Percentage

In this chapter, we will be discussing the below-mentioned topics in detail:

• Fractions, Ratio, Decimals and Percentage
• Convert a non-integer percentage to fraction/ decimal and vice versa
• Percentages greater than \(100\%\) or less than \(1\%\)
• Percentage of quantities 
  • Percentage of a quantity 
  • Express one quantity as a percentage of another quantity
  • Compare two quantities by percentage
• Percentage change

Fractions, Ratio, Decimals and Percentage

Let’s understand this with the help of some examples:

Percentages Greater Than \(100\%\) or Less Than \(1\%\)

Let’s understand this with the help of some examples:

Question 1:

Without using a calculator, express each of the following percentages as a fraction.

1. \(105\%\)
2. \(0.04\%\) 

Solution: 

\(\begin{align*} 105\% &= \frac{105}{100}\\ &= \frac{21}{20}\\ &= \frac{11}{20} \end{align*}\)

\(\begin{align*} 0.04\% &= \frac{0.04 × 100}{100 × 100}\\ &= \frac{4}{10000}\\ &= \frac{1}{2500} \end{align*}\)

Question 2:

Without using a calculator, express each of the following percentages as a decimal.

1. \(274\%\)
2. \(0.38\%\)

Solution: 

\(\begin{align*} 274\% &=  \frac{274}{100}\\ \\                   &= 2.74 \\ \\ \end{align*}\)

\(\begin{align*} 0.38\% &= \frac{0.38}{100}\\ \\                     &= 0.0038\\ \end{align*}\)

Percentage Of A Quantity

Let’s understand this with the help of some examples:

Question 3:

A box contains \(60\) balls, of which \(16\) are red.

1. If \(40\%\) of the balls are green, find the number of green balls.
2. The rest of the balls in the box are yellow.

If \(70\%\) of the yellow balls are removed, find the number of yellow balls left.

Solution: 

1. Number of green balls\(\begin{align} = \frac{40}{100} \times 60\\ \end{align}\)

\(\qquad\qquad\qquad\qquad\qquad= 24 \;balls\)

OR

Number of green balls \(= 0.4 × 60\)

 \(\qquad\qquad\qquad\qquad\quad\;\;= 24 \;balls\)

2. Number of yellow balls at first\(\begin{align*} &= 60 \;– 16 \;– 24 \end{align*}\)

\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad= 20\)

Number of yellow balls remaining \(= \frac{30}{100} × 20\)

\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad= 6 \;balls\)

OR

Number of yellow balls remaining \(= 0.3 × 20\) 

\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad= 6 \;balls\)\(\)

Comparing two quantities by percentage

Let’s understand this with the help of some examples:

Question 4:

Shop \(A\) sold \(225\) out of \(1000\) vinyl records in January. Shop \(B\) sold \(400\) out of \(1500\) vinyl records in the same month. Which shop had a higher proportion of sales in January?

Solution: 

Percentage of sales by Shop\(\begin{align} A=\frac{225}{1000} \times 100 \end{align}\)

\(\begin{align} \qquad\qquad\qquad\qquad\qquad\qquad\qquad=22.5\% \end{align}\)

Percentage of sales by Shop\(\begin{align} B= \frac{400}{1500} \times 100 \end{align}\)

\(\begin{align} \qquad\qquad\qquad\qquad\qquad\qquad\qquad= 26 \frac{2}{3}\% \end{align}\)

Shop \(B\) had a higher proportion of sales in January.