Question

Three rational number between 2/3 and 4/5

Answer 1

Answer :-

• Three rational numbers between 2/3 and 4/5 are 41/60, 42/60 and 43/60.

Given :-

• Two rational numbers.

To find :-

• Three rational numbers between them.

Step-by-step explanation :-

• Here, we are asked to find three rational numbers between 2/3 and 4/5. So first we have to make their denominator same. For this, we have to find the LCM of their denominators [3, 5]

LCM of 3 and 5 :-

$\begin{array}{c | c} \underline3 & \underline{3, 5} \\ \underline5 & \underline{1, 5} \\ & 1, 1\end{array}$

Hence,

• The LCM of 3 and 5 is 3 × 5 = 15.

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Now, the given numbers can also be written as :-

$\dfrac{2}{3} = \dfrac{2 \times 5}{3 \times 5} = \dfrac{10}{15}$

$\dfrac{4}{5} = \dfrac{4 \times 3}{5 \times 3} = \dfrac{12}{15}$

We notice that there are only two integers between 10/15 and 12/15. Thus, writing the given rational numbers with denominator 15 is not sufficient.

So, to insert 3 rational numbers, we will multiply both the numerator and denominator of each rational number by (3 + 1) that is, 4.

We have :-

$\dfrac{10}{15} = \dfrac{10 \times 4}{15 \times 4} = \dfrac{40}{60}$

$\dfrac{12}{15} = \dfrac{12 \times 4}{15 \times 4} = \dfrac{48}{60}$

Now,

$\bf \dfrac{40}{60} , \dfrac{41}{60} , \dfrac{42}{60} , \dfrac{43}{60} \: .... \: \dfrac{48}{60}$

We can choose any three rational numbers from here.

Hence :-

• Three rational numbers between 2/3 and 4/5 are 41/60, 42/60 and 43/60.

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Answer 2

To Find :- Three rational number between 2/3 and 4/5 .

Concept used :- A rational number between a and b is equal to (a + b) ÷ 2 .

Solution :-

First rational number between 2/3 and 4/5 :-

$\rightarrow\sf \: \frac{( \frac{2}{3} + \frac{4}{5})}{2} \\ \\ \rightarrow\sf \: \frac{ \frac{10 + 12}{15} }{2} \\ \\ \rightarrow\sf \: \frac{ \frac{22}{15} }{2} \\ \\ \rightarrow\sf \: \frac{22}{15} \times \frac{1}{2} \\ \\ \rightarrow\boxed{\frac{22}{30}}$

Now, Second rational number between 2/3 and 4/5 = A rational number between 2/3 and 22/30 :-

$\rightarrow\sf \: \frac{( \frac{2}{3} + \frac{22}{30})}{2} \\ \\ \rightarrow\sf \: \frac{ \frac{20 + 22}{30} }{2} \\ \\ \rightarrow\sf \: \frac{ \frac{42}{30} }{2} \\ \\ \rightarrow\sf \: \frac{42}{30} \times \frac{1}{2} \\ \\ \rightarrow\boxed{\frac{42}{60}}$

Now, Third rational number between 2/3 and 4/5 = A rational number between 22/30 and 4/5 :-

$\rightarrow\sf \: \frac{( \frac{4}{5} + \frac{22}{30})}{2} \\ \\ \rightarrow\sf \: \frac{ \frac{24 + 22}{30} }{2} \\ \\ \rightarrow\sf \: \frac{ \frac{46}{30} }{2} \\ \\ \rightarrow\sf \: \frac{46}{30} \times \frac{1}{2} \\ \\ \rightarrow\boxed{\frac{46}{60}}$

therefore, Three rational numbers between 2/3 and 4/5 are :-

$\red\rightarrow\sf \large\boxed{\bf\frac{42}{60}, \frac{22}{30} \: \: and \: \:\frac{46}{60}}$

Learn more :-

let a and b positive integers such that 90 less than a+b less than 99 and 0.9 less than a/b less than 0.91. Find ab/46

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