Practice Problems (Junior Cert Mathematics): Revision Notes

Practice Problems

Problems:

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Problem 1:

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Problem 2:

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Problem 3:

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Solutions:

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Problem 1:

Step 1: Calculate the Tax at the Standard Rate

• First, we calculate how much of Joan's income is taxed at the standard rate. This is the first €31,000 of her salary.
• Calculation: $31,000 \times 0.20 = €6,200$
• Explanation: We multiply €31,000 by 0.20 because 20% is the standard tax rate. This means Joan pays €6,200 in tax on the first €31,000 of her income.

Step 2: Calculate the Tax at the Higher Rate

• Next, we calculate the tax on the remaining income, which is taxed at the higher rate of 40%. The remaining income is: $39,000 - 31,000 = €8,000$
• Calculation: $8,000 \times 0.40 = €3,200$
• Explanation: The income over €31,000, which is €8,000, is taxed at 40%. We multiply €8,000 by 0.40 (the higher tax rate) to get €3,200.

Step 3: Add the Taxes Together

• Add the taxes from the standard rate and higher rate: $6,200 + 3,200 = €9,400$
• Explanation: To find out how much tax Joan owes before subtracting her tax credits, we add the tax from the standard rate (€6,200) and the higher rate (€3,200), which totals €9,400.

Step 4: Subtract the Tax Credits

• Finally, subtract Joan's tax credits from her total tax: $9,400 - 4,750 = €4,650$
• Explanation: Tax credits reduce the amount of tax Joan has to pay. We subtract her tax credits (€4,750) from the total tax (€9,400) to find the final amount of tax she needs to pay, which is €4,650.

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Problem 2:

Step 1: Set Up the Equations

• Cecil's income is split into two parts: one part taxed at 20% and the rest at 35%. Let's call his total salary €S.
• The tax on the first €30,000 is: $30,000 \times 0.20 = €6,000$
• Explanation: We calculate the tax on the first €30,000 of Cecil's income by multiplying it by 0.20 (the standard tax rate). This gives us €6,000.
• The remaining part of his income, which we'll call S - 30,000, is taxed at 35%. So, the tax on this part is: $(S - 30,000) \times 0.35$
• Explanation: The part of Cecil's income above €30,000 is taxed at 35%. We don't know this amount yet, so we express it as S - $30,000$ and multiply it by 0.35 (the higher tax rate).

Step 2: Calculate Total Tax Before Credits

• Add the tax from the standard rate and the higher rate: $6,000 + 0.35(S - 30,000)$
• Explanation: We add the tax on the first €30,000 (€6,000) to the tax on the remaining income (S - $30,000$) × $0.35$. This gives us the total tax before applying the tax credits.
• Subtract Cecil's tax credits to find the total tax payable: $6,000 + 0.35(S - 30,000) - 3,600 = 7,650$
• Explanation: We subtract Cecil's tax credits (€3,600) from the total tax. We know that the final tax Cecil pays is €7,650, so we set the equation equal to €7,650.

Step 3: Solve for the Gross Salary

• To solve for S, add the tax credit back and rearrange the equation: $6,000 + 0.35(S - 30,000) = 7,650 + 3,600$
• Explanation: To isolate S, we add the tax credit (€3,600) back to the final tax (€7,650).
• Simplify the equation: $6,000 + 0.35S - 10,500 = 11,250$ $0.35S - 4,500 = 11,250$ $0.35S = 15,750$ $S = \frac{15,750}{0.35} = €45,000$
• Explanation: We simplify the equation step by step to solve for S, which represents Cecil's gross salary. Finally, we find that his gross salary is €45,000.

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Problem 3:

Step 1: Set Up the Equation

• Let the gross weekly wage be €W. The tax on his wages is: $W \times 0.20 = \text{Tax before credits}$
• Explanation: We start by calculating the tax on his gross wage W. Since the tax rate is 20%, we multiply W by 0.20.
• Subtract the tax credit to find the net tax payable: $W \times 0.20 - 60 = 140$
• Explanation: The tax credit reduces the amount of tax he has to pay. We subtract the tax credit (€60) from the tax before credits and set this equal to the net tax he actually pays (€140).

Step 2: Solve for the Gross Weekly Wage

• Add the tax credit back and rearrange the equation: $W \times 0.20 = 140 + 60 = 200$

• Explanation: To solve for W, we add the tax credit (€60) back to the net tax (€140) to find the tax before credits. $W = \frac{200}{0.20} = €1,000$

• Explanation: We then divide by 0.20 to find W, the gross weekly wage. The bus driver's gross weekly wage is €1,000.