Question

In each of the following pairs of decimal numbers, state which number is smaller: \[37.701\] or \[37.71\].

Answer 1

Hint: In order to solve this problem, first of all we will convert the given decimal numbers into fractions of equal denominators. Then we will check which number is smaller using the condition that in a fraction if the denominator is the same then the fraction having less number in the numerator is the smallest among them. And hence, we will get the required result.

Complete step by step answer:
We are given the two numbers as \[37.701\] or \[37.71\]
Let us first consider the number \[37.701\] as
\[x = 37.701\]
Now on converting the decimal number to the fraction we get
\[ \Rightarrow x = \dfrac{{37701}}{{1000}}{\text{ }} - - - \left( i \right)\]
Now let us consider the number \[37.71\] as,
\[y = 37.71\]

So, on converting the decimal number to the fraction we get,
\[ \Rightarrow y = \dfrac{{3771}}{{100}}\]
Here we can see that the denominators in the given two numbers are different. So, let’s convert them into the same denominator. Here, the denominator in the first number is \[1000\] and the denominator in the second number is \[100\]. So, by multiplying the second number with \[10\] in numerator and denominator both, we get
\[ \Rightarrow y = \dfrac{{3771 \times 10}}{{100 \times 10}}\]
\[ \Rightarrow y = \dfrac{{37710}}{{1000}}{\text{ }} - - - \left( {ii} \right)\]

Now we can see that the denominators in both the numbers in the equation \[\left( i \right)\] and \[\left( {ii} \right)\] are the same. Now we know that in a fraction if the denominator is the same then the fraction having less number in the numerator is the smallest among them. Therefore, from the equation \[\left( i \right)\] and \[\left( {ii} \right)\] we can see that \[37701\] is less than \[37710\].
\[ \Rightarrow \dfrac{{37701}}{{1000}} < \dfrac{{37710}}{{1000}}\]
\[ \Rightarrow x < y\]
By back substituting the original values, we get
\[ \therefore 37.701 < 37.71\]

Hence, we conclude that \[37.701\] is smaller among the two.

Note: We can solve this problem by another method i.e., by checking the digits.
We have the condition that while comparing the digits from the left side the number having the less digit will be the smallest number. We have given the two numbers as \[37.701\] or \[37.71\]. Here, we can see that the starting two digits of both the numbers are the same. So now we will check the digits after the decimal point and apply the same condition. Now after the decimal point, the first digit is also the same in both the numbers. So, let’s check the second digit. Now the second digit after the decimal point in the first number is \[0\] and in the second number is \[1\]. So, by comparing the digits we get \[0 < 1\].Hence, we can conclude that,
\[ \Rightarrow 37.701 < 37.71\]
Hence, we get the result.