LET Elementary Mathematics — Measurement and Problem SolvingDetailed Explanation

The Metric (SI) System: Structure and Conversions

The metric system is the official system of measurement in the Philippines and is the primary system taught in the K–12 BEC from Grade 1 onward. Its greatest advantage is that it is a base-10 system, meaning every conversion is simply a matter of multiplying or dividing by a power of 10 — the same as shifting the decimal point. The three base units are the METER (m) for length, the GRAM (g) for mass, and the LITER (L) for capacity. All other metric units are formed by attaching a PREFIX to one of these base units. The seven standard prefixes, arranged from largest to smallest, are: KILO (k) = 1,000 times the base, HECTO (h) = 100 times, DEKA (da) = 10 times, BASE (no prefix) = 1, DECI (d) = 0.1 of the base, CENTI (c) = 0.01, and MILLI (m) = 0.001. A useful mnemonic is 'King Henry Died By Drinking Cold Milk' (Kilo, Hecto, Deka, Base, Deci, Centi, Milli). The CONVERSION RULE is simple: moving DOWN the ladder (from a larger unit to a smaller unit) means MULTIPLYING; moving UP the ladder (from a smaller unit to a larger unit) means DIVIDING. Each step on the ladder is a factor of 10. For example, converting 3.5 km to meters means moving 3 steps down (kilo to base), so multiply by 10 × 10 × 10 = 1,000: 3.5 × 1,000 = 3,500 m. Converting 450 cm to meters means moving 2 steps up (centi to base), so divide by 100: 450 ÷ 100 = 4.5 m. The most essential metric facts to memorize for the LET are: 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm; 1 kg = 1,000 g; 1 g = 1,000 mg; 1 metric ton = 1,000 kg; 1 L = 1,000 mL; 1 mL = 1 cm³ (a very useful equivalence for volume-capacity problems). Derived units — area and volume — require special care. Because 1 m = 100 cm, squaring both sides gives 1 m² = 10,000 cm² (NOT 100 cm²), and cubing gives 1 m³ = 1,000,000 cm³. This 'squaring/cubing the conversion factor' trap is one of the most commonly missed item types on the LET.

The English (Customary) System and Cross-System Conversions

While the metric system is standard in Philippine schools and government use, the LET also tests the English (customary) system because it appears in many real-world contexts — imported goods are labeled in pounds and ounces, fabric is sold by the yard, and Filipino Americans often describe measurements in feet and miles. Unlike the metric system, the English system is NOT base-10, so every conversion factor must be memorized individually. Key English conversions: LENGTH — 1 foot (ft) = 12 inches (in), 1 yard (yd) = 3 feet = 36 inches, 1 mile = 5,280 feet = 1,760 yards. WEIGHT — 1 pound (lb) = 16 ounces (oz), 1 (short) ton = 2,000 pounds. CAPACITY — 1 gallon (gal) = 4 quarts (qt), 1 quart = 2 pints (pt), 1 pint = 2 cups, 1 cup = 8 fluid ounces (fl oz). Cross-system (metric–English) BRIDGE CONVERSIONS appear frequently on the LET: 1 inch = 2.54 cm (exact), 1 kg ≈ 2.2 lb, 1 mile ≈ 1.6 km, 1 L ≈ 1.06 quarts. The technique for ALL conversions — within a system or across systems — is DIMENSIONAL ANALYSIS (also called the factor-label method or unit-fraction method). In dimensional analysis, you multiply the given measurement by one or more UNIT FRACTIONS — fractions whose numerator and denominator are equal in value but expressed in different units, making the fraction equal to 1. The key is to set up the fraction so that the UNIT YOU WANT TO ELIMINATE is on the bottom, and the UNIT YOU WANT TO KEEP is on the top. The unwanted units cancel like variables in algebra, leaving only the desired unit. Example: Convert 5 feet to centimeters. Step 1: 5 ft × (12 in / 1 ft) = 60 in [feet cancel]. Step 2: 60 in × (2.54 cm / 1 in) = 152.4 cm [inches cancel]. Answer: 152.4 cm. This two-step chain is more reliable than memorizing a single direct conversion because each step uses a fact you already know.

Measurement of Time and Temperature

Time and temperature conversions are tested regularly on the LET, especially in multi-step word problems involving scheduling, elapsed time, and science-related contexts. TIME uses MIXED BASES that are not decimal: 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week, approximately 30 or 31 days = 1 month, 365 days = 1 year (366 for a leap year). Because these bases are not powers of 10, you cannot shift decimals for time conversions. Instead, divide by the conversion factor when converting from a smaller to a larger unit (e.g., 150 seconds ÷ 60 = 2.5 minutes = 2 minutes 30 seconds), and multiply when going smaller (e.g., 3 hours = 3 × 60 = 180 minutes). A critical skill for LET word problems is computing ELAPSED TIME — the duration between a start and end time — and ADDING DURATIONS to a start time to find the end time. The method: (1) Convert the total duration to hours and minutes. (2) Add the hours first, then the minutes. (3) If minutes exceed 60, regroup (add 1 to hours, subtract 60 from minutes). Remember that clock time uses 12-hour cycles (AM/PM) or 24-hour format. TEMPERATURE: The Philippines uses CELSIUS (°C) as the official unit. The LET tests conversions between Celsius and Fahrenheit (°F). The two formulas are: Celsius to Fahrenheit: F = (9/5)C + 32 (or equivalently F = 1.8C + 32). Fahrenheit to Celsius: C = (5/9)(F − 32). Anchor points to VERIFY your formula: Water freezes at 0°C = 32°F; water boils at 100°C = 212°F; normal human body temperature = 37°C = 98.6°F; a comfortable room temperature is about 25°C = 77°F. These anchor points also serve as reasonableness checks. If your answer for 'a hot day in Manila' comes out as −10°C, something went wrong.

Rate, Speed, and Work Problems

Rate problems are the most frequently appearing multi-step word problem type on the LET Mathematics subtest for Elementary level. The MASTER FORMULA for all rate problems is: AMOUNT = RATE × TIME (A = R × T). This one formula rearranges into three versions: Rate = Amount ÷ Time, and Time = Amount ÷ Rate. DISTANCE-SPEED-TIME problems are the most common form: Distance = Speed × Time (D = S × T); so Speed = D ÷ T and Time = D ÷ S. For example, a jeepney traveling at 40 km/h for 2.5 hours covers 40 × 2.5 = 100 km. SPEED UNIT CONVERSION is a derived-unit skill: converting km/h to m/s requires converting both the distance (km to m) and the time (hours to seconds) simultaneously. The shortcut is: divide km/h by 3.6 to get m/s (because 1 km/h = 1,000 m ÷ 3,600 s = 1/3.6 m/s). So 72 km/h ÷ 3.6 = 20 m/s. To go the other direction, multiply m/s by 3.6. WORK PROBLEMS apply the same rate logic to jobs done per unit time. If one worker can complete a job in 'n' hours, their WORK RATE is 1/n of the job per hour. This is the single most important insight: work rates are ADDED, not the times. When two workers or two pipes work together: Combined Rate = Rate₁ + Rate₂ = 1/n₁ + 1/n₂. The combined time to finish = 1 ÷ Combined Rate = 1 ÷ (1/n₁ + 1/n₂). Example: Pipe A fills a tank in 4 hours (rate = 1/4 per hour); Pipe B fills it in 6 hours (rate = 1/6 per hour). Combined rate = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 per hour. Time = 1 ÷ (5/12) = 12/5 = 2.4 hours = 2 hours 24 minutes. MIXTURE and FUEL CONSUMPTION problems also use A = R × T: fuel consumed = fuel rate (liters per km) × distance. These problems reward careful unit labeling at every step.

Polya's Four-Step Problem-Solving Heuristic

George Polya's four-step framework, introduced in his 1945 book 'How to Solve It,' is the OFFICIAL PROBLEM-SOLVING FRAMEWORK taught in the K–12 Mathematics curriculum and tested in the LET. As future Grade 1–6 teachers, you must know these steps thoroughly — not only to solve problems yourself, but also because the LET often asks you to IDENTIFY which step is being performed in a given scenario, or to choose the appropriate STRATEGY for a specific type of problem. The four steps are: STEP 1 — UNDERSTAND THE PROBLEM. Read the problem carefully. Identify (a) what is GIVEN (the known information), (b) what is ASKED (the unknown or goal), and (c) the CONDITIONS or constraints. Restate the problem in your own words. Identify the units involved. Ask: 'Have I seen a problem like this before?' STEP 2 — DEVISE A PLAN. Select a problem-solving STRATEGY appropriate for the problem type. Common strategies include: Draw a picture or diagram (for geometry, arrangement, and distance problems); Work backward (when the final state is known); Look for a pattern (for number sequences); Guess, check, and revise (for constrained problems); Write an equation (to translate quantitative relationships); Make an organized list or table (to count cases systematically); Solve a simpler version (to build insight); and Use logical reasoning. STEP 3 — CARRY OUT THE PLAN. Execute the chosen strategy, showing each step clearly. Track units throughout. If the plan does not work, go back to Step 2 and choose a different strategy. Persistence and flexibility are key. STEP 4 — LOOK BACK. Verify the answer against the original conditions. Check: Is the answer in the correct units? Is the value numerically reasonable? Does it satisfy ALL conditions stated in the problem? Ask: 'Is there another way to solve this?' The LET frequently presents multi-step NON-ROUTINE problems — problems that cannot be solved by a single remembered formula — where Polya's framework is the only reliable guide. The Board also tests whether candidates can correctly identify the step being described and match a strategy to a problem type.

Tiling, Fencing, and Filling: Perimeter, Area, and Volume

A large category of LET measurement word problems involves practical contexts — fencing a yard, tiling a floor, painting a wall, filling a container — where the primary skill is correctly identifying whether to compute PERIMETER, AREA, or VOLUME, and then applying the correct formula with correct units. PERIMETER (P) is the total length around a figure — a one-dimensional linear measure. It is used whenever a problem involves ENCLOSING or BORDERING something: fencing a lot, putting a frame around a picture, lining the edges of a garden with stones, or calculating the length of wire needed around a boundary. Perimeter is measured in LINEAR UNITS (cm, m, ft). Key formulas: Rectangle P = 2(l + w); Square P = 4s; Triangle P = a + b + c. AREA (A) is the measure of the surface enclosed by a figure — a two-dimensional measure. It is used when COVERING a surface: tiling a floor, painting a wall, carpeting a room, or buying fabric. Area is measured in SQUARE UNITS (cm², m², ft²). Key formulas: Rectangle A = l × w; Square A = s²; Triangle A = (1/2)bh; Circle A = πr². VOLUME (V) is the measure of three-dimensional space — used when FILLING a container: finding how many liters of water a tank holds, how much soil to fill a garden bed, or how much air is in a room. Volume is measured in CUBIC UNITS (cm³, m³) or capacity units (mL, L). Key formulas: Rectangular prism V = l × w × h; Cube V = s³; Cylinder V = πr²h. The critical connection is 1 mL = 1 cm³ — so a box 10 cm × 5 cm × 4 cm holds 200 cm³ = 200 mL of liquid. The most common LET trap is confusing perimeter and area: fencing a rectangular lot uses PERIMETER (not area), but the COST of concrete tiles to cover the lot uses AREA. Reading the problem carefully is the key.

Use Amount = Rate × Time, rearranged as Time = Amount ÷ Rate. Units: liters ÷ (liters/min) = minutes. Reasonableness: 15 × 12 = 180 ✓.

Or directly: 1 km = 100,000 cm, so 5.4 × 100,000 = 540,000 cm. Moving 5 steps down the metric ladder (km to cm) means multiplying by 10⁵ = 100,000.

Always add rates, never times. LCM of 8 and 12 is 24: 3/24 + 2/24 = 5/24. Reciprocal = 24/5 = 4.8 hours. Convert to hours and minutes: 0.8 × 60 = 48 minutes. So 4 hours 48 minutes.

Step 1: Multiply 18 by 9/5: 18 × 9 = 162; 162 ÷ 5 = 32.4. Step 2: Add 32: 32.4 + 32 = 64.4°F. This is a cool temperature for the Philippines — 64.4°F is consistent with Baguio's climate. Reasonableness check passed.

This three-part problem tests whether you correctly switch between perimeter (for fencing) and area (for tiling). Fencing uses the PERIMETER (linear measure); tiling uses AREA (square measure). The costs are dramatically different, illustrating why the distinction matters.

When two objects move toward each other, add their speeds (relative speed). Time = Distance ÷ Combined Speed = 390 ÷ 130 = 3 hours. Look back: In 3 hours, bus covers 60 × 3 = 180 km; car covers 70 × 3 = 210 km. 180 + 210 = 390 km ✓.

Volume = length × width × height. Use 1 mL = 1 cm³ and 1 L = 1,000 mL. 60,000 mL ÷ 1,000 = 60 L. This is approximately the capacity of a small household water container — a reasonable answer.

Strategy: Work backward (Polya Step 2). Reverse each operation: the final addition (received ₱200) becomes subtraction going backward; the subtractions (spent, gave) become additions. Check: 700 − 350 − 120 + 200 = 430 ✓.

Use the shortcut: km/h ÷ 3.6 = m/s. Alternatively, convert distance (km to m: × 1,000) and time (hours to seconds: × 3,600) separately, then divide. Both methods yield 30 m/s.

Convert each part of the mixed English weight separately using the given conversion factors, then add. Alternatively: 11 lb 8 oz = 11.5 lb (since 8 oz = 0.5 lb), and 11.5 × 453.6 = 5,216.4 g. Both approaches give the same answer.