A furniture assembler builds chairs and tables. Let $C$ represent the rate at which the assembler builds chairs (chairs per hour), and $T$ represent the rate at which the assembler builds tables (tables per hour). The assembler spends $H_C$ hours building chairs and $H_T$ hours building tables. The total number of items built, $N$, is given by the formula $N=C\cdot H_C+T\cdot H_T$.

If the assembler's rate for building chairs ($C$) increases, while the total number of items built ($N$) and the hours spent building tables ($H_T$) remain constant, how must the hours spent building chairs ($H_C$) change?