Ali, age $13,$ has a lawn$-$mowing business with $10$ customers. For each customer, he cuts the grass $3$ times a year, which earns him $$50$ for each mowing. He has just paid $$200$ for a new mower. The maintenance and operating cost of the mower is $$120$ for the first year in service and increases by $20\%$ a year thereafter. A $1-$year old mower has a resale value of $$150,$ which decreases by $10\%$ a year thereafter. Ali, who plans to keep his business until he is $16,$ thinks that it is more economical to keep the new mower for all the planned work period. Is his decision justified?