Let \(X_n\) denote the chain \(\{1,2,\ldots ,n\}\) under its natural order. We denote the semigroups consisting of all order-preserving transformations and all orientation-preserving transformations on \(X_n\) by \(\mathcal{O}_n\) and \(\mathcal{OP}_n\), respectively. We denote by \(E(U)\) the set of all idempotents of a subset \(U\) of a semigroup \(S\). The fix and image sets of a transformation \(\alpha\) are defined and denoted by \(\operatorname{fix}(\alpha) = \{x \in X_n : x\alpha = x\}\) and \(\operatorname{im}(\alpha) = \{x\alpha : x \in X_n\}\), respectively. Let \begin{align*} E_r(\mathcal{O}_n)&=\{\alpha\in E(\mathcal{O}_n):|\operatorname{im}(\alpha)|=|\operatorname{fix}(\alpha)|=r\},\\ E_r^*(\mathcal{O}_n)&=\{\alpha\in E_r(On):1,n\in\operatorname{fix}(\alpha)\},\\ E_r(\mathcal{OP}_n) &=\{\alpha\in E(OPn):|\operatorname{fix}(\alpha)|=r\}\text{ and}\\ E_r^*(\mathcal{OP}_n) &=\{\alpha\in E_r(\mathcal{OP}_n):n\in\operatorname{fix}(\alpha)\} \end{align*} where \(1\leq r\leq n\). In Section 2, the authors determine the cardinalities of \(E_r(\mathcal{O}_n),\) \(E_r^*(\mathcal{O}_n),\) \(E_r(\mathcal{OP}_n)\) and \(E_r^*(\mathcal{OP}_n)\). Using these results, the authors determine the numbers of idempotents in \(\mathcal{O}_n\) and \(\mathcal{OP}_n\) by a new method. Next, let \(\mathcal{OP}^-_n\) denote the semigroup of all orientation-preserving and order-decreasing transformations on \(X_n\). In Section 3, the authors determine the cardinalities of \(\mathcal{OP}^-_n\), \(\mathcal{OP}^-_{n,Y}=\{\alpha\in\mathcal{OP}^-_n:\operatorname{fix}(\alpha)=Y\}\) for any nonempty subset \(Y\) of \(X_n\) and \(\mathcal{OP}^-_{n,r}=\{\alpha\in\mathcal{OP}^-_n:|\operatorname{fix}(\alpha)|=r\}\) for \(1\leq r\leq n\). Also, the authors determine the number of idempotents in \(\mathcal{OP}^-_n\) and the number of nilpotents in \(\mathcal{OP}^-_n\).