two vendors are competing for your busness.vendor A is offering you a 6/8/12 discount and vendor B is offering a discount of 7/9/10.what is the single equivalent discount being offered by each vendor(round to the nearest hundredth percent)
compare
1 - (.94)(.92)(.88)
with
1 - (.93)(.91)(.9)
A is better
To determine the single equivalent discount offered by each vendor, we first need to understand what the 6/8/12 and 7/9/10 discounts represent.
The fractions such as 6/8/12 and 7/9/10 are usually used to denote a series of discounts applied in sequence. Each fraction represents a percentage discount applied one after another.
In the case of vendor A with a discount of 6/8/12, it means three discounts are applied in sequence: first a 6% discount, then an 8% discount on the remaining amount after the first discount, and finally a 12% discount on the remaining amount after the first two discounts.
Similarly, for vendor B with a discount of 7/9/10, it means three discounts are applied in sequence: first a 7% discount, then a 9% discount on the remaining amount after the first discount, and finally a 10% discount on the remaining amount after the first two discounts.
To calculate the single equivalent discounts, we need to find the combined discount percentage that gives the same overall reduction as the sequence of discounts.
Let's find the single equivalent discount for vendor A:
1. First, find the effective discount after the first discount of 6%:
Discounted amount = Original amount * (1 - discount) = Original amount * (1 - 0.06) = Original amount * 0.94
2. Second, find the effective discount after the second discount of 8%:
Discounted amount after the first discount = Original amount * 0.94
Discounted amount after the second discount = (Original amount * 0.94) * (1 - 0.08) = Original amount * 0.94 * 0.92
3. Finally, find the effective discount after the third discount of 12%:
Discounted amount after the second discount = Original amount * 0.94 * 0.92
Discounted amount after the third discount = (Original amount * 0.94 * 0.92) * (1 - 0.12) = Original amount * 0.94 * 0.92 * 0.88
So, the single equivalent discount offered by vendor A is: 1 - (Original amount * 0.94 * 0.92 * 0.88 / Original amount) = 1 - (0.94 * 0.92 * 0.88) = 1 - 0.7681 ≈ 0.2319, or approximately 23.19%.
Now, let's find the single equivalent discount for vendor B using the same calculations:
1. First, find the effective discount after the first discount of 7%:
Discounted amount = Original amount * (1 - discount) = Original amount * (1 - 0.07) = Original amount * 0.93
2. Second, find the effective discount after the second discount of 9%:
Discounted amount after the first discount = Original amount * 0.93
Discounted amount after the second discount = (Original amount * 0.93) * (1 - 0.09) = Original amount * 0.93 * 0.91
3. Finally, find the effective discount after the third discount of 10%:
Discounted amount after the second discount = Original amount * 0.93 * 0.91
Discounted amount after the third discount = (Original amount * 0.93 * 0.91) * (1 - 0.10) = Original amount * 0.93 * 0.91 * 0.90
So, the single equivalent discount offered by vendor B is: 1 - (Original amount * 0.93 * 0.91 * 0.90 / Original amount) = 1 - (0.93 * 0.91 * 0.90) = 1 - 0.7707 ≈ 0.2293, or approximately 22.93%.
Therefore, vendor A is offering a single equivalent discount of approximately 23.19%, while vendor B is offering a single equivalent discount of approximately 22.93%.