Question

What is the multiplicative inverse of \[\dfrac{4}{15}?\]

Answer 1

Hint: We have to find the multiplicative inverse of \[\dfrac{4}{15},\] for that we will first define what multiplicative inverse is. We get y in the multiplicative inverse of x if their product is 1, i.e. xy = 1. So, we will consider that x is the multiplicative inverse of \[\dfrac{4}{15}.\] That means, \[x\times \dfrac{4}{15}=1.\] Then we will solve for x and get our required answer.

Complete step-by-step answer:
We are asked to find the multiplicative inverse of \[\dfrac{4}{15}.\] First, we need to understand what multiplicative inverse is. In simple words, we describe the multiplicative inverse of any number (say x) as the number which when multiplied by x will give as 1. This means for any number x, another number y is called the multiplicative inverse if its product with x becomes 1. If \[x\times y=1,\] then y is the multiplicative inverse of x.
Let us assume that the multiplicative inverse of \[\dfrac{4}{15}\] is x. Then as per definition, the product of x and \[\dfrac{4}{15}\] must be 1.
\[\Rightarrow x\times \dfrac{4}{15}=1\]
On simplifying, we get,
\[\Rightarrow \dfrac{4x}{15}=1\]
Now, multiplying both the sides by 15, we get,
\[\Rightarrow \dfrac{4x}{15}\times 15=15\]
\[\Rightarrow 4x=15\]
Now dividing both sides by 4, we get,
\[\Rightarrow \dfrac{4x}{4}=\dfrac{15}{4}\]
\[\Rightarrow x=\dfrac{15}{4}\]
So, we get our required multiplicative inverse of \[\dfrac{4}{15}\] as \[\dfrac{15}{4}.\]

Note: We can find the multiplicative inverse of any number except 0 as the product of 0 cannot give us 1. In the simplest terms, the multiplicative inverse of any number is just its reciprocal. So, for \[\dfrac{4}{15},\] the multiplicative inverse of \[\dfrac{4}{15}\] is the reciprocal of \[\dfrac{4}{15}\] which is \[\dfrac{15}{4}.\]