
Find 5 rational numbers between 1 and 2. We can approach this problem in at least two ways.
Answer
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Hint: A rational number is a number which can be expressed in the form $\dfrac{a}{b}$ where a and b are integers and $b\ne 0$ .
Rational numbers are either terminating or non-terminating and repeating in decimal form.
e.g. 3.4, 5.7777... = $5.\bar{7}$, 8.13 etc.
A number Z, which is greater than X and smaller than Y, is said to be between X and Y.
Complete step-by-step answer:
Method 1:
The numbers 1 and 2 have a denominator 1 and can be written as $\dfrac{1}{1}$ and $\dfrac{2}{1}$ respectively.
The numerators differ by 1 here. Since we have to find 5 rational numbers between them, let us change their denominators to such numbers so that the difference between the numerators becomes at least 6.
$\dfrac{1}{1}=\dfrac{1\times 10}{1\times 10}=\dfrac{10}{10}$ and $\dfrac{2}{1}=\dfrac{2\times 10}{1\times 10}=\dfrac{20}{10}$ .
Now, using the properties of numbers, we find that $\dfrac{11}{10},\dfrac{12}{10},\dfrac{13}{10},\dfrac{14}{10},\dfrac{15}{10}$ etc. are all between $\dfrac{10}{10}$ and $\dfrac{20}{10}$ .
Method 2:
The numbers 1 and 2 can also be written as 1.0 and 2.0. Now a number which is greater than 1.0 and less than 2.0 will be between them.
In other words, a number whose units place is 1 and the decimal part is more than 0, will be more than 1.0.
Therefore, 1.1, 1.2, 1.3, 1.4, 1.5 etc. are all between 1.0 and 2.0.
Note: There are infinitely many numbers between any two distinct real numbers.
$0.\bar{9}$ and 1 are not distinct numbers, they are equal.
The denominators can be changed to any number but it is easier to convert into multiples of 10.
Rational numbers are either terminating or non-terminating and repeating in decimal form.
e.g. 3.4, 5.7777... = $5.\bar{7}$, 8.13 etc.
A number Z, which is greater than X and smaller than Y, is said to be between X and Y.
Complete step-by-step answer:
Method 1:
The numbers 1 and 2 have a denominator 1 and can be written as $\dfrac{1}{1}$ and $\dfrac{2}{1}$ respectively.
The numerators differ by 1 here. Since we have to find 5 rational numbers between them, let us change their denominators to such numbers so that the difference between the numerators becomes at least 6.
$\dfrac{1}{1}=\dfrac{1\times 10}{1\times 10}=\dfrac{10}{10}$ and $\dfrac{2}{1}=\dfrac{2\times 10}{1\times 10}=\dfrac{20}{10}$ .
Now, using the properties of numbers, we find that $\dfrac{11}{10},\dfrac{12}{10},\dfrac{13}{10},\dfrac{14}{10},\dfrac{15}{10}$ etc. are all between $\dfrac{10}{10}$ and $\dfrac{20}{10}$ .
Method 2:
The numbers 1 and 2 can also be written as 1.0 and 2.0. Now a number which is greater than 1.0 and less than 2.0 will be between them.
In other words, a number whose units place is 1 and the decimal part is more than 0, will be more than 1.0.
Therefore, 1.1, 1.2, 1.3, 1.4, 1.5 etc. are all between 1.0 and 2.0.
Note: There are infinitely many numbers between any two distinct real numbers.
$0.\bar{9}$ and 1 are not distinct numbers, they are equal.
The denominators can be changed to any number but it is easier to convert into multiples of 10.
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