Next-Gen

Percentages

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by KaralaL
Last updated 2 years ago

Discipline:
Math
Subject:
Percentages
Grade:
5,6,7,8,9,10

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PERCENTAGES

CalculationsThe percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, first compute the ratio 50/1250 = 0.04, and then multiply by 100 to obtain 4%. The percent value can also be found by multiplying first, so in this example the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:(50/100) × (40/100) = 0.50 × 0.40 = 0.20 = 20/100 = 20%.It is not correct to divide by 100 and use the percent sign at the same time. (E.g. 25% = 25/100 = 0.25, not 25% / 100, which actually is (25/100) / 100 = 0.0025. A term such as (100/100)% would also be incorrect, this would be read as (1) percent even if the intent was to say 100%.)Whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what is the total that corresponds to 100%.

Compounding percentagesIt is important to understand that percent changes, as they have been discussed here, do not add in the usual way, if applied sequentially. For example, if the 10% increase in price considered earlier (on the $200 item, raising its price to $220) is followed by a 10% decrease in the price (a decrease of $22), the final price will be $198, not the original price of $200. The reason for the apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to different quantities ($200 and $220, respectively), and thus do not "cancel out".In general, if an increase of x percent is followed by a decrease of x percent, and the initial amount was p, the final amount is p((1+0.01x)(1-0.01x))=p(1-(0.01x)^2); thus the net change is an overall decrease by x percent of x percent (the square of the original percent change when expressed as a decimal number). Thus, in the above example, after an increase and decrease of x=10 percent, the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200. The net change is the same for a decrease of x percent followed by a increase of x percent; the final amount is p((1-0.01x)(1+0.01x))=p(1-(0.01x)^2).This can be expanded for a case where you do not have the same percent change. If the initial percent change is x and the second percent change is y, and the initial amount was p, then the final amount is p((1+0.01x)(1+0.01y)). To change the above example, after an increase of x=10 and decrease of y=-5 percent, the final amount, $209, is 4.5% more than the initial amount of $200.As shown above, percent changes can be applied in any order and have the same effect.In the case of interest rates, it is a common practice to state the percent change differently. If an interest rate rises from 10% to 15%, for example, it is typical to say, "The interest rate increased by 5%" — rather than by 50%, which would be correct when measured as a percentage of the initial rate (i.e., from 0.10 to 0.15 is an increase of 50%). Such ambiguity can be avoided by using the term "percentage points" (pp). In the previous example, the interest rate "increased by 5 pp" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.

DefinitionIn mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviation "pct."; sometimes the abbreviation "pc" is used in the case of quantities in economics. A percentage is a dimensionless number (pure number).

Percentage increase and decreaseSometimes due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).Some other examples of percent changes:An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).A decrease of 100% means the final amount is zero (100% − 100% = 0%).In general, a change of x percent in a quantity results in a final amount that is 100+x percent of the original amount (equivalently, 1+0.01x times the original amount).

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