To break down the total of 675.25 including taxes with a 5% TPS (federal tax) and a 9.975% TVQ (Quebec sales tax), we need to work backwards to find the pre-tax amount.

Let the pre-tax amount be \( x \). The equation for the total price after adding both taxes is:

\[
x \times (1 + \text{TPS} + \text{TVQ}) = 675.25
\]

Where:
- TPS = 5% = 0.05
- TVQ = 9.975% = 0.09975

This gives us the equation:

\[
x \times (1 + 0.05 + 0.09975) = 675.25
\]

\[
x \times 1.14975 = 675.25
\]

Now, we can solve for \( x \):

\[
x = \frac{675.25}{1.14975}
\]

Let's calculate that.The pre-tax amount is approximately \( 587.30 \).

To verify, let's calculate the taxes:
- TPS (5%) on \( 587.30 \)
- TVQ (9.975%) on \( 587.30 \)

Would you like a detailed breakdown of these taxes or have any questions?

Here are 5 related questions to deepen understanding:
1. How would the total change if only one of the taxes were applied?
2. What would the pre-tax price be with a higher TPS, say 7%?
3. How do you calculate taxes in reverse when given the total?
4. What happens if we apply taxes in a different order (TPS first or TVQ first)?
5. Can you determine the individual amounts of each tax on the total?

**Tip:** To reverse-engineer totals with taxes, divide the total by \(1 + \text{tax rates}\).

Learn how to reverse-engineer a total price of 675.25, which includes a 5% TPS and 9.975% TVQ. This step-by-step solution helps you calculate the pre-tax amount and break down the taxes using basic algebra and percentage concepts. Ideal for students in Grades 10-12.

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