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To determine the total cost [tex]\( C \)[/tex] of an item with an original price [tex]\( p \)[/tex], considering a 15% discount and an 8% sales tax, we need to calculate the discounted price first and then apply the sales tax to the discounted price.
Let's examine each equation option:
### Option A: [tex]\( C = 1.08 p - 0.15 p \)[/tex]
1. This expression considers adding 8% tax to the original price [tex]\( p \)[/tex], then subtracting the 15% discount from the taxed amount.
2. Simplifying:
[tex]\[ C = 1.08 p - 0.15 p = (1.08 - 0.15) p = 0.93 p \][/tex]
This option simplifies correctly to [tex]\( 0.93p \)[/tex], indicating a consistent result.
### Option B: [tex]\( C = 1.15 p + 0.08 p \)[/tex]
1. This expression seems to incorrectly add 15% and 8% directly to the original price [tex]\( p \)[/tex].
2. Simplifying:
[tex]\[ C = 1.15 p + 0.08 p = (1.15 + 0.08) p = 1.23 p \][/tex]
This result is inconsistent with the actual process of first applying the discount and then the sales tax.
### Option C: [tex]\( C = 1.08(0.85 p) \)[/tex]
1. This expression properly applies the 8% tax to the discounted price [tex]\( 0.85p \)[/tex] (with 15% off).
2. Simplifying:
[tex]\[ C = 1.08(0.85 p) = 1.08 \times 0.85 p = 0.918 p \][/tex]
This option simplifies correctly to [tex]\( 0.918p \)[/tex], indicating a consistent result.
### Option D: [tex]\( C = 0.85 p + 0.08(0.85 p) \)[/tex]
1. This expression first calculates the discounted price [tex]\( 0.85p \)[/tex] and then adds 8% of this discounted price.
2. Simplifying:
[tex]\[ C = 0.85 p + 0.08(0.85 p) = 0.85 p + 0.068 p = (0.85 + 0.068) p = 0.918 p \][/tex]
This option simplifies correctly to [tex]\( 0.918p \)[/tex], indicating a consistent result.
### Option E: [tex]\( C = p - 0.15 p + 0.08 p \)[/tex]
1. This expression seems to subtract 15% from the original price and then add the 8% tax to the discounted price.
2. Simplifying:
[tex]\[ C = p - 0.15 p + 0.08 p = (1 - 0.15 + 0.08) p = (1 - 0.07) p = 0.93 p \][/tex]
This option simplifies correctly to [tex]\( 0.93p \)[/tex], indicating a consistent result.
Based on the simplifications above, the equations that correctly determine the total cost of buying an item with an original price of [tex]\( p \)[/tex] after applying a 15% discount and an 8% sales tax are:
- Option A: [tex]\( C = 1.08 p - 0.15 p \)[/tex]
- Option C: [tex]\( C = 1.08(0.85 p) \)[/tex]
- Option D: [tex]\( C = 0.85 p + 0.08(0.85 p) \)[/tex]
- Option E: [tex]\( C = p - 0.15 p + 0.08 p \)[/tex]
Let's examine each equation option:
### Option A: [tex]\( C = 1.08 p - 0.15 p \)[/tex]
1. This expression considers adding 8% tax to the original price [tex]\( p \)[/tex], then subtracting the 15% discount from the taxed amount.
2. Simplifying:
[tex]\[ C = 1.08 p - 0.15 p = (1.08 - 0.15) p = 0.93 p \][/tex]
This option simplifies correctly to [tex]\( 0.93p \)[/tex], indicating a consistent result.
### Option B: [tex]\( C = 1.15 p + 0.08 p \)[/tex]
1. This expression seems to incorrectly add 15% and 8% directly to the original price [tex]\( p \)[/tex].
2. Simplifying:
[tex]\[ C = 1.15 p + 0.08 p = (1.15 + 0.08) p = 1.23 p \][/tex]
This result is inconsistent with the actual process of first applying the discount and then the sales tax.
### Option C: [tex]\( C = 1.08(0.85 p) \)[/tex]
1. This expression properly applies the 8% tax to the discounted price [tex]\( 0.85p \)[/tex] (with 15% off).
2. Simplifying:
[tex]\[ C = 1.08(0.85 p) = 1.08 \times 0.85 p = 0.918 p \][/tex]
This option simplifies correctly to [tex]\( 0.918p \)[/tex], indicating a consistent result.
### Option D: [tex]\( C = 0.85 p + 0.08(0.85 p) \)[/tex]
1. This expression first calculates the discounted price [tex]\( 0.85p \)[/tex] and then adds 8% of this discounted price.
2. Simplifying:
[tex]\[ C = 0.85 p + 0.08(0.85 p) = 0.85 p + 0.068 p = (0.85 + 0.068) p = 0.918 p \][/tex]
This option simplifies correctly to [tex]\( 0.918p \)[/tex], indicating a consistent result.
### Option E: [tex]\( C = p - 0.15 p + 0.08 p \)[/tex]
1. This expression seems to subtract 15% from the original price and then add the 8% tax to the discounted price.
2. Simplifying:
[tex]\[ C = p - 0.15 p + 0.08 p = (1 - 0.15 + 0.08) p = (1 - 0.07) p = 0.93 p \][/tex]
This option simplifies correctly to [tex]\( 0.93p \)[/tex], indicating a consistent result.
Based on the simplifications above, the equations that correctly determine the total cost of buying an item with an original price of [tex]\( p \)[/tex] after applying a 15% discount and an 8% sales tax are:
- Option A: [tex]\( C = 1.08 p - 0.15 p \)[/tex]
- Option C: [tex]\( C = 1.08(0.85 p) \)[/tex]
- Option D: [tex]\( C = 0.85 p + 0.08(0.85 p) \)[/tex]
- Option E: [tex]\( C = p - 0.15 p + 0.08 p \)[/tex]
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