Arithmetic Foundations course

1. 1.Order of Operations

   An expression like $8 - 2 \times 3$ only has one correct value because everyone agrees on the order in which operations are carried out. Without that agreement, the same line of symbols could mean 18 or 2, and calculators, spreadsheets and textbooks would disagree.

   • Grouping symbols include parentheses, brackets, fraction bars and the bar of a square root.
   • Within one level, always move left to right: $12 \div 3 \times 2 = 8$, not $2$.
   • A negative sign in front of a power is applied after the power: $-3^2 = -9$, while $(-3)^2 = 9$.
   Quick check: Evaluate $3 + 6 \times 2^2$.

   $27$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study order of operations

2. 2.Fractions

   A fraction $\frac{a}{b}$ names $a$ equal parts of a whole split into $b$ parts. Two fractions are equal when they describe the same amount, which is why multiplying the top and bottom by the same nonzero number never changes a fraction's value. That one idea powers simplifying, finding common denominators and comparing sizes.

   • Simplify by dividing top and bottom by their greatest common factor.
   • Use the least common multiple of the denominators as the common denominator to keep numbers small.
   • Convert mixed numbers to improper fractions before multiplying or dividing.
   Quick check: Compute $\frac{5}{6} + \frac{1}{4}$.

   $\frac{13}{12}$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study fractions

3. 3.Percentages

   Percent means per hundred, so $35\%$ is the fraction $\frac{35}{100}$ or the decimal $0.35$. Almost every percentage question is one of three forms: find a percent of a number, find what percent one number is of another, or find the original number before a percentage change.

   • To increase by $p\%$, multiply by $1 + \frac{p}{100}$; to decrease, multiply by $1 - \frac{p}{100}$.
   • Reverse percentage: divide the final value by the multiplier to recover the original.
   • Percentage points measure the difference between two percentages; percent change measures relative change.
   Quick check: What is $12\%$ of $250$?

   $30$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study percentages

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Factors and Multiples

Prime factorization, greatest common factor and least common multiple.

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