- Topic
- Continuity
- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
Continuity practice question 22
A tax rule charges 15% of income on incomes up to . On larger incomes it charges 25% of the whole income, minus a fixed deduction of dollars. is chosen so that the tax on an income of dollars is a continuous function: crossing the threshold never makes the tax jump. Find , and then enter the tax owed on an income of .
Each rule on its own is a straight line, so the tax could only jump at one income. Which one, and what must the two rules do there?
Continuity at : the second rule, approached from above, must give the same tax as the first rule at .
Solve , then use the second rule at .
A whole number of dollars, such as 4250.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Each rule is a line, so the only place the tax could jump is the threshold. Continuity there is one equation, which fixes D; then evaluate the rule that applies to $80,000.
Working
At the threshold itself the first rule applies.
Just above the threshold the second rule applies.
Continuity: the limit from above must equal the value at the threshold.
$80,000 is above the threshold, so use the second rule.
A check that also explains the rule: 15% on the first $50,000 and 25% only on the part above it, 7500 + 7500 = 15,000.
Answer
The answer is D is 5000 dollars, and the tax on 80,000 dollars is 15,000 dollars.
Check
$15,000 is 18.75% of $80,000, between the two rates of 15% and 25%, as an average rate must be.
Where this goes wrong
Stopping at D = 5000, or charging 25% on the whole income and forgetting to subtract D. Finding D is the continuity step; the question then asks for the tax that rule gives at $80,000.
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