Shortcut Methods
JEE Main & CBSE Board Exams:
1. Determining Number of Protons, Neutrons, Electrons:

Atomic number (Z) represents the number of protons in an atom.

Mass number (A) represents the total number of protons (Z) and neutrons (N) in an atom.
$$\text{Number of neutrons (N) = Mass number (A)  Atomic number (Z)}$$
 Number of electrons (E) is equal to the number of protons (Z) for a neutral atom.
2. Average Atomic Mass:
The average atomic mass of an element is a weighted average of the masses of its naturally occurring isotopes, taking into account their relative abundances.
Formula:
$$Average \ Atomic \ Mass = \frac{(Mass \ of \ Isotope \ 1 \times Abundance \ of \ Isotope \ 1) + (Mass \ of \ Isotope \ 2 \times Abundance \ of \ Isotope \ 2) + …}{Total \ Abundance}$$
3. Bohr’s Model Calculations:
 Radius of orbit (r) for the nth orbit:
$$r_n = \frac{n^2 h^2}{4\pi^2 m_ek Z}$$
 \(n\) is the principal quantum number.
 \(h\) is the Planck's constant.
 \(k\) is the Coulomb's constant.
 \(m_e\) is the electron mass.
 \(Z\) is the atomic number.
 Energy of the electron in the nth orbit (E):
$$E_n = \frac{Z e^2}{8\pi\varepsilon_0 r_n}$$
 \(e\) is the elementary charge.
 \(\varepsilon_0\) is the permittivity of free space.
 Wavelength of emitted photon ((\lambda)) during transition between two orbits (n1 to n2):
$$\frac{1}{\lambda}=R_H\left(\frac{1}{n_1^2}  \frac{1}{n_2^2}\right)$$
 \(R_H\) is the Rydberg constant.