Question

If 76 is divided into four parts proportional to \[7\] , \[5\] , \[3\] , \[4\] then the smallest part is:
\[\left( A \right){\text{ }}12\]
\[\left( B \right){\text{ }}15\]
\[\left( C \right){\text{ }}16\]
\[\left( D \right){\text{ }}19\]

Answer 1

Hint: Change the ratio \[7:5:3:4\] into \[7x\] , \[5x\] , \[3x\] and \[4x\] then add these terms and let them equal to \[76\] because this number is divided into the ratio \[7:5:3:4\] . After adding and on further solving you will get the value of x. Put that value of x in \[3x\] to find out the value of the smallest part.

Complete step-by-step solution:
It is given to us that the number seventy six is divided into four parts of ratio \[7:5:3:4\] and we have to find the value of the smallest part. According to the question we have to find the smallest part so we have to first write the ratio in the form of terms. We know that we can only find the smallest part if it is written in the form of terms. So we will use the terms of the ratio and a non zero variable x to express the numbers. Therefore,
\[7x\] , \[5x\] , \[3x\] and \[4x\] are the terms and the sum of these terms is \[76\] . So we can write them as
 \[ \Rightarrow 7x + 5x + 3x + 4x = 76\]
On doing addition in the above expression we get
\[ \Rightarrow 19x = 76\]
On shifting nineteen to the right hand side we get
\[ \Rightarrow x = \dfrac{{76}}{{19}}\]
When we multiply nineteen by four we get seventy six. Therefore,
\[ \Rightarrow x = 4\]
Therefore, the required value of x is \[4\] .
Because we have to find out the smallest part therefore we will put this value of x in \[3x\] .By doing this we get
\[ \Rightarrow 3x\]
\[ = 3\left( 4 \right)\]
\[ = 12\]
Hence, the correct option is \[\left( A \right){\text{ }}12\]

Note: Keep in mind that in this case you have to first change the ratio in the terms including a non zero variable x. Also remember that you have to put the value of x in the term with the smallest coefficient because they are asking you the value of the smallest part. And if in the question they are asking you to find the largest part then put the value of x in terms of having the greatest coefficient among all.