Question
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?
Answers
GPT-4o mini
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.**
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.**