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Find five rational numbers between
(2)/(5) and
(3)/(4) at equal intervals?

Find five rational numbers between $\frac{2}{5}$ and $\frac{3}{4}$ at equal intervals?

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Least common multiple of three or four numbers

Full solution

Q. Find five rational numbers between

\frac{2}{5}

and

\frac{3}{4}

at equal intervals?

Convert Fractions to Common Denominator: Convert the fractions to have a common denominator to make it easier to find numbers between them. $\newline$ The least common multiple (LCM) of the denominators $5$ and $4$ is $20$ . $\newline$ Convert $\frac{2}{5}$ to a fraction with a denominator of $20$ : $\left(\frac{2}{5}\right) * \left(\frac{4}{4}\right) = \frac{8}{20}$ . $\newline$ Convert $\frac{3}{4}$ to a fraction with a denominator of $20$ : $\left(\frac{3}{4}\right) * \left(\frac{5}{5}\right) = \frac{15}{20}$ .
Find Difference Between Fractions: Determine the difference between the two new fractions. $\newline$ The difference between $\frac{15}{20}$ and $\frac{8}{20}$ is $\frac{7}{20}$ .
Calculate Interval Size: Divide the difference by the number of intervals plus one to find the interval size. We want $5$ numbers, so we need $6$ intervals. Interval size: $(\frac{7}{20}) / 6 = \frac{7}{120}$ .
Find First Rational Number: Add the interval size to the smaller fraction to find the first rational number. $\newline$ First number: $\frac{8}{20} + \frac{7}{120} = \left(\frac{48}{120}\right) + \left(\frac{7}{120}\right) = \frac{55}{120}$ .
Find Second Rational Number: Add the interval size to the first number to find the second rational number. $\newline$ Second number: $\frac{55}{120} + \frac{7}{120} = \frac{62}{120}.$
Find Third Rational Number: Add the interval size to the second number to find the third rational number. $\newline$ Third number: $\frac{62}{120} + \frac{7}{120} = \frac{69}{120}$ .
Find Fourth Rational Number: Add the interval size to the third number to find the fourth rational number. $\newline$ Fourth number: $\frac{69}{120} + \frac{7}{120} = \frac{76}{120}$ .
Find Fifth Rational Number: Add the interval size to the fourth number to find the fifth rational number. $\newline$ Fifth number: $\frac{76}{120} + \frac{7}{120} = \frac{83}{120}$ .
Simplify Fractions: Simplify the fractions if possible. $\newline$ First number: $\frac{55}{120}$ cannot be simplified further. $\newline$ Second number: $\frac{62}{120}$ can be simplified to $\frac{31}{60}$ . $\newline$ Third number: $\frac{69}{120}$ can be simplified to $\frac{23}{40}$ . $\newline$ Fourth number: $\frac{76}{120}$ can be simplified to $\frac{19}{30}$ . $\newline$ Fifth number: $\frac{83}{120}$ cannot be simplified further.

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