Fractions, Decimals, Percent, Ratio and Proportion - Revision Notes
This chapter covers the numerical core of the LET Mathematics test. Fractions, decimals, percent, ratio, and proportion are five ways of expressing the same idea: a part compared with a whole. In the Philippine classroom, these concepts appear in everyday transactions — jeepney fares, school canteen prices, DepEd budget allocations, and pupil score interpretations. As a future elementary teacher, you must be able to solve these problems quickly and accurately, AND explain the reasoning clearly to Grades 1–6 pupils. The Board exam tests both computational skill and word-problem translation. Mastering the conversions among the four forms, the three percentage cases, and the cross-multiplication rule will allow you to answer the majority of LET Mathematics items with speed and confidence.
Operations on Fractions
Formulas
a/b + c/b = (a+c)/b
Example
3/7 + 2/7 = 5/7
Variables
a, c = numerators; b = common denominator
Application
Adding or subtracting fractions with the SAME denominator
a/b + c/d = (ad + bc) / bd [then simplify]
Example
2/3 + 1/4: LCD=12; 8/12 + 3/12 = 11/12
Variables
a/b and c/d are fractions with DIFFERENT denominators
Application
Adding fractions with different denominators using the LCD method
a/b × c/d = ac / bd
Example
3/4 × 8/9 = 24/36 = 2/3 (cancel first: 3/4 × 8/9 → 1/4 × 8/3 → 2/3)
Variables
a/b and c/d are any two fractions
Application
Multiplying two fractions
a/b ÷ c/d = a/b × d/c = ad / bc
Example
3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10
Variables
a/b = dividend; c/d = divisor; d/c = reciprocal of divisor
Application
Dividing one fraction by another
Mixed number to improper fraction: (W × D + N) / D
Example
2 1/3 = (2×3 + 1)/3 = 7/3
Variables
W = whole number part; D = denominator; N = numerator
Application
Converting before multiplication or division of mixed numbers
Exam tips
- LET word problems often phrase fraction operations using Filipino contexts (allowance, cloth, rice). Identify the operation first before computing.
- Cancel common factors BEFORE multiplying to keep numbers small and avoid errors.
- For LCD with large denominators, use prime factorization — it is faster and more reliable.
- Check reasonableness: if you added two positive fractions and got something smaller than either, you made an error.
- Mixed-number addition/subtraction can be done by keeping whole and fractional parts separate — no need to convert if adding only.
Key points
- A fraction a/b represents a part of a whole; 'a' is the numerator (part) and 'b' is the denominator (whole).
- A fraction is in lowest terms when GCF(numerator, denominator) = 1.
- Adding and subtracting fractions REQUIRES a common denominator (LCD).
- Multiplying fractions: multiply numerators together AND denominators together, then simplify.
- Dividing fractions: multiply the dividend by the RECIPROCAL of the divisor (KFC — Keep, Flip, Change).
- Mixed numbers must be converted to improper fractions before multiplying or dividing.
- To convert a mixed number to an improper fraction: (whole × denominator + numerator) / denominator.
- Always simplify your final answer to lowest terms.
Definitions
Proper Fraction
A fraction where the numerator is less than the denominator (e.g., 3/5).
Importance
Value is always between 0 and 1; appears in part-of-a-whole problems.
Improper Fraction
A fraction where the numerator is greater than or equal to the denominator (e.g., 7/3).
Importance
Needed before performing multiplication or division on mixed numbers.
Mixed Number
A whole number combined with a proper fraction (e.g., 2 1/3).
Importance
Final answers in word problems are usually expressed as mixed numbers.
Least Common Denominator (LCD)
The smallest common multiple of two or more denominators.
Importance
Required for adding and subtracting fractions with unlike denominators.
Reciprocal
The fraction obtained by flipping numerator and denominator (e.g., reciprocal of 2/7 is 7/2).
Importance
Division of fractions is performed by multiplying by the reciprocal of the divisor.
Common mistakes
- Adding or subtracting fractions without finding the LCD first (e.g., wrongly writing 1/2 + 1/3 = 2/5).
- Forgetting to convert mixed numbers to improper fractions before multiplying or dividing.
- Not simplifying the final answer to lowest terms.
- When dividing, flipping the WRONG fraction — always flip the DIVISOR (the second fraction), not the dividend.
- Canceling across an addition or subtraction sign (canceling is only valid in multiplication).
Operations on Decimals
Formulas
Product decimal places = decimal places in Factor 1 + decimal places in Factor 2
Example
0.6 × 0.4: 6×4=24; 1 place + 1 place = 2 places → 0.24
Variables
Count the digits to the right of the decimal point in each factor
Application
Placing the decimal point correctly after multiplying
a.bc ÷ 0.d → move point: ab.c ÷ d
Example
4.8 ÷ 0.6 → 48 ÷ 6 = 8
Variables
Move both decimal points the same number of places to eliminate the divisor's decimal
Application
Simplifying decimal division
Exam tips
- Always write a zero before the decimal point for clarity: write 0.25, not .25.
- When the LET gives peso amounts, set up as decimals and align carefully.
- Double-check your decimal multiplication by estimating: 0.6 × 0.4 ≈ 0.5 × 0.4 = 0.2, confirming 0.24 is reasonable.
- Memorize common decimal-fraction equivalents (see Conversions section) to speed up computation.
Key points
- Adding and subtracting decimals: ALIGN the decimal points, then operate as with whole numbers.
- Multiplying decimals: ignore the points, multiply as whole numbers, then count decimal places.
- The number of decimal places in the PRODUCT equals the SUM of the decimal places in both factors.
- Dividing decimals: move the divisor's decimal point to make it a whole number; move the dividend's point the same number of places.
- Zeros may be annexed to the right of a decimal without changing its value (0.5 = 0.50 = 0.500).
- Comparing decimals: compare digit by digit from left to right after aligning decimal points.
Definitions
Decimal Point
A dot separating the whole-number part from the fractional part of a number.
Importance
Correct placement is essential; misplacing it by one position changes the value by a factor of 10.
Terminating Decimal
A decimal that ends after a finite number of digits (e.g., 0.25, 0.75).
Importance
Results from fractions whose denominators have only factors of 2 and/or 5.
Repeating Decimal
A decimal with a digit or group of digits that repeats infinitely (e.g., 1/3 = 0.333...).
Importance
Fractions like 1/3 and 1/6 produce repeating decimals; use the fraction form for exact computation.
Common mistakes
- Failing to align decimal points when adding or subtracting (e.g., 2.5 + 0.35 written as 2.5 + 0.35 misaligned giving 2.85 correct but 2.6 if aligned wrongly).
- Forgetting to count decimal places correctly in multiplication.
- Moving only one decimal point when dividing instead of moving BOTH.
- Dropping a zero placeholder (e.g., 0.06 × 0.3: 6×3=18, 3 decimal places → 0.018, NOT 0.18).
Converting Among Fractions, Decimals, and Percent
Formulas
Fraction → Decimal: a/b = a ÷ b
Example
3/8 = 3 ÷ 8 = 0.375
Variables
a = numerator, b = denominator
Application
Convert any fraction to its decimal equivalent
Decimal → Percent: d × 100 = P%
Example
0.375 × 100 = 37.5%
Variables
d = decimal value, P = percent value
Application
Express a decimal as a percent
Percent → Decimal: P% ÷ 100 = d
Example
37.5% ÷ 100 = 0.375
Variables
P = percent value, d = decimal value
Application
Convert percent to decimal for calculations
Percent → Fraction: P% = P/100 (then simplify)
Example
25% = 25/100 = 1/4
Variables
P = percent value
Application
Express a percent as a fraction in lowest terms
Exam tips
- Write the conversion rules on your scratch paper at the start of the exam: ×100 (decimal→%), ÷100 (%→decimal).
- LET often mixes forms in one problem (e.g., '55%, 3/5, and 0.6 — arrange from least to greatest'). Convert ALL to decimals first.
- For 'of' problems, the moment you see 'of', write a multiplication sign.
Key points
- Percent means 'per hundred' — it is a fraction with denominator 100.
- Fraction to Decimal: DIVIDE the numerator by the denominator.
- Decimal to Percent: MULTIPLY by 100 (move decimal point TWO places to the RIGHT).
- Percent to Decimal: DIVIDE by 100 (move decimal point TWO places to the LEFT).
- Percent to Fraction: Write the percent over 100, then SIMPLIFY.
- Fraction to Percent: First convert to decimal, then multiply by 100.
- Memorize the table of benchmark fractions, decimals, and percents for speed.
- The word 'OF' in problems means MULTIPLY (e.g., 3/4 of 20 = 3/4 × 20 = 15).
Definitions
Percent (%)
A ratio that compares a number to 100. The symbol % means 'divided by 100' or 'per hundred'.
Importance
All three percentage cases (find part, rate, base) depend on fluently converting percent to decimal.
Benchmark Fractions
Commonly used fractions with known decimal and percent equivalents: 1/2=0.5=50%, 1/4=0.25=25%, 3/4=0.75=75%, 1/3≈0.333=33⅓%, 1/5=0.2=20%, 1/8=0.125=12.5%, 1/10=0.1=10%.
Importance
Memorizing these eliminates computation steps and prevents errors under exam time pressure.
Common mistakes
- Moving the decimal in the WRONG direction when converting (e.g., writing 25% as 25.0 instead of 0.25).
- Confusing 'multiply by 100' and 'divide by 100' for the two conversion directions.
- Not simplifying percent-to-fraction conversions (leaving 20/100 instead of 1/5).
- Treating 33% as exactly 1/3 — the exact value is 33 1/3% or 0.3333...; using 33% introduces a rounding error.
The Three Percentage Cases
Formulas
Part = Rate × Base
Example
What is 20% of 450? Part = 0.20 × 450 = 90
Variables
Rate = percent expressed as decimal; Base = original whole; Part = resulting amount
Application
Case 1: Finding what a percent of a number is
Rate = Part ÷ Base (then × 100 for %)
Example
40 is what % of 160? Rate = 40 ÷ 160 = 0.25 = 25%
Variables
Part = given amount; Base = original whole
Application
Case 2: Finding what percent one number is of another
Base = Part ÷ Rate
Example
15 is 25% of what? Base = 15 ÷ 0.25 = 60
Variables
Part = given amount; Rate = percent as decimal
Application
Case 3: Finding the original whole when a part and its percent are known
Percent Change = (New − Original) ÷ Original × 100
Example
Score rose from 40 to 50: (50−40)÷40 × 100 = 25% increase
Variables
New = changed value; Original = starting value (the base)
Application
Finding percent increase (positive result) or percent decrease (negative result)
Successive Percent: Final = Original × (1 ± r1) × (1 ± r2)
Example
₱1,000 with 10% off, then 20% off: 1000 × 0.90 × 0.80 = ₱720 (28% total discount, NOT 30%)
Variables
r1, r2 = rates as decimals; + for increase, − for decrease
Application
Applying two percent changes in sequence
Discount Price = Original × (1 − discount rate)
Example
20% off ₱450: 450 × 0.80 = ₱360
Variables
discount rate as decimal
Application
Shortcut for finding sale price
Total with VAT = Price × 1.12
Example
₱500 meal + 12% VAT: 500 × 1.12 = ₱560
Variables
12% = Philippine VAT rate
Application
Finding the total bill including 12% VAT
Exam tips
- Memorize the three formulas as a triangle: Part on top, Rate and Base on the bottom. Cover the unknown to see the operation.
- The word 'is' signals the PART; the word 'of' signals the BASE; the percent sign signals the RATE.
- For VAT problems: 12% VAT is added, so multiply by 1.12. For a discount: subtract the percent, so multiply by (1 − rate).
- Percent increase trap: 'A salary was increased by 20% to ₱12,000. What was the original salary?' → Base = 12,000 ÷ 1.20 = ₱10,000.
- Successive discounts: NEVER add the percentages. Always compute step by step on the running total.
Key points
- Every percentage problem fits one of THREE cases based on the formula: Rate × Base = Part (Percentage).
- CASE 1 — Find the Part (Percentage): Part = Rate × Base. The rate and base are given.
- CASE 2 — Find the Rate: Rate = Part ÷ Base. Express as a percent by multiplying by 100.
- CASE 3 — Find the Base: Base = Part ÷ Rate. The part and rate are given.
- The BASE is always the original whole (100%) — the number that follows 'of' or 'out of'.
- The RATE is the percent (convert to decimal before computing).
- The PART is the result — the portion of the base indicated by the rate.
- Percent increase/decrease: (change ÷ original) × 100. The ORIGINAL is always the base.
- Successive percents act on DIFFERENT bases — you cannot simply add the two percents.
Definitions
Base
The original whole amount — the number that represents 100%. It follows the word 'of' or 'out of' in a problem.
Importance
Identifying the base correctly is the most critical step; using the wrong base is the #1 source of errors.
Rate
The percent value, always converted to a decimal before computing.
Importance
Forgetting to convert percent to decimal (e.g., using 20 instead of 0.20) gives an answer 100× too large.
Part (Percentage)
The portion of the base indicated by the rate; the result of applying the percent to the base.
Importance
The 'part' is what you find in Case 1; it is given in Cases 2 and 3.
Percent Increase
A positive percent change; the new value is greater than the original.
Importance
The original (smaller) value is the base — a common exam trap uses the new value as the base instead.
Successive Percents
Two or more percent changes applied in sequence, each acting on the RESULT of the previous step.
Importance
The two rates CANNOT be simply added because they act on different bases — a classic LET trap.
Common mistakes
- Using the NEW value instead of the ORIGINAL as the base in percent-change problems.
- Adding successive percents (e.g., 10% + 20% = 30% discount) instead of computing step by step.
- Forgetting to convert the percent to a decimal before multiplying (using 25 instead of 0.25).
- Confusing which case applies — draw a ratio box: identify which of Part/Rate/Base is missing.
- In Case 3, dividing the base by the part instead of the part by the rate.
Ratio and Proportion
Formulas
Cross-multiplication: a/b = c/d → a×d = b×c
Example
2/6 = x/15 → 6x = 30 → x = 5 cups of rice for 15 servings
Variables
a, b, c, d are the four terms of a proportion; one is usually unknown (x)
Application
Solving any proportion for the missing term
Inverse proportion: a₁ × b₁ = a₂ × b₂ (product is constant)
Example
4 workers × 6 days = 3 workers × d → d = 24/3 = 8 days
Variables
a = first quantity (e.g., workers), b = second quantity (e.g., days)
Application
Problems where more of one quantity means less of the other
Partitive proportion: each part = total ÷ sum of ratio terms, then × ratio number
Example
Share ₱1,500 in ratio 2:3:5 → sum=10; one part=₱150; shares: ₱300, ₱450, ₱750
Variables
total = the whole amount; ratio terms = numbers in the given ratio
Application
Dividing a total into parts according to a given ratio
Map/Scale: Drawing length / Actual length = Scale
Example
Scale 1:50,000; map distance 4 cm → actual = 4 × 50,000 = 200,000 cm = 2 km
Variables
Scale is given as 1:n (1 unit on the map = n units in reality)
Application
Finding actual distances from map distances or vice versa
Exam tips
- ALWAYS check: does more of A mean more of B (direct) or less of B (inverse)? The answer determines which method to use.
- For map scales, always convert to the SAME UNIT before computing, then convert back to the required unit at the end.
- In partitive proportion, ALWAYS verify: the shares must add up to the total. Use this as your check.
- Set up proportions in the SAME ORDER: if the left side is 'servings/cups', the right side must also be 'servings/cups'.
- If a ratio has THREE terms (e.g., 2:3:5), the sum of parts = 2+3+5 = 10, not 3.
Key points
- A RATIO compares two quantities of the same kind; written as a:b or a/b.
- A ratio must always be simplified to LOWEST TERMS (12:18 = 2:3).
- Ratios must compare the SAME UNITS — convert before writing the ratio.
- A PROPORTION states that two ratios are equal: a/b = c/d.
- Cross-multiplication: if a/b = c/d, then a×d = b×c (the cross-products are equal).
- DIRECT proportion: as one quantity increases, the other also increases (same-direction change).
- INVERSE proportion: as one quantity increases, the other decreases (opposite-direction change); the PRODUCT is constant.
- PARTITIVE proportion: a whole is divided into parts according to a given ratio.
- Scale drawings and maps are proportions: scale = drawing length : actual length.
- The key cue word 'OF' means multiply; set up ratios in the SAME ORDER on both sides.
Definitions
Ratio
A comparison of two quantities of the same kind, expressed as a:b or a/b.
Importance
The foundation of proportion; must be in lowest terms and same units.
Proportion
A mathematical statement that two ratios are equal (a:b = c:d or a/b = c/d).
Importance
Solved by cross-multiplication; underlies direct, inverse, and partitive proportion types.
Direct Proportion
A relationship where two quantities increase or decrease together at the same rate. Ratio of corresponding values is constant.
Importance
Used in recipe scaling, material estimation, speed-distance problems at constant speed.
Inverse Proportion
A relationship where one quantity increases as the other decreases; their PRODUCT is constant.
Importance
Used in worker-time problems; the key distinguisher from direct proportion.
Partitive Proportion
Dividing a whole into parts according to a specified ratio.
Importance
Common in sharing problems (allowance, inheritance, mixture ingredients).
Scale (Map)
A ratio comparing a length on a drawing/map to the corresponding actual length.
Importance
Scale problems are proportions in disguise; cross-multiplication solves them directly.
Common mistakes
- Setting up the ratios in DIFFERENT orders on the two sides of the proportion (e.g., putting 'old:new' on the left and 'new:old' on the right).
- Treating an inverse proportion as direct proportion (setting up ratios instead of products).
- Comparing quantities in DIFFERENT UNITS without converting first (e.g., 50 cm to 2 m without converting 2 m to 200 cm first).
- In partitive proportion, forgetting to divide the total by the SUM of ratio terms.
- Leaving a ratio in non-simplified form (e.g., 12:18 instead of 2:3).
Rate and Mixture Applications
Formulas
D = R × T
Example
A bus travels at 60 km/h for 2.5 hours: D = 60 × 2.5 = 150 km
Variables
D = distance (km or m); R = rate/speed (km/h); T = time (hours)
Application
Solving any speed-distance-time problem
Unit Rate = Total Cost ÷ Total Quantity
Example
500g pack ₱75: ₱75÷500 = ₱0.15/g vs 750g pack ₱108: ₱108÷750 = ₱0.144/g → 750g is better
Variables
Total Cost in pesos; Total Quantity in units (kg, L, pcs)
Application
Comparing prices to find the best buy
Average Speed = 2 × r1 × r2 / (r1 + r2)
Example
Going at 40 km/h, returning at 60 km/h: Avg = 2×40×60/(40+60) = 4800/100 = 48 km/h
Variables
r1 and r2 are the two different speeds for equal distances
Application
Finding average speed when the same distance is covered twice at different speeds
Exam tips
- Memorize D = R × T as a triangle: cover the unknown (D, R, or T) to see the formula.
- For best-buy problems, ALWAYS state your answer with units: 'the 750g pack at ₱0.144/g is the better buy'.
- Average speed trap: if you travel equal DISTANCES (not equal times), you CANNOT average the speeds arithmetically.
- In time problems: convert minutes to hours before using the formula (30 minutes = 0.5 hours).
Key points
- A RATE compares quantities of DIFFERENT kinds (km per hour, pesos per kilo, pupils per teacher).
- A UNIT RATE has a denominator of 1; find it by dividing (₱165 for 3 kg → ₱55/kg).
- For best-buy comparisons, find the UNIT RATE for each option; the smaller unit cost is the better buy.
- Speed problems use the formula: Distance = Rate × Time (D = R × T).
- Rearrangements: R = D ÷ T and T = D ÷ R.
- For mixture problems, find the unit rate (cost per unit) and compare or blend proportionally.
- Average speed (when same distance is covered at two different speeds): Average Speed = 2 × r1 × r2 / (r1 + r2).
Definitions
Rate
A ratio that compares two quantities measured in DIFFERENT units (e.g., km per hour, peso per kilogram).
Importance
Distinguishes from ratio (same units); unit rates enable direct comparison.
Unit Rate
A rate with a denominator of 1 (e.g., ₱55 per kg, 80 km per hour).
Importance
Standard form for comparing rates; divide total by quantity to find it.
Best Buy
The option with the LOWEST unit cost for the same quality product.
Importance
Common in LET problems comparing market prices; always compute unit rates before comparing.
Common mistakes
- Using the TOTAL distance instead of the distance for each leg in average-speed problems.
- Simply averaging two speeds arithmetically (e.g., (40+60)/2 = 50) when the harmonic mean formula is needed for equal distances.
- Comparing total prices instead of unit prices when packages have different sizes.
- Confusing RATE with RATIO — rate compares different units; ratio compares same units.
Comparing and Ordering Fractions and Decimals
Formulas
Cross-multiply comparison: a/b vs c/d → compare a×d with b×c
Example
Compare 3/8 vs 5/12: 3×12=36, 5×8=40; since 40>36, 5/12 > 3/8
Variables
a/b and c/d are the two fractions being compared
Application
Quickly comparing two fractions without finding LCD
Exam tips
- For sets of 3+ fractions, convert all to decimals — it is faster than finding the LCD of multiple denominators.
- Use benchmark 1/2 = 0.5: fractions below 0.5 are less than half; above 0.5 are greater than half. Use this to eliminate wrong options.
- In ordering problems (least to greatest), write the decimal equivalents on your scratch paper, order them, then match back to the original form.
Key points
- Three reliable methods to compare fractions: (1) convert to DECIMAL, (2) rewrite with a COMMON DENOMINATOR, (3) CROSS-MULTIPLY.
- Cross-multiply to compare a/b and c/d: compute a×d and b×c; the LARGER product is over the LARGER fraction.
- When ordering a mix of fractions, decimals, and percents: convert ALL to DECIMALS first.
- Use benchmark values (0, 1/2, 1) for quick checks — is the fraction closer to 0, to 1/2, or to 1?
- Denominators alone do NOT determine size: 1/8 < 1/3 even though 8 > 3.
Definitions
Cross-multiplication (for comparison)
Multiplying the numerator of each fraction by the denominator of the other to compare sizes.
Importance
Faster than finding LCD when comparing exactly two fractions; the larger cross-product is associated with the larger fraction.
Common mistakes
- Comparing fractions by numerators alone or denominators alone (1/3 vs 1/4 — students wrongly say 1/4 is bigger because 4>3).
- In cross-multiplication for comparison, associating the larger product with the WRONG fraction (the product b×c goes with a/b, and a×d goes with c/d).
- Not converting all values to the same form before ordering mixed sets.
Connections
- Fractions, decimals, and percents are the same relationship expressed in three different forms — converting fluently among them is the master skill that unlocks all other topics in this chapter.
- The three percentage cases (find part, find rate, find base) all derive from the single formula Rate × Base = Part; knowing which quantity is missing determines which operation to use.
- Ratio and proportion are extensions of fraction equality: a proportion a/b = c/d is simply two equivalent fractions, and cross-multiplication is the same operation used to add unlike fractions.
- Direct proportion is related to multiplication of fractions: 'more servings need proportionally more rice' reflects the linear relationship y = kx, where k is the unit rate.
- Inverse proportion connects to the concept of equivalent products (a₁b₁ = a₂b₂), which is the same as the cross-products of a proportion rearranged.
- Percent change problems and partitive proportion both require identifying the BASE correctly — misidentifying the base is the common error in both topic areas.
- Rate (distance/time, price/unit) is a specialized ratio comparing different units; the unit rate concept bridges ratio-proportion and the real-world applications section.
- Successive percent changes use repeated multiplication of decimals, directly applying decimal multiplication skills from the earlier section.
- Comparing fractions using cross-multiplication is the same operation as solving a proportion — recognizing this connection reduces the number of formulas to memorize.
- In the K–12 BEC Mathematics curriculum, these concepts are taught progressively from Grade 3 (fractions) through Grade 6 (ratio, proportion, percent), reflecting their hierarchical relationship.
Quick review questions
A student spent 1/3 of his allowance on food and 1/4 on transportation. What fraction of his allowance was left?
Answer
5/12
Find the total spent: 1/3 + 1/4. LCD = 12. Rewrite: 4/12 + 3/12 = 7/12 spent. Left = 1 − 7/12 = 12/12 − 7/12 = 5/12.
A shirt priced at ₱450 is on sale at 20% off. What is the sale price?
Answer
₱360
A 20% discount means you pay 80% of the price. Sale price = 0.80 × 450 = ₱360. OR: Discount = 0.20 × 450 = ₱90; Sale price = 450 − 90 = ₱360.
A class passed 45 students, which was 90% of those who took the test. How many students took the test?
Answer
50 students
Case 3 — Find the Base. Base = Part ÷ Rate = 45 ÷ 0.90 = 50 students.
A recipe for 6 servings of arroz caldo requires 2 cups of rice. How many cups are needed for 15 servings?
Answer
5 cups
Direct proportion. Set up: 2/6 = x/15. Cross-multiply: 6x = 2 × 15 = 30. x = 30 ÷ 6 = 5 cups.
Four workers can finish painting a classroom in 6 days. How long will 3 workers take at the same pace?
Answer
8 days
Inverse proportion: workers × days = constant. 4 × 6 = 3 × d. 24 = 3d. d = 8 days. Fewer workers means more days — answer is reasonable.
Divide ₱1,500 among three children in the ratio 2:3:5. How much does each child receive?
Answer
₱300, ₱450, and ₱750
Partitive proportion. Sum of parts = 2+3+5 = 10. One part = ₱1,500 ÷ 10 = ₱150. Child 1: 2×₱150 = ₱300; Child 2: 3×₱150 = ₱450; Child 3: 5×₱150 = ₱750. Check: 300+450+750 = ₱1,500 ✓
An appliance priced at ₱1,000 receives two successive discounts: 10% off, then 20% off the reduced price. What is the final price and the overall percent discount?
Answer
Final price: ₱720; Overall discount: 28%
First discount: 1,000 × 0.90 = ₱900. Second discount: 900 × 0.80 = ₱720. Overall discount = (1,000 − 720) ÷ 1,000 × 100 = 280/1,000 × 100 = 28%. NOT 30% — the two percents act on different bases.
A 500g pack of rice costs ₱75 and a 750g pack costs ₱108. Which is the better buy?
Answer
The 750g pack
Unit rate for 500g: ₱75 ÷ 500 = ₱0.150/g. Unit rate for 750g: ₱108 ÷ 750 = ₱0.144/g. Since ₱0.144 < ₱0.150, the 750g pack is cheaper per gram and is the better buy.
On a map with scale 1:50,000, two barangay halls are 4 cm apart. What is the actual distance in kilometers?
Answer
2 km
Actual distance = 4 × 50,000 = 200,000 cm. Convert to km: 200,000 cm ÷ 100,000 cm/km = 2 km.
What is 40 expressed as a percent of 160?
Answer
25%
Case 2 — Find the Rate. Rate = Part ÷ Base = 40 ÷ 160 = 0.25. Convert to percent: 0.25 × 100 = 25%.
Compare 3/8 and 5/12. Which is greater?
Answer
5/12 is greater
Cross-multiply: 3×12 = 36 and 5×8 = 40. Since 40 > 36, and 40 corresponds to 5/12, therefore 5/12 > 3/8.
A dressmaker uses 2 1/3 meters of cloth for a uniform and 1 1/2 meters for a skirt. How much cloth is used in total?
Answer
3 5/6 meters
Convert: 2 1/3 = 7/3 and 1 1/2 = 3/2. LCD = 6. Add: 14/6 + 9/6 = 23/6 = 3 5/6 meters.
Diagrams
Chapter Overview: Fractions, Decimals, Percent, Ratio and Proportion
A comprehensive mind map showing all major topics and sub-topics in this chapter, helping LET reviewees see how fractions, decimals, percent, ratio, and proportion interconnect before diving into details.
Decision Tree: Identifying the Three Percentage Cases
A decision-tree flowchart guiding LET examinees to identify which of the three percentage cases applies and showing the correct formula and a worked example for each case, with a verification step.
Step-by-Step: Solving Proportion Problems (Direct, Inverse, Partitive)
A flowchart helping LET examinees distinguish between the three types of proportion problems, set up the correct equation for each, and verify their answer — the most common source of errors in this topic.
Converting Among Fractions, Decimals, and Percent
A triangular flowchart showing all six conversion paths among fractions, decimals, and percents with the rule for each direction, serving as a quick-reference visual for LET examinees.
Distribution of Percentage Case Types in LET-Style Problems
A pie chart showing the approximate distribution of percentage question types encountered in LET-style Mathematics examinations, helping examinees prioritize their study time and practice efforts.
4-Step Problem-Solving Routine for LET Word Problems
A comprehensive step-by-step flowchart of the recommended 4-step problem-solving routine for LET Mathematics word problems, covering all major sub-topics of this chapter and emphasizing verification.
Exam strategy
Start the LET Mathematics section by quickly scanning all items and tagging them by type: fraction operation, percentage case, proportion type, or rate problem. This prevents spending too long on any one item. For every word problem, follow this 4-step routine: (1) READ — identify the given quantities and what is asked; (2) TRANSLATE — convert 'of' to ×, identify Part/Rate/Base or set up the proportion; (3) COMPUTE — apply the formula, converting percent to decimal first; (4) CHECK — verify the answer is reasonable in context (a discounted price is lower; more workers means fewer days). Memorize the benchmark fraction-decimal-percent equivalents (1/2, 1/3, 1/4, 1/5, 1/8, 1/10) on scratch paper at the start of the exam to speed up conversions. For proportion problems, always set up ratios in the SAME ORDER on both sides before cross-multiplying. For successive percent problems, NEVER add the percents — always compute step by step. When a problem mixes fractions, decimals, and percents, convert everything to decimals first. Budget approximately 1–1.5 minutes per item; if an item requires more than two computation steps, re-read to ensure you have not missed a simpler approach.