An important category of percentage exercises is markup and markdown problems.
For these, you calculate the markup or markdown of the price or cost in absolute terms (you find by how much the price or cost changed), and then you calculate the percent change relative to the original value. So they're really just another form of "increase - decrease" exercises, but generally restricted to monetary values.
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The markup is 40% of the $25 cost, so the markup is:
(0.40)(25) = 10
Then the selling price, being the cost plus markup, is:
25 + 10 = 35
I was adding prices in terms of dollars, so I'll need to remember to put the appropriate unit on my answer.
The item sold for $35.
We can tell which is which in a given exercise by noting the term that they use:
First, I'll calculate the markup in absolute terms:
75 – 40 = 35
Then I'll find the markup rate (that is, the relative markup over the original price) by using the standard percent-of equation:
I'll convert this equation from English to algebra:
35 = (x)(40)
Solving for the markup rate, I get:
35 ÷ 40 = x = 0.875
Since x stands for a percentage, I need to remember to convert this decimal value to the corresponding percentage, complete with the percent sign.
The markup rate is 87.5%.
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This problem is somewhat backwards. They gave me the selling price, which is cost plus markup, and they gave me the markup rate, but they didn't tell me the actual cost or markup amount. So I have to be clever to solve this.
I will let "x" be the cost. Then the markup, being 40% of the cost, is 0.40x. And the selling price of $63 is the sum of the cost and markup, so:
63 = x + 0.40x
63 = 1x + 0.40x
63 = 1.40x
63 ÷ 1.40 = x= 45
The shoes cost the store $45.
First, I'll find the markdown. The markdown is 25% of the original price of $55, so:
x = (0.25)(55) = 13.75
By subtracting this markdown from the original price, I can find the sale price:
55 – 13.75 = 41.25
The sale price is $41.25.
First, I'll find the amount of the markdown:
425 – 318.75 = 106.25
Then I'll calculate "the markdown over the original price", or the markdown rate: ($106.25) is (some percent) of ($425), so:
106.25 = (x)(425)
...and the relative markdown over the original price is:
x = 106.25 ÷ 425 = 0.25
Since the "x" stands for a percentage, I need to remember to convert this decimal to percentage form.
The markdown rate is 25%.
This problem is backwards. They gave me the sale price ($127.46) and the markdown rate (15%), but neither the markdown amount nor the original price. I will let "x" stand for the original price. Then the markdown, being 15% of this price, was 0.15x. And the sale price is the original price, less the markdown, so I get:
x – 0.15x = 127.46
1x – 0.15x = 127.46
0.85x = 127.46
x = 127.46 ÷ 0.85 = 149.952941176...
This problem didn't state how to round the final answer, but dollars-and-cents is always written with two decimal places, so:
The original price was $149.95.
Note in this last problem that I ended up, in the third line of calculations, with an equation that said "eighty-five percent of the original price is $127.46". You can save yourself some time if you think of discounts in this way: if the price is 15% off, then you're only actually paying 85%. Similarly, if the price is 25% off, then you're paying 75%; if the price is 30% off, then you're paying 70%; and so on.