When a problem asks in what year will this happen, I don't know what to do with fractional years.

Question

Often I encounter word problems that use a year as the base and then ask me to say in what year something first occurs. For example, LP's sold 100,000 units in 1984 and CD's sold 10,000. LP's are decreasing by 10,000 each year and CD's are increasing by 7500. In what year will CD's sold exceed LP's sold? 

 

My answer is 5.14 years later or 1984 + 5.14 =1989.14. Is this around the middle of Feb. 1989? Or is this in 1990?

 

I always seem to be off by a year. 

Answer 1

We can make an arithmetic sequence formula for each company.  We can use this method because the consistent change each year makes a common difference between the units.

 

units for LD:

 

L_n = 100000 - 10000(n - 1)                      because decreasing

L_n = -10000n + 110000

 

 

units for CD:

 

C_n = 10000 + 7500(n - 1)                         because increasing

C_n = 7500n + 2500

 

where     n=1 represents the year 1984

 

Set the inequality

 

C_n > L_n

 

7500n + 2500 >  -10000n + 110000

 

Solve this inequality for n.