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:
Fractions |
Ratio |
Percentage |
Decimals |
---|---|---|---|
\(\begin{align} \frac{2}{5} \end{align}\) | \(2:5\) | \(\begin{align} \frac{2}{5} \times 100 = 40\% \end{align}\) | \(\begin{align*} 40\% &= \frac{40}{100}\\ \\ &= 0.4 \end{align*}\) |
\(\begin{align} 25\% &= \frac{25}{100}\\\\ &= \frac{1}{4} \end{align}\) | \(1:4\) | \(25\%\) | \(0.25\) |
\(\begin{align} \frac{90}{100} = \frac{9}{10} \end{align}\) | \(9:10\) | \(\begin{align} 0.9 \times 100\% = 90\% \end{align}\) | \(0.9\) |
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.
- \(105\%\)
- \(0.04\%\)
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.
- \(274\%\)
- \(0.38\%\)
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.
- If \(40\%\) of the balls are green, find the number of green balls.
- 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:
- 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 \(= 0.4 × 60\)
\(\qquad\qquad\qquad\qquad\quad\;\;= 24 \;balls\)
- 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.