In a system designed to work out the tax to be paid: An employee has $4000 of salary tax free. The next $1500 is taxed at 10%. The next $28000 after that is taxed at 22%. Any further amount is taxed at 40%. To the nearest whole pound, which of these groups of numbers fall into three DIFFERENT equivalence classes?

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  1. $4000; $5000; $5500

  2. $32001; $34000; $36500

  3. $28000; $28001; $32001

  4. $4000; $4200; $5600

Show answer
Correct Option: D

AI Explanation

To determine which groups of numbers fall into three different equivalence classes, we need to consider the tax brackets and how they affect the tax calculation for each salary amount.

Let's analyze each option:

Option A) $4000; $5000; $5500

  • The first $4000 is tax-free.
  • The next $1000 ($5000 - $4000) is taxed at 10%.
  • The next $500 ($5500 - $5000) is also taxed at 10%.

Since all three numbers fall within the same tax bracket and are subject to the same tax rate, they belong to the same equivalence class. Therefore, Option A is not the answer.

Option B) $32001; $34000; $36500

  • The first $4000 is tax-free.
  • The next $1500 is taxed at 10%.
  • The next $28000 is taxed at 22%.
  • Any further amount is taxed at 40%.

The three numbers in this option fall into different tax brackets, so they are likely to belong to different equivalence classes. However, we need to check if any two of them fall into the same tax bracket.

  • $32001: This amount falls within the 22% tax bracket.
  • $34000: This amount also falls within the 22% tax bracket.
  • $36500: This amount falls within the 40% tax bracket.

Since $32001 and $34000 fall into the same tax bracket, they belong to the same equivalence class. Therefore, Option B is not the answer.

Option C) $28000; $28001; $32001

  • The first $4000 is tax-free.
  • The next $1500 is taxed at 10%.
  • The next $28000 is taxed at 22%.
  • Any further amount is taxed at 40%.

The three numbers in this option fall into different tax brackets, so they are likely to belong to different equivalence classes. However, we need to check if any two of them fall into the same tax bracket.

  • $28000: This amount falls within the 22% tax bracket.
  • $28001: This amount also falls within the 22% tax bracket.
  • $32001: This amount falls within the 40% tax bracket.

Since $28000 and $28001 fall into the same tax bracket, they belong to the same equivalence class. Therefore, Option C is not the answer.

Option D) $4000; $4200; $5600

  • The first $4000 is tax-free.
  • The next $200 ($4200 - $4000) is taxed at 10%.
  • The next $1400 ($5600 - $4200) is taxed at 22%.

The three numbers in this option fall into different tax brackets. None of them fall into the same tax bracket, so they belong to different equivalence classes. Therefore, Option D is the correct answer.

To the nearest whole pound, the numbers in Option D fall into three different equivalence classes.

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