If Jane gives 18 beads to Doris, both of them will have the same numbe of beads. If Doris gives 12 beads to Jane, Doris will have half as man beads as Jane. How many beads does Jane have at first?

Let's solve the problem step by step by setting up equations based on the information provided. 1. **Let:** - \( J \) = Initial number of beads Jane has. - \( D \) = Initial number of beads Doris has. 2. **First Scenario:** - Jane gives 18 beads to Doris. - **After the transaction:** - Jane has \( J - 18 \) beads. - Doris has \( D + 18 \) beads. - According to the problem, after this transaction, they have the same number of beads: \[ J - 18 = D + 18 \quad \text{(Equation 1)} \] 3. **Second Scenario:** - Doris gives 12 beads to Jane. - **After the transaction:** - Doris has \( D - 12 \) beads. - Jane has \( J + 12 \) beads. - According to the problem, Doris will have half as many beads as Jane: \[ D - 12 = \frac{1}{2}(J + 12) \quad \text{(Equation 2)} \] 4. **Solving the Equations:** - **From Equation 1:** \[ D = J - 18 - 18 = J - 36 \] - **Substitute \( D = J - 36 \) into Equation 2:** \[ (J - 36) - 12 = \frac{1}{2}(J + 12) \] \[ J - 48 = \frac{1}{2}J + 6 \] \[ J - \frac{1}{2}J = 6 + 48 \] \[ \frac{1}{2}J = 54 \] \[ J = 108 \] **Answer:** Jane initially has **108 beads**.