Dr. Jones lives in a country with a progressive tax system. That is, he does not pay any taxes on the first $20,000 in income he makes, he pays 5% taxes on the next $25,000, he pays 10% taxes on the next $35,000, he pays 15% on the next $50,000, and he pays 20% on every additional dollar onwards. If Dr. Jones pays $10,000 in taxes, then how much income does he make?

Answer 1

To determine Dr. Jones's income based on the taxes he pays, we can break down the tax brackets and calculate the total income step by step.

1. Income from $0 to $20,000: Taxed at 0%.

   • Tax: $0
   • Income: $20,000

2. Income from $20,001 to $45,000 ($25,000 additional): Taxed at 5%.

   • Tax: \(0.05 \times 25,000 = $1,250\)
   • Income: $45,000 \((20,000 + 25,000)\)

3. Income from $45,001 to $80,000 ($35,000 additional): Taxed at 10%.

   • Tax: \(0.10 \times 35,000 = $3,500\)
   • Income: $80,000 \((45,000 + 35,000)\)

4. Income from $80,001 to $130,000 ($50,000 additional): Taxed at 15%.

   • Tax: \(0.15 \times 50,000 = $7,500\)
   • Income: $130,000 \((80,000 + 50,000)\)

5. At this point, we have accumulated the following tax:

   • Tax from the first bracket: $0
   • Tax from the second bracket: $1,250
   • Tax from the third bracket: $3,500
   • Tax from the fourth bracket: $7,500

   Total Tax up to $130,000 income: \[ 0 + 1,250 + 3,500 + 7,500 = 12,250 \]

Since Dr. Jones only pays $10,000 in taxes, let's analyze his income in the last bracket.

6. Determining the income over $130,000:

   • Let \(x\) be the income above $130,000.
   • Then, the total income will be \(130,000 + x\).
   • Tax on the amount over $130,000 is taxed at 20%, giving us an additional \(0.20x\).

7. Setting up the tax equation: We know the total tax paid is $10,000, so: \[ 12,250 + 0.20x = 10,000 \]

8. Solving for \(x\): \[ 0.20x = 10,000 - 12,250 \] \[ 0.20x = -2,250 \]

   This indicates a miscalculation, as we expected \(x\) to yield a positive number.

Since the cumulative tax of $12,250 has exceeded $10,000, the entire income must be within the allowable tax brackets without reaching the fourth tier.

We'll re-evaluate how much tax to subtract at $80,000, creating a new base income.

Let’s assess again only up to $80,000:

1. Cumulative tax at $80,000: \[ 0 + 1,250 + 3,500 = 4,750 \]

2. Next, income from $80,001 with 15% tax on up to a point where total tax is $10,000: \[ 5,000 (next tier after $80,000) \] Since he paid already $4,750, we look how much of the \(0.15\) bracket he has used:

   \[ 10,000 - 4,750 = 5,250 \implies \frac{5,250}{0.15} = 35,000 \] Therefore, starting from $80,000 + 35,000 results at $115,000.

In summary, Dr. Jones makes $115,000.