Percentages, Fractions (Copy)

Vulgar Fractions

I quite simple terms, a vulgar fraction is a fraction that is expressed by p/q. In other words, a fraction expressed using a numerator over a denominator is a vulgar fraction.

• Vulgar fractions are NOT expressed in decimals
• Vulgar fractions are also called common fractions.
• The P and Q are always integers in a vulgar fraction, and Q is never zero. If Q is zero, the equation becomes undefined.
• Examples can be 500/200, 102528/242 and 299/1 etc.
• It must be remembered that any number just written as an integer, for example, 2, -25, 11 can be converted to a vulgar fraction by simply putting 1 as the denominator. So, 2 is the same as 2/1.

Decimal Fractions

The decimal fraction is a fraction where the numerator has decimal points involved. Thus, any decimal number is a decimal fraction. For example 2.852 or -5.3285 are decimal fractions. A simple decimal number can be easily converted into a vulgar fraction as well.. For example,

• If you have to convert 5.096 into vulgar fraction
  • Step 1: Remove the decimal point and add a 1 in the denominator. So we have 5096/1
  • Step 2: Add zeros equal to the amount of numbers after the decimal in the original value. Thus, in the original value, there were 3 numbers “096” after the decimal, so we add three zeros to the denominator.
  • Our answer becomes 5096/1000
  • Now simplify the equation
  • We are left with 637/125.

Thus, the vulgar fraction equivalent of 5.096 is 637/125. Don’t believe me? Just take your calculator and pop in this number, you will get the same answer!

Percentages

Percentage means something expressed as a fraction of 100. Thus, the denominator of a finally calculated percentage is 100.

• The percentage is represented by the % sign.
• Thus, 25%, for example, means 25/100.
• 112% means 112/100 and -82% means  -82/100.

How To Calculate A Percentage.

If you are asked to calculate x percent of something, the method is as follows:

• Convert the percentage to fractional form, i.e. denominator will be 100.
• Multiply the percentage with the number.

Thus, if you are asked to calculate 46 percent of 2852 it would be as follows

• 
• If you are solving paper 1, simplify to get your answer. For paper 2, simply pop-in the values in your calculator.
• Thus, it shall become 1311.92

Expressing One Quantity As A Percentage of Another

In some cases, you are required to express x as a percentage of y. There are two ways the examiner can ask this:

• Express x as a percentage of y (where x and y are numbers)
• What percentage of y is x (again x and y are numbers)

Please remember, in the first method, the number that comes before i.e. is x is the numerator. In the second method, the first number, i.e. y, is the denominator. Let’s check out examples of both:

• Express 121 as a percentage of 1100.
  • First number 121, second number 1100.
  • So  (the multiplication with 100 is to convert to percentage)
  • So we have 11%. DO NOT forget the percent sign.
• What percentage of 1100 is 121.
  • In this method, first number is 1100 and the second number is 121.
  • Thus, we have 1100 as denominator, and 121 as numerator.
  • Thus,
  • 
  • Which gives us the same 11 %.

How To Calculate Percentage Increase or Percentage Decrease?

Please remember the formula thoroughly for this, as the students make SUPER mistakes in this formula.

• Remember, the change in value MUST have a sign, otherwise, your answer is wrong. if there has been an increase the sign shall be positive, if there has been a decrease, the sign will be negative. You MAY NOT put a sign if there has been an increase, but do not forget the sign if there has been a decrease.
• The original value is the base value. DO NOT take the new value. That is the common mistake I was referring to earlier.

Thus, if the question is as follows:

The sales in 2017 were $1000. The sales in this year have been $2890. What is the percentage increase or decrease in sales.

You will solve as follows:

Sales in Base Year (THE ORIGINAL VALUE) = 1000
Sales This Year (New Value) = 2890

Change in sales = New Value – Original Value
Change in Sales = 2890-1000 = 1890

Now the Formula

So, we have

189 %  increase.

Reverse Percentages

This part is the trickiest out of all the parts that have been considered up to this point.

Basically, here, you are given a percentage value that you have to use in order to track the original value.

For example, the question says

The price of a computer after a 40 percent decrease is $ 120. What was the original value of the computer.

We know that the formula to find the percentage decrease or increase was

We also know that in this case, the answer to this formula is 40, the new value is 120. We do not know the original value or the change in value. So, we will use the formula in reverse.

Remember, in the reverse formula, the answer to the equation is already as a percentage, so we will NOT add a 100 as denominator to 40.

Now we solve it algebraically

Original Value = 12000/60

Original Value = $200

Now Let Us Cross Check The Answer

80 Dollars is the decreased amount.

As a result, the new value is $120.

Sometimes, you only need to find the decrease in the value, thus, the formula becomes even shorter.

In order to start learning about addition and subtraction of fractions, please go over the following topics first:

1- L.C.M

2- Prime Factors

Steps for adding fractions

1- First check if the denominators are the same. If yes, simply add/subtract the numerators. If, however, the denominators are not the same, take the L.C.M of the denominator

2- Divide the L.C.M with each denominator and multiply the quotient with the numerator.

3- After creating a single fraction, do the addition and subtraction

4- Simplify in the end if possible.

Example

2/3 + 5/3 – 8/3

Denominators are the same so

= (2+5-8)/3

= -1/3

Example 2

1/2 + 7/6 – 8/9

Denominators are not the same

Take L.C.M of denominators

L.C.M = 18

Now Divide 18 by each denominator and multiply with the numerator to get new numerators.

18/2 x 1 = 9

18/6 x 7 = 21

18/9 x 8 = 16

Once you get these, make the new equation

(9+21-16)/18

Do the addition/ subtraction

14/18

Simplify

7/9

There is no L.C.M or prime factors involved in multiplication or division of fraction. This is a common mistake that must be avoided.

In multiplication of matrices the following steps are to be followed

1- Simplify the numerators with the denominators

2- Multiply the numerators with the numerators and the denominators with the denominators

Example

Please see the video attached for an informative example.

Division of Fractions

A dividing fraction must be converted in a multiplying fraction to solve the question. This is done by reciprocal. In other words we make the numerator into the denominator and the denominator into the numerator. Or a/b becomes b/a. After reciprocation, the division sign changes into the multiply sign. So in order to solve

1/2 ÷ 3/4

We reciprocate 3/4 into 4/3

It becomes

1/2 x 4/3

After that we use the same steps as used in the multiplication of fractions.