Calculating GST, Discounts, and Simple Interest in New Zealand Dollars

Here we are at the final lesson of our first mathematics module! We've covered a lot of ground, from fractions and integers to ratios and rates. In our last session, we explored the relationship between speed, distance, and time, which is a great example of a proportional relationship. Today, we'll continue applying proportional reasoning to some very practical, everyday situations involving money.

This lesson focuses on a key financial literacy skill: how to calculate GST, discounts, and simple interest in New Zealand dollar contexts. By the end of our session, you'll be able to confidently figure out the price of an item on sale, calculate the GST on a purchase, and understand how simple interest works on savings or loans. These are skills you will use throughout your life.

1. Calculating Discounts

Almost everyone loves a good sale. A discount is simply a percentage decrease in the original price of an item. Let's say you want to buy a pair of headphones that originally cost $68, but they are now on sale for 15% off. How do you find the new price?

There are two common ways to do this. The first method involves two steps:

1. Calculate the dollar amount of the discount.
2. Subtract that amount from the original price.

The following video clearly demonstrates this two-step process.

While this method works perfectly, there is a more efficient way to get the answer in a single step using a decimal multiplier.

If an item has a 15% discount, it means you are paying for what's left after 15% is taken away. Since the original price is 100%, you are paying for:
$100\%-15\%=85\%$
So, you just need to calculate 85% of the original price. To do this, you convert 85% to its decimal form, 0.85, and multiply. This number, 0.85, is your decimal multiplier.

For the $68 headphones with a 15% discount:
$\$68\times 0.85=\$57.80$
This gets you the final price in one calculation. The following resource explains this multiplier method in more detail.

2. Goods and Services Tax (GST) in New Zealand

In New Zealand, most goods and services have a 15% tax added to their price. This is called the Goods and Services Tax, or GST. Understanding how to work with GST is essential for budgeting and checking prices.

There are two main GST calculations you'll need to do:

1. Adding 15% GST to a price.
2. Finding out how much GST was included in a total price.

Adding GST

This is a percentage increase. If a price is listed as "$100 + GST," you need to add 15% to it. Just like with discounts, you can do this in two steps (find 15% of $100, which is $15, then add it to get $115) or you can use a multiplier.

Since you are adding 15%, the new price will be 100% + 15% = 115% of the original.
Your decimal multiplier is 1.15.

So, for the $100 item:
$\$100\times 1.15=\$115$

Finding the GST in a Price

This is a bit trickier. If a shop receipt says the total cost was $115 including GST, how do you find out what the price was before tax?

You can't just subtract 15% from $115, because the 15% was calculated on the original, lower price. To solve this, we can use a special rule. The official Inland Revenue Department website explains the standard method used in New Zealand.

As the IRD resource shows, the rule is: To find the GST portion of a GST-inclusive price, you multiply the total price by 3 and then divide by 23.

Why does this work? A price including 15% GST is 115% of the original price. The GST amount is 15% of the original. The fraction of the total price that is GST is therefore $\frac{15}{115}$. This fraction simplifies to $\frac{3}{23}$.

3. Simple Interest

Interest is a fee paid for borrowing money, or alternatively, money earned for lending it (like when you deposit money in a bank). Simple interest is the most basic type, where the interest is calculated only on the original amount of money.

The formula for simple interest is:
$I=P\times R\times T$
Where:

• $I$ is the Interest amount (in dollars).
• $P$ is the Principal – the starting amount of money.
• $R$ is the interest Rate per time period, always expressed as a decimal.
• $T$ is the Time – the number of periods the money is borrowed or invested for.

It's crucial that the time units for Rate and Time match. If the interest rate is per year, then the time must be in years.

Example:
You deposit $500 (P) into a savings account that pays 3% simple interest per year (R). How much interest will you earn in 2 years (T)?

1. Identify the values:
   • P = $500
   • R = 3% = 0.03 (remember to convert the percentage to a decimal)
   • T = 2 years
2. Apply the formula:
   $I=500\times 0.03\times 2$
   $I=\$30$
3. Answer: You will earn $30 in interest over two years. Your total balance would be $500 + $30 = $530.

Practice Problems

Let's test your new skills!

1. A video game costs $95, but is on sale with a 20% discount. What is the sale price?
2. A cafe meal costs $24.50 before GST. What is the total price you have to pay, including 15% GST?
3. You buy a new phone for $1,380, and the price is GST-inclusive. How much of that price was GST?
4. You borrow $4,000 from a relative to buy a car. They charge you 4% simple interest per year. How much interest will you owe after 3 years?

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Solutions:

1. Discount: 100% - 20% = 80%. The multiplier is 0.80.
   Sale Price = $95 × 0.80 = $76.
2. Adding GST: The multiplier is 1.15.
   Total Price = $24.50 × 1.15 = $28.18 (rounded to the nearest cent).
3. Finding GST: Use the 3/23 rule.
   GST Amount = ($1380 × 3) ÷ 23 = $4140 ÷ 23 = $180.
4. Simple Interest: I = P × R × T
   Interest = $4000 × 0.04 × 3 = $480.

Key Takeaways

Congratulations on completing the first module! You've just mastered three fundamental financial calculations that you will encounter frequently.

• Discounts and GST increases can be calculated quickly using a decimal multiplier. For a discount of $x\%$, multiply by $(1-x/100)$. For a GST increase of 15%, multiply by 1.15.
• To find the GST component of a final price in New Zealand, multiply the total by 3 and divide by 23.
• Simple interest is calculated on the original principal amount using the formula $I=P\times R\times T$.

This brings our module on Number, Proportion, and Financial Reasoning to a close. You've built a strong foundation in how numbers and rates work.

In our next lesson, we will begin a new module on Algebra. We'll start by learning how to collect like terms to simplify an algebraic expression. This involves shifting from working only with numbers to using letters to represent unknown or changing values, a powerful next step in your mathematical journey.