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Two rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{3}$ using the Average Method.

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Last updated date: 12th Jul 2024
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Answer
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Hint: We solve this question by using the average method for obtaining a set of numbers between two numbers. This method can be done by using the formula $\dfrac{a+b}{2}.$ Using this twice for the set of numbers a and b, we can obtain the numbers.

Complete step by step answer:
We are required to find two rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{3}$ using the Average Method. To do so, we use the average formula for the two numbers given by a and b,
$\Rightarrow \dfrac{a+b}{2}$
The two numbers given are $\dfrac{1}{4}$ and $\dfrac{1}{3}$ , using them in the above formula,
\[~\Rightarrow \dfrac{\dfrac{1}{4}+\dfrac{1}{3}}{2}\]
To add the two terms in the numerator, take the LCM of the denominators,
$\Rightarrow LCM\left( 3,4 \right)=12$
Using this in the above equation by multiplying both the numerator and denominator by 3 for the first term and by 4 for the second term,
\[~\Rightarrow \dfrac{\dfrac{1\times 3}{4\times 3}+\dfrac{1\times 4}{3\times 4}}{2}\]
Multiplying the terms,
\[~\Rightarrow \dfrac{\dfrac{3}{12}+\dfrac{4}{12}}{2}\]
Adding the two numerators since the denominator terms are same,
\[~\Rightarrow \dfrac{\dfrac{7}{12}}{2}\]
The denominator can be represented as $\dfrac{2}{1},$ and this term can be reciprocated and multiplied with the numerator as,
\[~\Rightarrow \dfrac{7}{12}\times \dfrac{1}{2}=\dfrac{7}{24}\]
Hence, one of the numbers is $\dfrac{7}{24}.$ Next, we use this as one of the numbers and say $\dfrac{1}{4}$ as the second number and calculate the average for these two numbers. Doing so gives us a number between the two numbers $\dfrac{7}{24}$ and $\dfrac{1}{4},$ which is still between the two numbers $\dfrac{1}{4}$ and $\dfrac{1}{3}$ .
Calculating the average for $\dfrac{7}{24}$ and $\dfrac{1}{4},$
$\Rightarrow \dfrac{\dfrac{7}{24}+\dfrac{1}{4}}{2}$
Now, we need to find the LCM of the denominators of the two terms in the numerator. This is found to be
$\Rightarrow LCM\left( 4,24 \right)=24$
Using this in the above equation by multiplying both the numerator and denominator of the first term by 1 and numerator and denominator of the second term by 6.
\[~\Rightarrow \dfrac{\dfrac{7\times 1}{24\times 1}+\dfrac{1\times 6}{4\times 6}}{2}\]
Multiplying the terms,
\[~\Rightarrow \dfrac{\dfrac{7}{24}+\dfrac{6}{24}}{2}\]
Adding the two numerators since the denominator terms are same,
\[~\Rightarrow \dfrac{\dfrac{13}{24}}{2}\]
The denominator can be represented as $\dfrac{2}{1},$ and this term can be reciprocated and multiplied with the numerator as,
\[~\Rightarrow \dfrac{13}{24}\times \dfrac{1}{2}=\dfrac{13}{48}\]
Hence, the other number is $\dfrac{13}{48}.$ Hence, the two rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{3}$ using Average Method are found to be $\dfrac{7}{24}$ and $\dfrac{13}{48}.$

Note: We need to note that concept of average method for finding rational numbers between two numbers. It is also important to know how to calculate the LCM of two numbers. We can even apply this concept of average method for integers too.