1.1 Arithmetic & Number Sense

Why Arithmetic Comes First

The Philippine Military Academy Entrance Examination (PMAEE) Mathematics subtest is answered without a calculator. That single rule reshapes how you should study: every problem is scored on whether you can compute cleanly by hand and quickly. The blueprint lists arithmetic and number sense first because algebra, geometry, and word problems all rest on it. If you lose points on integer signs or order of operations, you lose them everywhere.

Integers and the Sign Rules

An integer is a whole number, positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...). Three rules cover almost every trap:

• Multiplying or dividing: same signs give a positive result; opposite signs give a negative. So (-6) x (-4) = 24, but (-6) x (4) = -24.
• Subtracting a negative turns into addition: 8 - (-5) = 8 + 5 = 13.
• Adding numbers with different signs: subtract the smaller absolute value from the larger and keep the sign of the larger. So -9 + 4 = -5.

Worked example. Simplify -3 - (-7) + (-2). Rewrite the double negative first: -3 + 7 + (-2) = 4 - 2 = 2.

Order of Operations (PEMDAS)

PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. The single most common PMAEE trap is treating multiplication as always before division, or addition always before subtraction. They are not ranked against each other: multiplication and division share one level and are done left to right; addition and subtraction share the next level and are also done left to right.

Worked example. Evaluate 6 + 2 x (3^2 - 4) / 2.

1. Parentheses/exponent inside: 3^2 = 9, so 9 - 4 = 5.
2. Expression becomes 6 + 2 x 5 / 2.
3. Multiply and divide left to right: 2 x 5 = 10, then 10 / 2 = 5.
4. Add: 6 + 5 = 11.

A test-taker who divides before multiplying might write 5 / 2 first and get a wrong answer. Always sweep left to right within the same rank.

Factors, Multiples, and Divisibility

A factor of a number divides it evenly; a multiple is the result of multiplying it by an integer. Because you cannot punch numbers into a calculator, memorize these divisibility tests to factor and reduce fractions fast:

The greatest common factor (GCF) is the largest number dividing two values; the least common multiple (LCM) is the smallest number both divide. For 12 and 18: GCF = 6, LCM = 36. You need the GCF to reduce fractions and the LCM to add unlike fractions.

Fractions and Decimals

To add or subtract fractions, use a common denominator. For 2/3 + 1/4, the LCM of 3 and 4 is 12: 8/12 + 3/12 = 11/12. To multiply, multiply across (2/3 x 3/5 = 6/15 = 2/5). To divide, multiply by the reciprocal (3/4 / 2/3 = 3/4 x 3/2 = 9/8).

Keep these decimal-fraction equivalents memorized for speed: 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 3/8 = 0.375, 1/3 = 0.333.... A frequent trap is placing the decimal point wrong when multiplying: 0.3 x 0.4 has two decimal places total, so the answer is 0.12, not 1.2.

Percent

Percent means per hundred, so 45% = 45/100 = 0.45. The fastest hand method is converting to a friendly fraction: 25% of 80 = (1/4)(80) = 20; 20% of 150 = (1/5)(150) = 30; 12.5% of 64 = (1/8)(64) = 8.

Percent increase/decrease uses (change / original) x 100. If a cadet's run time drops from 40 minutes to 34, that is 6/40 = 0.15 = a 15% decrease.

Ratio and Proportion

A ratio compares quantities (3:5); a proportion states two ratios are equal (3/5 = x/20). Solve a proportion by cross-multiplying: 3 x 20 = 5x, so 60 = 5x and x = 12.

When a total is split by a ratio, add the ratio numbers first. Worked example. Divide 240 in the ratio 3:5. The parts sum to 8, so each part is 240 / 8 = 30. The shares are 3 x 30 = 90 and 5 x 30 = 150. The trap is dividing 240 by a single ratio number (like 240 / 3) instead of by the sum of the ratio numbers.