6.1 Operations, Order of Operations & Number Sense

Working With Integers, Fractions and Decimals

The Numerical Ability subtest of the CSE-PPT Professional exam is a no-calculator section, so every computation must be done by hand or in your head. Fast, accurate handling of the three number forms, integers (whole numbers and their negatives), fractions, and decimals, is the foundation for every word problem later in the exam. Master these first and the harder items become mechanical.

Signed-number (integer) rules

When adding or subtracting signed numbers, picture movement along a number line. Same signs: add the absolute values and keep the sign. Different signs: subtract the smaller absolute value from the larger and keep the sign of the larger number. For multiplication and division, like signs give a positive, unlike signs give a negative.

Worked example: -8 + 5 = -3 (different signs, 8 - 5 = 3, keep the negative). And -6 x -4 = +24 (like signs). A classic trap is the double negative: 10 - (-3) = 10 + 3 = 13, not 7.

Worked example (basic chain): 248 + 367 - 195. Work left to right: 248 + 367 = 615; then 615 - 195 = 420. To check mentally, round: 250 + 370 - 195 is about 425, close to 420, so no gross error slipped in.

Fraction operations

To add or subtract fractions, convert to a common denominator (the least common denominator, LCD), combine the numerators, and simplify. To multiply, multiply numerators and denominators straight across, cancelling common factors first to keep the numbers small. To divide, multiply by the reciprocal, a rule reviewers call 'keep-change-flip'.

Worked example: 3/4 + 1/6. The LCD of 4 and 6 is 12. Convert: 3/4 = 9/12 and 1/6 = 2/12. Add: 9/12 + 2/12 = 11/12. Because 11 and 12 share no common factor, 11/12 is already in lowest terms.

Worked example (multiply and divide): 2/3 x 3/8 = (2 x 3)/(3 x 8) = 6/24 = 1/4 (or cancel the 3s first: 2/1 x 1/8 = 2/8 = 1/4). And 5/6 divided by 2/3 = 5/6 x 3/2 = 15/12 = 5/4.

Mixed numbers and comparing fractions

Convert a mixed number to an improper fraction before computing: 2 3/4 = (2 x 4 + 3)/4 = 11/4. To compare fractions without a calculator, cross-multiply: for 3/5 versus 5/8, compare 3 x 8 = 24 against 5 x 5 = 25; since 24 is less than 25, 3/5 is smaller. This trick is faster than finding a common denominator when you only need to know which is larger.

Common mistake: adding fractions by adding the numerators and denominators straight across (1/2 + 1/3 is not 2/5). Always convert to the LCD first; 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Decimals and fraction-decimal conversion

Line up the decimal points when adding or subtracting. When multiplying, ignore the points, multiply as whole numbers, then place the point so the answer has as many decimal places as the two factors combined. To convert a decimal to a fraction, write the digits over the matching place value and simplify.

Worked example: 0.625 to a fraction. 0.625 = 625/1000. Divide top and bottom by 125: 625 divided by 125 = 5 and 1000 divided by 125 = 8, so 0.625 = 5/8. Memorize the common equivalents; they appear constantly:

Order of operations (PEMDAS / GEMDAS)

Evaluate in this fixed order: Parentheses (grouping), Exponents, Multiplication and Division (left to right, equal rank), then Addition and Subtraction (left to right). Filipino reviewers often shorten the last four to MDAS. The single biggest error is doing addition before an earlier multiplication or division.

Worked example: 12 + 6 / 3 x 2 - 4. No parentheses or exponents, so handle division and multiplication left to right first: 6 / 3 = 2, then 2 x 2 = 4. The expression is now 12 + 4 - 4 = 12. A student who added 12 + 6 first would get a different value, and that wrong path is exactly what the distractor options are built from.

Factors, multiples and divisibility

A factor divides a number exactly; a multiple is the product of a number and an integer. Quick divisibility rules save time with no calculator: divisible by 2 if the last digit is even; by 3 if the digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 5 if it ends in 0 or 5; by 6 if divisible by both 2 and 3; by 9 if the digit sum is divisible by 9; by 10 if it ends in 0.

Example: Is 342 divisible by 6? It is even (passes 2) and 3 + 4 + 2 = 9 is divisible by 3, so yes, 342 / 6 = 57.

Rounding and estimation

To round to a place value, look at the digit to the right: 5 or more rounds up, 4 or less rounds down. Estimation lets you eliminate impossible choices fast. If a problem asks for 48 x 21, estimate 50 x 20 = 1,000, so options of 108 or 10,080 must be wrong while the real answer 1,008 sits right beside the estimate. On a timed, no-calculator test, always sanity-check your exact answer against a rounded estimate before you shade the circle.