) is greater than the denominator (the bottom number).
For example,\[\dfrac{7}{5},\dfrac{3}{2}\] etc.
Mixed fraction:
A whole number and a proper fraction combined into one “Mixed fraction”.
For example,\[5\dfrac{1}{2},7\dfrac{1}{5}\] etc.