Let a = 1/12 and b = 5/6. Express both a and b in the form Class 9

Let a = 1/12 and b = 5/6. Express both a and b in the form Class 9

Question 1.

Let a = $\frac{7}{12}$ and b = $\frac{5}{6}$. Express both a and b in the form $\frac{k_1}{m}$ and $\frac{k_2}{m}$ where k₁, k₂ and m are integers and k₂ - k₁ > 6. Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition k₂ - k₁ > n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.

Solution:

Given,

a = $\frac{7}{12}$, b = $\frac{5}{6}$

LCM(12, 6) = 12

a = $\frac{7}{12}$ and b = $\frac{5}{6}$ = $\frac{10}{12}$

Here,

k₁ = 7, k₂ = 10, m = 12

But k₂ - k₁ = 10 - 7 = 3

which is not greater than 6.

Again, multiplying numerator and denominator by 3, we get

$\begin{aligned} & a=\frac{7}{12}=\frac{21}{36} \\ & b=\frac{10}{12}=\frac{30}{36}\end{aligned}$

Now,

k₁ = 21, k₂ = 30, m = 36

and k₂ - k₁ = 30 - 21 = 9 > 6

Thus, the condition is satisfied.

Now, the fractions between $\frac{21}{36}$ and $\frac{30}{36}$ are $\frac{22}{36}, \frac{23}{36}, \frac{24}{36}, \frac{25}{36}, \frac{26}{36}$

These are five distinct rational numbers between a and b.

Now, suppose the two rational numbers are $\frac{k_1}{m}$ and $\frac{k_2}{m}$.

The rational numbers lying strictly between them are $\frac{k_1+1}{m}, \frac{k_1+2}{m}, \ldots, \frac{k_2-1}{m}$

The total number of such rational numbers is k₂ > - k₁ - 1

To get exactly n rational numbers,

k₂ - k₁ - 1 ≥ n

∴ k₂ - k₁ ≥ n + l

Hence, this condition is necessary.