Chapter 0.5, Problem 30E

To determine

The reciprocal of the number $3\frac{3}{4}$ .

Answer to Problem 30E

The reciprocal of the number $3\frac{3}{4}$ is $\frac{4}{15}$ .

Explanation of Solution

Given information:

The number $3\frac{3}{4}$ .

Formula used:

Mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

Reciprocal of any real number (except 0) is obtained when 1 is divided by the number.

  $\begin{array}{c}\frac{a}{b}=1÷\frac{a}{b}\\ =1\cdot \frac{b}{a}\\ =\frac{b}{a}\end{array}$

Here, $\frac{b}{a}$ is called the multiplicative inverse of $\frac{a}{b}$ .

When a number is multiplied with its reciprocal, then the result is always 1.

  $\frac{a}{b}\cdot \frac{b}{a}=1$

Calculation:

Consider the expression $3\frac{3}{4}$ .

Recall that mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

So, $3\frac{3}{4}$ will be converted as,

  $\begin{array}{c}3\frac{3}{4}=\frac{\left(4\times 3\right)+3}{4}\\ =\frac{15}{4}\end{array}$

Reciprocal of any real number (except 0) is obtained when 1 is divided by the number.

  $\begin{array}{c}\frac{a}{b}=1÷\frac{a}{b}\\ =1\cdot \frac{b}{a}\\ =\frac{b}{a}\end{array}$

Here, $\frac{b}{a}$ is called the multiplicative inverse of $\frac{a}{b}$ .

So, the reciprocal of $\frac{15}{4}$ is evaluated as,

  $\begin{array}{c}\frac{15}{4}=1÷\left(\frac{15}{4}\right)\\ =1\cdot \left(\frac{4}{15}\right)\\ =\frac{4}{15}\end{array}$

Recall when a number is multiplied with its reciprocal, then the result is always 1.

  $\frac{a}{b}\cdot \frac{b}{a}=1$

So, when $\frac{15}{4}$ is multiplied by $\frac{4}{15}$ , the result should be 1.

  $\frac{15}{4}\cdot \left(\frac{4}{15}\right)=1$

Thus, the reciprocal of the number $\frac{15}{4}$ is $\frac{4}{15}$ .