Question

Solve the following:
 \[7\div 3.5\]

Answer 1

Hint: The given problem involves integers as well as a number with a decimal point. In order to solve it, we have to make it simpler. We do that by converting \[3.5\]into integer-by-integer form. Then the problem becomes easy, as it involves only integers.

Complete step by step answer:
Any system of numbers with \[10\]as the base, with \[10\]different numerals \[0,1,2,3,4,5,6,7,8,9\]and ‘.’ called dot or decimal point is called a decimal number system. Every real number has a decimal representation, which consists of two parts namely integral part and fractional part. For example, if we write the number \[7\]as \[7.0\], \[7\]is the integral part and \[0.0\]is the fractional part. In the number \[1.25\], \[1\]is the integral part and \[0.25\]is the fractional part.
Any rational number of the form \[\dfrac{p}{q}\]where \[p\]and \[q\]are integers and \[q\ne 0\]can be transformed into a decimal number and any decimal number with a finite fractional part, can be transformed into an integer-by-integer form number. In fact, any decimal number with terminating fractional part or non-terminating recurring fractional part can be transformed into an integer-by-integer form number. For example, \[0.25\]can be written as \[\dfrac{1}{4}\]and \[0.\overline{3}\]can be written as \[\dfrac{1}{3}\]. Thus, in the real number system, natural numbers, whole numbers, integers and rational numbers can be written in this way. Irrational numbers are the only numbers which cannot be written in an integer-by-integer form.
Consider our problem:
 \[7\div 3.5\]…… ( \[1\])
Here we have to divide the integer \[7\]by the number \[3.5\].
We first convert \[3.5\]to \[\dfrac{p}{q}\]from where \[p\]and \[q\]are integers and \[q\ne 0\], to make the problem simple. For that, we multiply and divide \[3.5\]by \[10\].
 \[\therefore 3.5=3.5\left( \dfrac{10}{10} \right)\]
 \[\Rightarrow 3.5=\dfrac{35}{10}\]
multiplying a decimal number by \[10\], shifts the decimal point one position to the right from the original position.
Substituting \[3.5=\dfrac{35}{10}\]in ( \[1\])
 \[\Rightarrow \] \[7\div 3.5\] \[=7\div \dfrac{35}{10}\]
 \[=7\left( \dfrac{10}{35} \right)\]
Since dividing by a number is nothing, but multiplying the reciprocal of that number. Hence
 \[7\div 3.5=\] \[\dfrac{70}{35}\]
 \[\therefore 7\div 3.5=2\]which is our required solution.

Note:
Like the decimal number system which we are using, there are other number systems as well. Binary number system with base \[2\]uses only the numbers \[0,1\]. Octal number system with base \[8\], uses the numbers \[0-7\]. Hexadecimal number system with base \[16\], uses symbols \[0-9\]and \[A-F\]. Compared to decimal number systems, these three number systems are more compatible with computer applications.