Certainly, let's break down the problem step by step:
Part (a):
Shufen has two schemes to choose from for her new salary.
1. Scheme A: Shufen gets a 12% increase in her current monthly salary.
2. Scheme B: Shufen gets an 8% increase in her current monthly salary plus a [tex]$1000 annual bonus.
Given:
- Shufen's current monthly salary is $[/tex]2400.
Calculation for Scheme A:
- Current monthly salary = [tex]$2400
- Increase under Scheme A = 12% of $[/tex]2400 = 0.12 * [tex]$2400 = $[/tex]288
- New monthly salary under Scheme A = [tex]$2400 + $[/tex]288 = [tex]$2688
Calculation for Scheme B:
- Current monthly salary = $[/tex]2400
- Increase under Scheme B = 8% of [tex]$2400 = 0.08 * $[/tex]2400 = [tex]$192
- New monthly salary excluding bonus = $[/tex]2400 + [tex]$192 = $[/tex]2592
- Total annual salary under Scheme B = Monthly salary for 12 months + [tex]$1000 bonus
= $[/tex]2592 * 12 + [tex]$1000 = $[/tex]31104 + [tex]$1000 = $[/tex]32104
- New effective monthly salary under Scheme B = [tex]$32104 / 12 ≈ $[/tex]2675.33
Comparison:
- New monthly salary under Scheme A = [tex]$2688
- New effective monthly salary under Scheme B ≈ $[/tex]2675.33
Since [tex]$2688 (Scheme A) is greater than $[/tex]2675.33 (Scheme B), Scheme A is the better option for Shufen.
Part (b):
To find the new current monthly salary that makes Scheme B better, let's set up the inequality where the annual salary with Scheme B should be more than the annual salary with Scheme A.
Recall:
- New monthly salary under Scheme A = [tex]$2688
- Annual salary with Scheme A = $[/tex]2688 * 12 = [tex]$32256
Let \( x \) be the new current monthly salary when Scheme B becomes better:
- Scheme B annual salary with \( x \):
- Monthly salary after 8% increase = \( x * 1.08 \)
- Annual salary under Scheme B = \( 12 (x 1.08) + 1000 \)
For Scheme B to be better than Scheme A:
\[ 12 (x 1.08) + 1000 > 32256 \]
Solve the inequality:
\[ 12 (x 1.08) + 1000 > 32256 \]
\[ 12 (x 1.08) > 32256 - 1000 \]
\[ 12 (x 1.08) > 31256 \]
\[ x * 1.08 > 31256 / 12 \]
\[ x * 1.08 > 2604.6667 \]
\[ x > 2604.6667 / 1.08 \]
\[ x > 2411.7284 \]
Thus, to make Scheme B the better option, Shufen's current monthly salary should be more than approximately $[/tex]2411.73.
So, a possible amount for Shufen's current monthly salary if she were to accept Scheme B to make it the better offer would be anything above approximately $2411.73.
