nment A company charges \( \$ 20 \) to make one monogrammed shirt, but reduces this cost by \( \$ 0.10 \) per shirt for each additional shirt ordered up to 100 shirts. If the of an order is \( \$ 846 \), how many shirts were ordered? If the cost of an order is \( \$ 846.2 .145 \), \( \square \) shirts were ordered. (Simplify your answer.)

We let \( x \) be the number of shirts ordered. The company’s pricing rule is that the cost to make one shirt is reduced by \(\$0.10\) for every additional shirt ordered. In other words, the cost per shirt when \( x \) shirts are ordered becomes \[ 20 - 0.10(x-1). \] Simplify the expression: \[ 20 - 0.10(x-1)=20-0.10x+0.10=20.10-0.10x. \] The total cost of the order, \( C \), is the number of shirts times the cost per shirt: \[ C = x(20.10-0.10x). \] For the first part of the problem the total cost is given as \(\$846\). Thus the equation is: \[ x(20.10-0.10x)=846. \] To clear decimals, multiply both sides by 10: \[ x(201- x)=8460. \] Expanding, we obtain: \[ 201x-x^2=8460. \] Writing in standard quadratic form: \[ x^2-201x+8460=0. \] Next, we compute the discriminant: \[ \Delta = 201^2-4\cdot 1\cdot8460. \] Calculate: \[ 201^2=40401, \] \[ 4\cdot8460=33840, \] so \[ \Delta=40401-33840=6561. \] Since \[ \sqrt{6561}=81, \] the solutions for \( x \) are given by: \[ x=\frac{201\pm 81}{2}. \] Thus, \[ x=\frac{201+81}{2}=\frac{282}{2}=141 \quad \text{or} \quad x=\frac{201-81}{2}=\frac{120}{2}=60. \] Because the discount applies only for orders up to 100 shirts, the value \( x=141 \) is not allowable. Therefore, the number of shirts ordered is \[ \boxed{60}. \]