Basic Operations and Fractions
The Basic Operations and Fractions domain tests whether you can evaluate numerical expressions in the correct order and handle fractions - including mixed numbers - without a calculator. Expect order-of-operations items, fraction addition and subtraction with unlike denominators, multiplication and division of fractions, simplifying, and converting between improper fractions and mixed numbers.
Order of operations
Work through an expression in this sequence: grouping symbols (parentheses, brackets, fraction bars), exponents and square roots, multiplication and division left to right, then addition and subtraction left to right. Multiplication does not automatically outrank division - whichever appears first, moving left to right, is done first. The same left-to-right rule applies to addition and subtraction.
Working with fractions
- Build equivalent fractions using the least common denominator before adding or subtracting; change only the numerator, never the denominator.
- To multiply, multiply numerators together and denominators together, then simplify by the greatest common factor.
- To divide, keep the first fraction, change ÷ to ×, and flip the second fraction (use its reciprocal).
- Convert every mixed number to an improper fraction before multiplying or dividing: multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
- When subtracting mixed numbers and the top fraction is smaller, borrow 1 from the whole number and rewrite it as denominator/denominator.
Worked example
Multiply 2 1/3 by 1 1/2.
- Convert: 2 1/3 = (2×3 + 1)/3 = 7/3; 1 1/2 = (1×2 + 1)/2 = 3/2.
- Multiply straight across: (7×3)/(3×2) = 21/6.
- Simplify by the GCF, 3: 21 ÷ 3 = 7 and 6 ÷ 3 = 2, giving 7/2.
- Convert back: 7 ÷ 2 = 3 remainder 1, so the answer is 3 1/2.
Traps that cost marks
Adding denominators (1/2 + 1/3 is not 2/5). Forgetting to invert the second fraction in division, or inverting the first one instead. Evaluating 4 + 2 × 3 as 18 rather than 10 by misapplying the order of operations. Leaving an answer unsimplified when the answer choices are simplified. Converting a fraction to a decimal and rounding too early in a multi-step item.