Depth · one concept from first principles · step 2 of 4
With the flat fee, the average is 20000 divided by the number of pens, which is smallest when the order is 100 pens.
Even at its cheapest, the flat-fee range averages 200 won per pen, so the answer must be above 100.
7.EE.B.4Eliminate PossibilitiesFor x pens with x ＞ 100, the extra pens number x - 100.
Past 100 pens, the cost is a fixed part plus 100 won per pen.
7.EE.B.4Identify SubproblemsAverage cost is total cost divided by x. Since x ＞ 0, multiply both sides by x.
Each extra pen costs only 100 won, which pulls the average down toward 100; it drops below 150 once there are more than 200 pens.
7.EE.B.4Convert To AlgebraFor more than pens, the average cost is below won exactly when .
The average cost per pen is the total cost shared equally over the pens, so it is .
Multiplying both sides by the positive count stretches the number line without turning it, so the direction of stays.
The smallest whole number greater than 200 is 201. Check it against 200.
At 200 pens the average is exactly 150, which is not less than 150.
7.EE.B.4Guess And CheckWhen a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.
A parent dashboard for the family lives at sensimlab.com.