Sure, let's solve the problem step-by-step.
1. Given problem statement:
- Let's denote the original positive number by [tex]\( x \)[/tex].
- According to the problem, you perform the following operations on [tex]\( x \)[/tex]:
- Add 7 to [tex]\( x \)[/tex]: [tex]\( x + 7 \)[/tex]
- Divide the result by 3: [tex]\(\frac{x + 7}{3}\)[/tex]
- Subtract 3 from the result: [tex]\(\frac{x + 7}{3} - 3\)[/tex]
- Square the result: \left(\frac{x + 7}{3} - 3\right)^2
- The final result is 16.
2. Set up the equation:
[tex]\[
\left(\frac{x + 7}{3} - 3\right)^2 = 16
\][/tex]
3. Take the square root of both sides to simplify:
[tex]\[
\frac{x + 7}{3} - 3 = \pm 4
\][/tex]
We get two possible equations from this:
[tex]\[
\frac{x + 7}{3} - 3 = 4 \quad \text{and} \quad \frac{x + 7}{3} - 3 = -4
\][/tex]
4. Solve both equations:
For the first equation:
[tex]\[
\frac{x + 7}{3} - 3 = 4
\][/tex]
[tex]\[
\frac{x + 7}{3} = 7
\][/tex]
[tex]\[
x + 7 = 21
\][/tex]
[tex]\[
x = 14
\][/tex]
For the second equation:
[tex]\[
\frac{x + 7}{3} - 3 = -4
\][/tex]
[tex]\[
\frac{x + 7}{3} = -1
\][/tex]
[tex]\[
x + 7 = -3
\][/tex]
[tex]\[
x = -10
\][/tex]
But since [tex]\( x \)[/tex] must be a positive number, we discard [tex]\( x = -10 \)[/tex] as a solution.
5. Conclusion:
The number you thought of is [tex]\( \boxed{14} \)[/tex].
