The concept of percentages is quite simple. It makes fractions seem more friendly and helps present relative differences and relations between numerical quantities. Here's an overview of concepts and techniques that will help you answer percentage questions more efficiently.
Here is a glimpse of our tutorial on percentages. This is merely a taster of the tutorial you get when you purchase one of our numerical packages.
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A quick reminder about percentages
We truly hope you are familiar with the following percentage formula: (if not, it's good to have you with us.)
% = Fraction X 100
This formula allows us to alternate between fractions and their percent form. Let's take a look at the following example:
If we decide to put 25 on the % side, we get the following equation:
25% = fraction X 100
If we divide the equation by 100, we get:
25/100 = 1/4 = Fraction
Therefore, 25% is merely another way of presenting 1/4. This formula is sometimes elaborated to include the components of a fraction (nominator and denominator), but for our purposes the formula above is satisfactory.
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Percentages and decimal numbers
Many people find it easier to calculate percent changes using decimal numbers. Since a percent is actually a fraction, 1% can be written as 0.01. Therefore, increasing a number by 1% means multiplying it by (1+0.01)= 1.01, and decreasing a number by 1% means multiplying it by (1-0.01)=0.99. Thus, a percent increase means multiplying by numbers greater than one, and a percent decrease means multiplying by numbers that are smaller than one.
How to calculate % changes without the calculator's % function
To speed up the calculation process, we shall use a different format of the above formula. From now on, when asked to calculate a percent increase in value, use the following formula:
% = [(value after change/value before change) - 1] X 100
When asked to calculate a decrease in value use this formula and then multiply by (-1), or use the +\- sign in your calculator. Take a look at the following simplified example:
"The price of X was 30 and is now 40. What is the percent difference between the two prices..."
In this case it is clear that we are looking for an increase change, so we are looking for the ratio between 40 and 30, which constitutes the fraction in our formula:
[(40/30) - 1] X 100 = 33.33%
If you feel comfortable with numbers you can always skip the multiplication by 100 to get 0.33, and thus conclude that this represents 33%. Here is another example with a decrease in value:
"The price of X was 40 and is now 30. What is the percent difference between the two prices..."
In this case, the new number has decreased in respect to the original. If we use the same formula, we get a negative number:
[(30/40) -1] X 100 = (- 25)%
This still represents the true absolute value we are looking for, so you can simply multiply by (-1) or use the +\- sign in your calculator to get the correct answer. In theory, we could also switch places in the formula:% = [1- (value before change/value after change)]X 100
However, this formula can slow down the calculation if we are using the simplest calculator, as it will require us for two steps because a subtraction precedes a multiplication.
How to use the calculator's % function?
The % function simply multiplies a given product by 100. Therefore, after dividing the initial and final values - prior to pressing the = sign - press the % sign and the fraction you have entered will be multiplied by 100.
? When measuring an increase:
Divide: (value after change/value before change)
Press the % sign
Deduct 100
? When measuring a decrease:
Divide: (value after change/value before change)
Press the %
Try dividing 40 by 30 and then pressing the % button. You will get 133 which stands for 133%. Now deduct 100 and you will get 33% which tells you that 40 is 33% percent greater than 30.
Implicit words that ask for percentage calculations
The most popular word that is used by test writers to imply a percent calculation is: "Relative change".
Reverse percentage questions
It is quite popular to present a quantity that has gone through a perecnt change and ask its original value. The fastest way to get these calculations is the follwoing: If a certain value is known to have been changed by a certain percentage, its original value can be found by dividing it by the decimal size it now comprises from the original:
[(Present value) / (Decimal size in relation to original value)] = (Original value before change)
Example:
Q: "A car's value was reduced by 10% and is now worth £900. What was its original price."
A: The car is now worth only 0.9 of its original value, as it experienced a 10% decrease. According to our formula:
[(900) / (0.9)] = 1000
Examine the following example:
Q: A car's value was increased by 10% and is now worth £550. What was its original price?
[(550)/(1.1)] = 500
Further reading on numerical reasoning tests:
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