Looking back at the cup of coffee problem, you see that the new price (84 cents) was higher than the original price (80 cents).
You can ask what percent of the original price is represented by the new price:
Thus, the new price is 105% of the original price. Remember that the percent change is 5%. That is, the new price is 5% higher than the original price. There is a fundamental relationship between these numbers, resulting from the simple idea that the Change equals the New value minus the Original value, or equivalently, Original + Change = New.
If a quantity is increased by x percent, then the new quantity is (100 + x)% of the original. Thus, a 15% increase produces a quantity that is 115% of the original.
You can write this relationship as:
In the case of the cup of coffee, you see that:
Likewise, in the shirt problem, you had a 20% decrease in the price of a $30 shirt, resulting in a new price of $24.
The new price is some percent of the old price. Calculate that percent:
Subsequently, the new price (20% less than the original price) is 80% of the original price.
If a quantity is decreased by x percent, then the new quantity is (100 − x)% of the original. Thus, a 15% decrease produces a quantity that is 85% of the original.
You can write this relationship thus:
In the case of the shirt,
.
These formulas are all just another way of saying Original ± Change = New.
Example 4: What number is 50% greater than 60?
The whole amount is the original value, which is 60. The percent change (i.e., the percent “greater than”) is 50%. To find the answer, you must first find the part, which is the amount of the increase:
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You know that Original ± Change = New. Therefore, the number that is 50% greater than 60 is 60 + 30, or 90, which is also 150% of 60.
You could also solve this problem using the formula:
Thus,
.
Example 5: What number is 150% greater than 60?
The whole amount is the original value, which is 60. The percent change (i.e., the percent “greater than”) is 150%. To find the answer, you must first find the part, which is the amount of the increase:
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Now, x is the Change, not the new value. It is easy to forget to add back the original amount when the percent change is more than 100%. Thus, the number that is 150% greater than 60 is 60 + 90, or 150, which is also 250% of 60.
You could also solve this problem using the formula:
. Thus,
.
A plant originally cost $35. The price is increased by 20%. What is the new price?
70 is 250% greater than what number?