To find 10 rational numbers between \(\frac{-3}{4}\) and \(\frac{5}{6}\), we can follow these steps:
1. Understand the Boundaries:
- The starting point (lower boundary) is \(\frac{-3}{4}\).
- The ending point (upper boundary) is \(\frac{5}{6}\).
2. Express the Boundaries in Decimal Form for Clarity:
- \(\frac{-3}{4} = -0.75\)
- \(\frac{5}{6} \approx 0.8333\)
3. Find the Intervals Between These Points:
- We need to find 10 equally spaced rational numbers between \(-0.75\) and \(0.8333\). This means there are 9 intervals between the total 10 numbers.
4. Calculate the Spacing:
- The difference between the two endpoints is \(0.8333 - (-0.75) = 1.5833\).
- Divide this interval into 9 equal parts: \( \frac{1.5833}{9} \approx 0.17592593\).
5. Generate the Rational Numbers:
- Start from \(-0.75\) and add the interval spacing successively.
So, the 10 rational numbers between \(\frac{-3}{4}\) and \(\frac{5}{6}\) are:
1. \(-0.75\)
2. \(-0.75 + 0.17592593 \approx -0.57407407\)
3. \(-0.57407407 + 0.17592593 \approx -0.39814815\)
4. \(-0.39814815 + 0.17592593 \approx -0.22222222\)
5. \(-0.22222222 + 0.17592593 \approx -0.0462963\)
6. \(-0.0462963 + 0.17592593 \approx 0.12962963\)
7. \(0.12962963 + 0.17592593 \approx 0.30555556\)
8. \(0.30555556 + 0.17592593 \approx 0.48148148\)
9. \(0.48148148 + 0.17592593 \approx 0.65740741\)
10. \(0.65740741 + 0.17592593 \approx 0.83333333\)
Thus, the 10 rational numbers between \(\frac{-3}{4}\) and \(\frac{5}{6}\) are:
[tex]\[
-0.75, -0.57407407, -0.39814815, -0.22222222, -0.0462963, 0.12962963, 0.30555556, 0.48148148, 0.65740741, 0.83333333
\][/tex]
