Simplify ((((C x 105.55) / 100) x 103.96) / 100) - C and state the result as a percentage of C.

Step-by-step solution

1. 1

   Rewrite each division by 100 as a growth multiplier. Dividing by 100 after multiplying by $105.55$ is the same as multiplying by

   $\frac{105.55}{100}=1.0555,\qquad \frac{103.96}{100}=1.0396$

   so the expression is simply $C\cdot 1.0555\cdot 1.0396-C$: an increase of $5.55\%$ followed by one of $3.96\%$, with the final amount compared against the original.

2. 2

   Multiply the two growth factors. Successive percentage changes multiply, they do not add:

   $1.0555\times 1.0396=1.0972978$

   This is exact, not rounded — both factors have four decimals, so the product has at most eight.

3. 3

   Factor out $C$. The expression is a difference of two multiples of the same quantity:

   $C\cdot 1.0972978-C\cdot 1=C\left(1.0972978-1\right)=0.0972978\,C$

   Factoring first means the answer stays valid for every value of $C$; there is no need to know the starting amount.

4. 4

   Read the result as a percentage.

   $0.0972978\,C=9.72978\%\ \text{of}\ C$

   So the two rises together add $9.72978\%$ to the original.

5. 5

   Compare with the naive sum to see the compounding. Simply adding gives $5.55\%+3.96\%=9.51\%$. The extra $0.21978\%$ is the second increase applied to the first increase, i.e. $0.0555\times 0.0396=0.0021978$. That cross term is exactly the gap, which confirms the multiplication was done correctly.