$3.$ Advanced Break$-$Even Analysis and Profitability

A digital consulting agency offers three services with different hourly rates:

$-$ Data Analysis: $$400$ per hour

$-$ Market Research: $$350$ per hour

$-$ Strategic Planning: $$500$ per hour

The variable costs for each service are:

$-$ Data Analysis: $$150$ per hour

$-$ Market Research: $$100$ per hour

$-$ Strategic Planning: $$200$ per hour

Fixed costs for the agency are $$3,000,000$ annually.

$$ a$)$ Calculate the break$-$even point in hours for each service individually, assuming only one service is provided.

$$ b$)$ If the sales mix is $40\%$ Data Analysis, $30\%$ Market Research, and $30\%$ Strategic Planning, determine the combined break$-$even point in hours.

$$ c$)$ The company expects to sell $2,000$ hours in total this year. Calculate the profit or loss

$$ d$)$ If fixed costs decrease by $20\%$ due to cost reductions, find the new break$-$even point for the sales mix in part $($b$).$