Question

Find 10 rational numbers between $\dfrac{3}{5}\And \dfrac{5}{12}$ using L.C.M method.

Answer 1

Hint: To find 10 rational numbers between $\dfrac{3}{5}\And \dfrac{5}{12}$ using L.C.M method, we are going to first of all make the denominators of the two fractions $\dfrac{3}{5}\And \dfrac{5}{12}$ as same. We can make the denominator the same by multiplying the denominator of $\dfrac{3}{5}$ by 12 and the denominator for $\dfrac{5}{12}$ by 5. Then to find the 10 rational numbers between them we are going to multiply the numerator and the denominator of these two fractions by 11 and after that find the rational numbers between them.

Complete step-by-step answer:
The two fractions given in the above problem of which we have to find the 10 rational numbers are as follows:
$\dfrac{3}{5}\And \dfrac{5}{12}$
Now, we are going to multiply the denominator of $\dfrac{3}{5}$ by 12 and the denominator for $\dfrac{5}{12}$ by 5 and we get,
$\begin{align}
  & \dfrac{3}{5\times 12}=\dfrac{3}{60}; \\
 & \dfrac{5}{12\times 5}=\dfrac{5}{60} \\
\end{align}$
To get 10 rational numbers between $\dfrac{3}{60}\And \dfrac{5}{60}$ we are going to multiply the numerator and the denominator by 11 in both the fractions and we get,
$\begin{align}
  & \dfrac{3\times 11}{60\times 11}=\dfrac{33}{660}; \\
 & \dfrac{5\times 11}{60\times 11}=\dfrac{55}{660} \\
\end{align}$
Now, writing the rational numbers between the above two fractions and we get,
$\dfrac{34}{660},\dfrac{35}{660},\dfrac{36}{660},\dfrac{37}{660},\dfrac{38}{660},\dfrac{39}{660},\dfrac{40}{660},\dfrac{41}{660},\dfrac{42}{660},\dfrac{43}{660}$
Hence, we have written the 10 rational numbers between $\dfrac{3}{5}\And \dfrac{5}{12}$.

Note: You might be thinking that we can write the 10 rational numbers as follows: $\dfrac{44}{660},\dfrac{45}{660},\dfrac{46}{660},\dfrac{47}{660},\dfrac{48}{660},\dfrac{49}{660},\dfrac{50}{660},\dfrac{51}{660},\dfrac{52}{660},\dfrac{53}{660},\dfrac{54}{660}$
So, the answer is we can yes, we can write these rational numbers. The above rational numbers are completely correct.
Also, just like we have multiplied numerator and denominator by 11, we can multiply numerator and denominator by 12 and find the rational numbers.
Similarly, you can multiply other numbers (like 13, 14, 15, 16…) also in the numerator and denominator to get the rational numbers.