Skip to main content
\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · MYP 1

Percentage

⌚ ~122 min ▢ 13 blocks ✓ 55 exercises ➣ Prerequisites : Fractions, Decimals
I Percentages and Their Forms
I.1 What a Percentage Is
Definition — Percentage
A percentage is a way of writing a fraction with denominator \(100\): \(42\pourcent\) is \(\dfrac{42}{100}\), that is \(42\) out of \(100\).
The symbol \(\pourcent\) means “percent”, which comes from the phrase “per centum”, meaning “out of one hundred”.
Example
This grid has 100 squares. Since \(\textcolor{colorprop}{23}\) out of \(\textcolor{colordef}{100}\) squares are coloured, we say that \(\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent}\) of the grid is coloured.
Figure
$$\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent} = \dfrac{\textcolor{colorprop}{23}}{\textcolor{colordef}{100}}$$
Example
A breakfast cereal contains \(42\pourcent\) of carbohydrates. What mass of carbohydrates is there in \(100\) g of cereal? In \(200\) g? In \(50\) g?

“\(42\pourcent\)” means “\(42\) for every \(100\)”: in \(100\) g of cereal there are \(42\) g of carbohydrates. The mass of carbohydrates is proportional to the mass of cereal:
\hline Cereal (g) \(100\) \(200\) \(50\)
Carbohydrates (g) \(42\) \(84\) \(21\)
\(200\) g is twice \(100\) g, so \(2 \times 42 = 84\) g; \(50\) g is half of \(100\) g, so \(42 \div 2 = 21\) g.

Skills to practice
  • Converting Fraction Models to Percentages
  • Reading a Percentage as a Rate for 100
I.2 Converting Between Forms

Percentages, fractions, and decimals are three different ways to talk about the same value. For example, \(\textcolor{colorprop}{50}\textcolor{colordef}{\pourcent}\), the fraction \(\dfrac{\textcolor{colorprop}{1}}{\textcolor{colordef}{2}}\), and the decimal \(0.5\) all mean “half”. Being able to convert between these forms is a very useful skill.
Remark: A very important idea is that \(100\pourcent = \dfrac{100}{100} = 1\).
Since \(100\pourcent\) is just another way of writing the number \(1\), you can multiply any number by \(100\pourcent\) without changing its value. This is a very useful trick for converting decimals and fractions into percentages.

Method — Percentage to Fraction
To convert a percentage to a fraction, write it as a fraction over 100, then simplify if possible.
Figure
Method — Fraction to Percentage
To convert a fraction like \(\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}}\) to a percentage, you have two common methods:
  • Method 1: Equal Fractions. Find an equal fraction with denominator \(100\) (when possible, in particular when the denominator divides \(100\); it is not possible for \(\dfrac{1}{3}\)).
    Figure
  • Method 2: Multiply by \(100\pourcent\). This works because multiplying by \(100\pourcent\) is the same as multiplying by 1. $$ \begin{aligned} \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} &= 0.75 \quad (\text{since } 3 \div 4 = 0.75) \\ &= 0.75 \times \textcolor{colordef}{100\pourcent} \\ &= \textcolor{colorprop}{75}\textcolor{colordef}{\pourcent} \end{aligned} $$
Method — Percentage to Decimal
To convert a percentage to a decimal, divide by 100. A quick way to do this is to move the decimal point two places to the left. $$ \textcolor{colorprop}{45}\textcolor{colordef}{\pourcent} = \textcolor{colorprop}{45} \div \textcolor{colordef}{100} = 0.45 $$
Method — Decimal to Percentage
To convert a decimal to a percentage, multiply by 100. A quick way to do this is to move the decimal point two places to the right and add the percent sign (\(\pourcent\)). $$ 0.68 = 0.68 \times 100\pourcent = \textcolor{colorprop}{68}\textcolor{colordef}{\pourcent} $$
Skills to practice
  • Converting Percentages into Decimals
II One Quantity as a Percentage of Another
II.1 Expressing a Part as a Percentage

Percentages are one of the best ways to compare a part to a whole.
For example, if a class of \(\textcolor{colordef}{20}\) students has \(\textcolor{colorprop}{12}\) girls, what percentage of the class is female? To solve this, we look for an equal fraction with denominator \(100\): $$\frac{\text{part}}{\text{whole}} = \frac{\textcolor{colorprop}{12}}{\textcolor{colordef}{20}} = \frac{\textcolor{colorprop}{12} \times 5}{\textcolor{colordef}{20} \times 5} = \frac{\textcolor{colorprop}{60}}{\textcolor{colordef}{100}} \qquad \text{(equal fractions: numerator and denominator multiplied by } 5\text{)}$$ The percentage of girls in the class is \(\textcolor{colorprop}{60}\textcolor{colordef}{\pourcent}\). This means for every \(\textcolor{colordef}{100}\) students, \(\textcolor{colorprop}{60}\) would be girls.

Method — Ratio to Percentage
To convert a part-to-whole ratio (the fraction part/whole) into a percentage, use the following formula: $$\text{Percentage}= \frac{\textcolor{colorprop}{\text{part}}}{\textcolor{colordef}{\text{whole}}} \times \textcolor{colordef}{100}\pourcent$$ Scaling the whole to \(\textcolor{colordef}{100}\) shows the percentage directly: for example, \(\textcolor{colorprop}{5}\) girls out of \(\textcolor{colordef}{25}\) scale up (\(\times 4\)) to \(\textcolor{colorprop}{20}\) out of \(\textcolor{colordef}{100}\).
Figure
Example
At a maths quiz, Leo answered \(\textcolor{colorprop}{21}\) questions correctly out of a total of \(\textcolor{colordef}{25}\) questions. Calculate his percentage score.

  • The \(\textcolor{colorprop}{\text{part}}\) is the number of correct answers: \(\textcolor{colorprop}{21}\).
  • The \(\textcolor{colordef}{\text{whole}}\) is the total number of questions: \(\textcolor{colordef}{25}\).
  • \(\begin{aligned}[t]\text{Percentage Score} &= \frac{\textcolor{colorprop}{21}}{\textcolor{colordef}{25}} \times \textcolor{colordef}{100}\pourcent\\ &= 0.84 \times 100\pourcent \\ &= \textcolor{colorprop}{84}\textcolor{colordef}{\pourcent}\end{aligned}\)

A Proportion Never Exceeds 1
A proportion is the ratio of a part to the whole: \(\dfrac{\text{part}}{\text{whole}}\). Since the part is never bigger than the whole, a proportion is always smaller than or equal to \(1\), and, as a percentage, smaller than or equal to \(100\pourcent\). In a class of \(25\) pupils with \(11\) girls, the proportion of girls is \(\dfrac{11}{25} = \dfrac{44}{100} = 44\pourcent\).
Report an error
Compare Like with Like
One quantity can be written as a percentage of another only when both are of the same kind (two lengths, two masses, two durations…) and written in the same unit: convert first, then divide. For example, to express \(35\)~cm as a percentage of \(7\)~m, write \(7\)~m \(=700\)~cm: $$\begin{aligned}[t] \text{Percentage}&=\dfrac{35\,\mathrm{cm}}{700\,\mathrm{cm}}\times100\pourcent &&\text{(same unit)}\\ &=\dfrac{5}{100}\times100\pourcent &&\text{(simplifying by }7\text{)}\\ &=5\pourcent. \end{aligned}$$ Without the conversion, \(\dfrac{35}{7}\times100\pourcent=500\pourcent\): impossible, since a part never exceeds its whole.
Report an error
Skills to practice
  • Expressing a Proportion as a Percentage
  • Finding the Percentage of a Ratio
  • Converting to the Same Unit First
II.2 Comparing with Percentages

In Parliament A, there are \(\textcolor{colorprop}{26}\) women out of \(\textcolor{colordef}{50}\) members. In Parliament B, there are \(\textcolor{colorprop}{60}\) women out of \(\textcolor{colordef}{200}\) members. Hugo says, “Since there are more women in Parliament B, women have better representation there.”
Is this statement a fair comparison?

To make a fair comparison, we must compare the percentages, not the absolute numbers.
  • Parliament A:
    \(\text{Percentage of women} = \frac{\textcolor{colorprop}{26}}{\textcolor{colordef}{50}} \times 100\pourcent = \textcolor{colorprop}{52}\textcolor{colordef}{\pourcent}\)
  • Parliament B:
    \(\text{Percentage of women} = \frac{\textcolor{colorprop}{60}}{\textcolor{colordef}{200}} \times 100\pourcent = \textcolor{colorprop}{30}\textcolor{colordef}{\pourcent}\)
Conclusion: Since \(\textcolor{colorprop}{52}\textcolor{colordef}{\pourcent} > \textcolor{colorprop}{30}\textcolor{colordef}{\pourcent}\), women are proportionally better represented in Parliament A.


Method — Comparing with Percentages
When comparing different part-to-whole ratios, converting them to percentages provides a common baseline (out of 100), which allows for a fair and direct comparison.
  • Step 1: Calculate the percentage for each group.
  • Step 2: Compare the percentages to draw a conclusion.
Skills to practice
  • Comparing Effectiveness Using Percentages
III A Percentage of a Quantity
III.1 Finding a Percentage of a Quantity
Method — Percentages to Know by Heart
Some percentages are fractions in disguise, and their value is found by a division:
  • \(50\pourcent = \dfrac{50}{100} = \dfrac{1}{2}\): half — divide by \(2\);
  • \(25\pourcent = \dfrac{1}{4}\): a quarter — divide by \(4\); \(75\pourcent = \dfrac{3}{4}\): three quarters;
  • \(10\pourcent = \dfrac{1}{10}\): a tenth — divide by \(10\); \(20\pourcent = \dfrac{1}{5}\): a fifth — divide by \(5\);
  • \(5\pourcent\) is half of \(10\pourcent\); \(100\pourcent\) is the whole quantity.
Example
Compute mentally: \(25\pourcent\) of \(32\), \(10\pourcent\) of \(250\), \(75\pourcent\) of \(4\), \(5\pourcent\) of \(60\).

$$\begin{aligned}[t] 25\pourcent \text{ of } 32 &= 32 \div 4 = 8 &&\text{(a quarter)}\\ 10\pourcent \text{ of } 250 &= 250 \div 10 = 25 &&\text{(a tenth)}\\ 75\pourcent \text{ of } 4 &= 3 \times (4 \div 4) = 3 &&\text{(three quarters)}\\ 5\pourcent \text{ of } 60 &= 6 \div 2 = 3 &&\text{(half of } 10\pourcent \text{ of } 60\text{)} \end{aligned}$$

Method — Finding the Part
To find a part of a total, multiply the percentage by the whole. $$\textcolor{colorprop}{\text{Part}} = \text{Percentage} \times \textcolor{colordef}{\text{Whole}}$$ Taking \(x\pourcent\) of a number means taking the fraction \(\dfrac{x}{100}\) of it. Remember to convert the percentage to a decimal or fraction before calculating.
Example
In a school with \(\textcolor{colordef}{200}\) students, \(\textcolor{colorprop}{60}\textcolor{colordef}{\pourcent}\) are girls. Calculate the number of girls.

Method 1: Using the formula $$\begin{aligned}\textcolor{colorprop}{\text{Number of girls}} &= \textcolor{colorprop}{60}\textcolor{colordef}{\pourcent} \times \textcolor{colordef}{200} \\ &= 0.60 \times \textcolor{colordef}{200} \\ &= \textcolor{colorprop}{120}\end{aligned}$$ There are \(\textcolor{colorprop}{120}\) girls in the school. Method 2: “for 100”
\(60\pourcent\) means \(60\) girls for every \(100\) pupils. Since \(200 = 2 \times 100\), there are \(2 \times 60 = 120\) girls.
\hline Pupils \(100\) \(200\)
Girls \(60\) \(120\)

Skills to practice
  • Calculating a Percentage Mentally
  • Finding a Part Using Percentages
III.2 Percentage Increase and Decrease

Quantities often change by a certain percentage. For example:
  • A store offers a 20\(\pourcent\) discount (a decrease).
  • A salary increases by 7\(\pourcent\).
  • A city's population grows by 10\(\pourcent\).

Method — Two-Step Method for Percentage Change
  1. Calculate the change amount: $$\text{Change} = \text{Percentage} \times \text{Original Value}$$
  2. Calculate the new value:
    • For an increase: $$\text{New Value} = \text{Original Value} + \text{Change}$$
    • For a decrease: $$\text{New Value} = \text{Original Value} - \text{Change}$$
Example
The original price of a shirt is \(\dollar\)50. Calculate the final price after a 20\(\pourcent\) discount.

  1. Calculate the decrease amount: $$\begin{aligned} \text{Decrease} &= 20\pourcent \text{ of }\dollar 50\\ &= 20\pourcent \times \dollar 50\\ & = 0.20 \times \dollar 50 \\ &= \dollar 10\\ \end{aligned} $$
  2. Calculate the new price: $$\text{New Price} = \dollar 50 - \dollar 10 = \dollar 40$$

Skills to practice
  • Finding the New Quantity after an Increase
  • Finding the New Quantity after a Decrease
IV Going further
Skills to practice
  • Solving Percentage Problems
  • Investigating Patterns in Percentages
  • Quiz