Question

What should be added to \[\left( {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}} \right)\] to get \[3\] ?

Answer 1

Hint: In order to solve this question, we will let the number that should be added is \[a\] . Now firstly we will simplify the given terms by taking L.C.M and find the sum. Then we will again take the sum of the obtained rational number with \[a\] and equate it with \[3\] to form a linear equation in \[a\] . After that we will solve the equation for the value of \[a\] and hence we will get the required result.

Complete step by step answer:
Here we have been asked to find the number that should be added to \[\left( {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}} \right)\] to get the resultant value as \[3\]. Now, let us assume that the number that should be added is \[a\]. Now first of all let’s find the sum of the given terms i.e., \[\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}\].Now on taking L.C.M of \[2,3\] and \[5\] which is \[30\] we get \[\dfrac{{15 + 10 + 6}}{{30}}\]. On adding, we get \[\dfrac{{31}}{{30}}\].

So according to the question we have to add \[a\] to \[\dfrac{{31}}{{30}}\] to get the result as \[3\]. Therefore, we have
\[ \Rightarrow a + \dfrac{{31}}{{30}} = 3\]
Clearly, this is a linear equation in \[a\] so we need to solve for the value of \[a\] .
Therefore, leaving \[a\] in the left-hand side and taking all the other terms to the right-hand side, we get
\[ \Rightarrow a = 3 - \dfrac{{31}}{{30}}\]
On taking L.C.M we get
\[ \Rightarrow a = \dfrac{{90 - 31}}{{30}}\]
On simplifying, we get
\[ \therefore a = \dfrac{{59}}{{30}}\]

Hence, the required number that must be added is \[\dfrac{{59}}{{30}}\].

Note: We must remember the basic thing of how to simplify rational numbers. And also note that to add fractions we take the L.C.M of their denominators and not the numerators. Also keep in mind that the L.C.M of prime numbers is simply their product. As here \[2,3\] and \[5\] all are prime numbers and hence their L.C.M is simply \[30\] . Here, we can also verify the result by just simply substituting the value of \[a\]
As we have
L.H.S. equal to
\[a + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}\]
On substituting we get
\[ \Rightarrow \dfrac{{59}}{{30}} + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}\]
On simplifying, we get
\[ \Rightarrow \dfrac{{59 + 15 + 10 + 6}}{{30}} = \dfrac{{90}}{{30}} = 3\]
which is equal to the R.H.S
Hence, we get the correct answer.