0.3.1 Fractions, percentages and order of operations

Order of operations

An expression like $2 + 3 \times 4$ could mean 20 or 14, depending on which operation you do first. Maths and Python agree on one order, so it always means 14.

Rule: Order of operations

\begin{gathered} \text{brackets} \;\to\; \text{powers} \\ \to\; \times\ \div \;\to\; +\ - \end{gathered}

Work out brackets first, then powers. Then multiply and divide, from left to right. Last, add and subtract, from left to right.

Expression	Value	Why
$2 + 3 \times 4$	14	multiply before adding
$(2 + 3) \times 4$	20	brackets first
$2 \times 3^2$	18	the power applies to the 3 only
$12 \div 4 \div 3$	1	divide left to right: $3 \div 3$
$-3^2$	$-9$	the power comes before the minus sign

A fraction bar groups everything above it and everything below it, as if in brackets. So $\dfrac{6 + 4}{2} = 5$ , and in code that is (6 + 4) / 2. Leave out the brackets and Python works out 6 + 4 / 2 = 8.

12 \div 4 \div 3

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Fractions

A fraction $\frac{a}{b}$ is $a$ divided by $b$ . The top is the numerator and the bottom the denominator. Multiplying and dividing are the easy cases.

Rule: Multiplying and dividing fractions

\begin{gathered} \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \\ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \end{gathered}

Multiply tops together and bottoms together. To divide, flip the second fraction and multiply. For example, $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ .

To add, the bottoms must match first. Rewrite each fraction over a common denominator, then add the tops. For $\frac{1}{4} + \frac{1}{6}$ , both 4 and 6 go into 12:

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

Rule: Adding fractions

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Bring both fractions over the same bottom, then add the tops. Simplify at the end by dividing top and bottom by any common factor: $\frac{10}{12} = \frac{5}{6}$ .

Try itOrdinary Python numbers are decimals with a tiny rounding error, so 0.1 + 0.2 is not exactly 0.3.Fraction from the standard library keeps fractions exact and simplifies them for you. It is a handy answer-checker.

from fractions import Fraction print(Fraction(2, 3) * Fraction(3, 4)) print(Fraction(2, 3) / Fraction(4, 5)) print(Fraction(1, 4) + Fraction(1, 6)) print(0.1 + 0.2, 0.1 + 0.2 == 0.3) print(Fraction(1, 10) + Fraction(2, 10) == Fraction(3, 10))

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Percentages

Percent means "per hundred", so 7% is $\frac{7}{100} = 0.07$ . Turn a percentage into a decimal before you calculate: 7% of 250 is $0.07 \times 250 = 17.5$ .

The change from an old value to a new one is usually given as a percentage of the old value.

Rule: Percentage change

\text{change} = \frac{\text{new} - \text{old}}{\text{old}}

From 80 to 92 is a change of $\frac{92 - 80}{80} = 0.15$ , a rise of 15%. Always divide by the value you started from.

Growing by a percentage $p$ multiplies by $1 + p$ , and falling by $p$ multiplies by $1 - p$ . So a rise of 25% multiplies by 1.25, and a fall of 20% multiplies by 0.80.

Rule: Undoing a percentage rise

\begin{gathered} \text{new} = \text{old} \times (1 + p) \\ \text{old} = \frac{\text{new}}{1 + p} \end{gathered}

Here $p$ is the rise as a decimal. To undo it, divide by $1 + p$ . Do not take $p$ off the new value. After a 20% rise a value is 120. It started at $120 / 1.2 = 100$ , not $120 \times 0.8 = 96$ .

1.25 \times 0.8 = 1

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Percent or percentage points?

If a rate moves from 4% to 5%, it rose by 1 percentage point: the plain difference, $5\% - 4\%$ . As a percentage change it rose by 25%, because $\frac{0.05 - 0.04}{0.04} = 0.25$ . Both statements are true, and people mix them up constantly. Say which one you mean.

Common mistakes

\frac{1}{2} + \frac{1}{3} = \frac{5}{6}

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Fractions, percentages and order of operations

0 of 8 correct

1.
Evaluate $3 + 4 \times 2$
2.
Evaluate $20 - 2 \times 3^2$
3.
Write $\dfrac{2}{3} + \dfrac{1}{6}$ as a single fraction
4.
Write $\dfrac{3}{4} \div \dfrac{9}{8}$ as a single fraction
5.
What is 15% of 80?
6.
A value rises from 50 to 56. What is the percentage change? (Type 12% or 0.12.)
7.
After a 25% rise a value is 150. What was it before the rise?
8.
A rate moves from 4% to 5%. By what percentage did it rise?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

\times\ \div

Fractions, percentages and order of operations

Order of operations

Rule: Order of operations

Fractions

Rule: Multiplying and dividing fractions

Rule: Adding fractions

Percentages

Rule: Percentage change

Rule: Undoing a percentage rise

Percent or percentage points?

Common mistakes

Fractions, percentages and order of operations

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