Percentages/Profit/Loss/Vat (Junior Cert Mathematics): Revision Notes

Practice Problems

Problems:

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Problem 1:

Explanation:

This problem helps you figure out the total number when you only know part of it, based on a given percentage. This is useful in scenarios like finding out how many students are in a class when you know how many represent a specific percentage.

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Problem 2:

Explanation:

This problem shows you how to calculate what a new amount will be after a percentage increase. It's common in situations where prices go up, like when an item becomes more expensive.

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Problem 3:

Explanation:

Here, you'll learn how to calculate the new price of something after a percentage decrease, which is important when dealing with sales or discounts.

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Problem 4:

Explanation:

This problem introduces you to the concept of percentage error, which is useful when you want to know how accurate a measurement is. It compares the difference between what you measured and the actual value.

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Problem 5:

Explanation:

This is a more complex problem that combines calculating VAT with a percentage discount. It's a practical example of how taxes and discounts work together to affect the final price of an item.

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Solutions:

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Problem 1:

Solution**:** To solve this problem, we need to find the total number of students in the class.

• We know that 25% of the class is 8 students.
• To find the total number of students (which is 100%), we divide 8 by 25% (or 0.25 in decimal form).
• The calculation is: $\frac{8}{0.25} = 32$
• Why does this work? Because dividing by the percentage (in decimal form) gives us the whole, just like how 25% of something means 1/4 of it. So, if 8 is 1/4, multiplying by 4 (or dividing by 0.25) gives the full amount.
• Answer: There are 32 students in the class.

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Problem 2:

Solution: To find out the new price after a 20% increase, follow these steps:

• First, find 20% of the original price (€50). We do this by multiplying €50 by 0.20 (since 20% as a decimal is 0.20). $50 \times 0.20 = 10$
• This tells us that the price is increasing by €10.
• Next, add this €10 to the original price to find the new price. $50 + 10 = €60$
• Why do we do this? The increase is added to the original price because the item is now worth more by that percentage.
• Answer: The new price is €60

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Problem 3:

Solution: To find the sale price after a 25% discount, follow these steps:

• First, calculate 25% of the original price (€80). We do this by multiplying €80 by 0.25. $80 \times 0.25 = 20$
• This €20 is the amount of the discount.
• Next, subtract this discount from the original price to find the new price. $80 - 20 = €60$
• Why do we do this? The discount amount is taken off the original price because the item is now cheaper by that percentage.
• Answer: The sale price is €60.

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Problem 4:

Solution: To find the percentage error, follow these steps:

• First, find the error by subtracting the actual length from the measured length. $105 \text{ cm} - 100 \text{ cm} = 5 \text{ cm}$
• This 5 cm is how much your measurement was off.
• Next, to find the percentage error, divide this error by the actual value and multiply by 100 to turn it into a percentage. $\frac{5 \text{ cm}}{100 \text{ cm}} \times 100 = 5\%$
• Why do we do this? The percentage error tells us how big the error is relative to the actual measurement. It's a way of showing how far off you were in a form that's easy to understand.
• Answer: The percentage error is 5%.

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Problem 5:

Solution: This problem involves two steps: adding VAT and then applying a discount.

• Step 1: First, calculate the VAT. The VAT rate is 20%, so first, we find 20% of €500. $500 \times 0.20 = €100$
• This €100 is the VAT that needs to be added to the original price.
• Now, add the VAT to the original price to get the price including VAT. $500 + 100 = €600$
• Step 2: Now that we have the price including VAT, we apply the 15% discount. First, calculate 15% of €600. $600 \times 0.15 = €90$
• This €90 is the discount amount.
• Subtract this discount from the price including VAT to get the final price. $600 - 90 = €510$
• Why do we do this? Adding VAT increases the price because of tax, and then applying the discount decreases it because of the sale. Doing the steps in order ensures that the final price is correct.
• Answer: The final price after VAT and discount is €510.

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