Question

By what rational number should \[\dfrac{-7}{12}\] be multiplied to get the product as \[\dfrac{5}{14}?\]

Answer 1

Hint: To solve this question, we will first assume a variable whose value is to be determined. Let it be x, then we will apply the given condition in the question that x multiplied by \[\dfrac{-7}{12}\] should give \[\dfrac{5}{14},\] then following the proper calculation, we will get the result.

Complete step by step answer:
Let us assume the number as x. Then according to the given condition in the question, x is a rational number and x multiplied by \[\dfrac{-7}{12}\] should give \[\dfrac{5}{14}.\] Now, let us write this mathematically, then we will have,
\[\Rightarrow x\left( \dfrac{-7}{12} \right)=\dfrac{5}{14}\]
Multiplying both the sides by 12, we get,
\[\Rightarrow -7x\times \dfrac{12}{12}=\dfrac{5}{14}\times 12\]
\[\Rightarrow -7x=\dfrac{5\times 12}{14}\]
Again dividing both the sides of the above equation by 7, we get,
\[\Rightarrow \dfrac{-7x}{7}=\dfrac{5\times 12}{14\times 7}\]
\[\Rightarrow -x=\dfrac{5\times 12}{14\times 7}\]
Again multiplying (– 1) both the sides and using – (– 1) = + 1, we have,
\[\Rightarrow x=\dfrac{-5\times 12}{14\times 7}\]
Finally, simplifying the RHS, we get,
\[\Rightarrow x=\dfrac{-5\times 6}{7\times 7}\]
\[\Rightarrow x=\dfrac{-30}{49}\]
So, the value of x is \[\dfrac{-30}{49}.\] Therefore, \[\dfrac{-30}{49}\] should be multiplied by \[\dfrac{-7}{12}\] to get the product as \[\dfrac{5}{14}.\]

Note: A key point to note here in this question is that we need a number \[\dfrac{5}{14}\] that is positive and the number on LHS \[\left( \dfrac{-7}{12} \right)\] is negative. Therefore, the product of two negative terms gives a positive number. Therefore, any number for the value of x should be negative. We are getting \[x=\dfrac{-30}{49}\] which is negative, so it is correct.