6.3 Measurement, Conversions & Basic Algebra

Measurement and Unit Conversions

The Philippines uses the metric system (SI), so the exam expects fluency with metric length, mass, and volume plus time conversions. The metric system is built on powers of ten, which makes most conversions a matter of moving the decimal point.

Metric prefixes and length

Each step on the metric ladder is a factor of ten. The core prefixes:

To convert from a larger unit to a smaller unit, multiply (the count of units gets bigger); from a smaller unit to a larger unit, divide.

Worked example: Convert 3.5 km to meters. Kilo means x 1,000, so 3.5 x 1,000 = 3,500 m. Convert 250 cm to meters: centi is / 100, so 250 / 100 = 2.5 m. Convert 2,000 g to kilograms: 2,000 / 1,000 = 2 kg. The same ladder governs litres for volume.

Time conversions

Time is not metric, so memorize the anchors: 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week.

Worked example: How many minutes are in 2 1/4 hours? 2.25 x 60 = 135 minutes. How many seconds in 3 minutes? 3 x 60 = 180 seconds. A pacing fact that doubles as a time problem: 170 items in 3 hours 10 minutes equals 190 minutes, roughly 1.1 minutes per item.

Perimeter and area

For the rectangle problems that recur on the exam, perimeter = 2(length + width) and area = length x width, where area carries square units.

Worked example: A rectangular lot is 15 m long and 8 m wide. Area = 15 x 8 = 120 square meters; perimeter = 2(15 + 8) = 46 m. Watch the units: area answers are always in square units (m squared), a detail the distractor options exploit by offering the perimeter value instead.

Basic Algebra: Expressions and Linear Equations

Algebra items ask you to translate words into symbols and solve for an unknown. A variable (usually x) stands for the unknown, and an equation states that two expressions are equal.

Translating words to algebra

Learn the keyword-to-operation map:

Be careful with 'less than': '5 less than a number' is x - 5, not 5 - x. And 'twice the sum of a number and 3' is 2(x + 3), with parentheses that change the answer.

Worked example: 'A number increased by 7 equals 20' becomes x + 7 = 20, so x = 13.

Solving linear equations

Isolate the variable by applying inverse operations to both sides, undoing addition or subtraction before multiplication or division.

Worked example: 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Divide both sides by 2: x = 4. Check by substituting: 2(4) + 3 = 11. Correct.

Worked example (variable on both sides): 5x - 4 = 3x + 8. Subtract 3x: 2x - 4 = 8. Add 4: 2x = 12. Divide: x = 6.

Worked example (word problem to equation): 'Three consecutive integers sum to 72; find them.' Let the integers be x, x + 1, and x + 2. Then x + (x + 1) + (x + 2) = 72, so 3x + 3 = 72, 3x = 69, and x = 23. The integers are 23, 24, and 25. Check: 23 + 24 + 25 = 72.

Evaluating and simplifying expressions

To evaluate, substitute the given value and follow the order of operations. To simplify, combine like terms, meaning terms with the same variable and power.

Worked example (evaluate): If x = 4, find 3x squared - 5 = 3(16) - 5 = 48 - 5 = 43. The exponent applies before the coefficient multiplies.

Worked example (simplify): 4x + 7 - x + 2 = (4x - x) + (7 + 2) = 3x + 9. Only like terms combine; 3x and 9 cannot merge.

Common traps

Distributing a negative: -(x - 3) = -x + 3, not -x - 3. Forgetting to apply an operation to both sides. Reversing the order in 'less than' or 'more than' phrases. On a no-calculator exam, always substitute your answer back into the original equation; a five-second check prevents a lost point.