Arithmetic and Number Sense: Operations, Fractions, Decimals, Percents, Ratios
Core skills
Order of operations governs every computation: grouping symbols first, then exponents, then multiplication and division strictly left to right, then addition and subtraction strictly left to right. For fractions, you need a common denominator to add or subtract, you multiply straight across to multiply, and you multiply by the reciprocal to divide. Always reduce the final answer, and convert improper fractions to mixed numbers if the answer choices are mixed numbers. With decimals, line up place values when adding or subtracting, and count total decimal places when multiplying. Percents are ratios out of 100, so convert by moving the decimal two places (35% = 0.35; 0.06 = 6%). Nearly every percent problem reduces to part = percent x whole. Ratios compare quantities, so keep track of whether a ratio is part-to-part (boys to girls) or part-to-whole (boys to all students), and solve proportions by cross-multiplying.
Traps
Multiplying before dividing when they should be done left to right; treating 3/4 divided by 2/3 as 3/4 times 2/3; forgetting to borrow when subtracting mixed numbers; assuming a percent increase and an equal percent decrease cancel out; and mixing units inside a ratio.
Worked example
Divide 3/4 by 2/3. Multiply by the reciprocal: 3/4 x 3/2 = 9/8 = 1 1/8. Rule of thumb: rewrite division of fractions as multiplication by the flipped second fraction before touching any numbers.