What is a Percent?
Percents are often used to express scores, discounts, and various statistics. The word “percent” means “one part in every hundred”.
In math, a percentage is a ratio or a number expressed as a fraction with the denominator of 100. The symbol used to denote percent is “%”.
In terms of algebra, finding out a “percent” is represented as,
\(x\%=\frac{x}{100}\)
Example: Represent 60 out of 100 as a percent.
60 out of 100, means \(\frac{60}{100}\)
Hence, \(\frac{60}{100}=60\% \) or 60 percent.
Fractions
A fraction is a portion or section of any quantity taken from the whole.
For example: Suppose there is a box with eight basketballs and you have taken one basketball from it. Write the fraction to represent the basketballs taken out of the box.
Of the total basketballs, the number of basketballs you have taken is:
one-eighth or \(\frac{1}{8}\) of the total.
Mixed Number
A mixed number is a combination of a whole number and a proper fraction. It usually denotes a number that falls between two whole numbers.
What are Models?
A model is a grid that helps in the representation of fractions and percents. It’s a grid with 10 columns and 10 rows. So in total we have 100 boxes, which means each box is a \(\frac{1}{100}\) fraction, or 1% of the total number of boxes.
So how can we represent fractions and percents on this model?
- Percent: Each box or cell in the grid represents 1%. So if you need to visualize 8% then you have to shade 8 cells of the grid.
- Fractions: Each box or cell is \(\frac{1}{100}\). So we first convert the fraction such that it has a denominator of 100. Then shade the number of boxes or cells as the value of the numerator.
Let’s understand this with the help of an example.
Example: Draw a grid for the number \(\frac{2}{25}\)
\(\frac{2}{25}=\frac{2 \times 4}{25 \times 4}=\frac{8}{100}\) Write as a fraction with denominator 100
So to visualise \(\frac{8}{100}\), we need to shade 8 cells in the 10-by-10 grid below.
What are Equivalent Fractions?
We know that two or more numbers that have the same value are equivalent to each other. So, two or more fractions that refer to the same part of a whole are called equivalent fractions. Equivalent fractions always represent the same point on the number line.
Example:
\(\frac{2}{3}\) is equivalent to \(\frac{4}{6}\) because they represent the same part of a whole.
That’s because, \(\frac{2}{3}=\frac{4}{6}\)
Example: Write \(\frac{5}{25}\) as a percent. Also write an equivalent fraction related to it.
We have to multiply 25 by 4 to get 100. So to represent the above fraction in percent, multiply the numerator and denominator by 4.
\(\frac{5}{25}=\frac{5 \times 4}{25 \times 4}=\frac{20}{100}=20\%\)
\(\frac{5}{25}=\frac{20}{100}\)
So, \(\frac{20}{100}\) is an equivalent fraction of \(\frac{5}{25}\).
Solved Percent to Fractions Examples
Example 1: In a soft drinks factory, for every 20 cans produced, 10 cans are not of good quality and are discarded. What is the percent of the discarded cans? Express this using models as well.
Number of cans produced = 20
Number of cans discarded = 10
Fraction of cans discarded = \(\frac{10}{20}\)
Expressed as a percent = \(\frac{10 \times 5}{20 \times 5}=\frac{50}{100}=50\%\)
So 50% of the cans are discarded.
In the 10-by-10 grid we need to shade 50 cells to represent
\(\frac{50}{100}\) or 50%.
Example 2: In a family of 16 members, four have digital watches. What percentage of the members have a digital watch?
Write a fraction that represents the members who have a digital watch to the total number of family members, and convert the obtained fraction into a percent.
\(\frac{\text{Members with a digital watch}}{\text{Total Number of Members}}=\frac{4}{16}\)
The denominator of 16 is not a factor of 100. But 4 is a factor of both 16 and 100. So, first write an equivalent fraction whose denominator is 4. Then convert that fraction to a percent.
So, \(\frac{4}{16}=\frac{1}{4}\) Divide the numerator and denominator by 4
\(\frac{1 \times\ 25}{4 \times\ 25}=\frac{25}{100}\) Multiply the numerator and denominator by 25
= 25 % Write the numerator with a percent symbol
So, 25 % of the members have a digital wrist watch.
The word cent denotes 100. So percent means for every hundred. A cent is also \(\frac{1}{100}\)th of a dollar.
Percentage refers to a general relationship rather than a specific measure like percent. “Percent” means “per hundred” and can be written or expressed with the percent symbol “%”. For example, a large percentage of the city participated in the race, but only 2 percent crossed the finish line!
To create equivalent fractions, we multiply or divide the numerator and denominator by the same number. When both the numerator and denominator of a fraction are multiplied by the same non-zero number, the fraction retains its value.
When both the numerator and denominator have no common factors that can be divided further, and when the numerator and the denominator are coprime, they are said to be in the simplest form or the fraction has the lowest terms.
Grids are used as a model for the representation of fractions and percents with the help of a diagram. It is a simple and creative way to teach students about fractions and percents of numbers.