Study Guides
Numerical Ability: Formula & Concept Reference
Every major Numerical Ability formula, each with a real worked example. Use this alongside practice questions — seeing the formula, then applying it, is how it actually sticks.
1. Number Problem
"A less than B" means → B − A
"A more than B" means → B + A
Digit problems
abc = 100a + 10b + c
sum of digits = a + b + c
Worked example — Word-rule example
A number is 8 less than twice another number. Their sum is 40. Find the numbers.
Explanation: "X less than Y" = Y − X, so the number = 2(other number) − 8.
- • x = 2y − 8
- • x + y = 40
- • (2y − 8) + y = 40 → 3y = 48
Answer: y = 16, x = 24
Check: 2(16) − 8 = 24 ✓, and 24 + 16 = 40 ✓
Worked example — Two-digit example
The sum of the digits of a two-digit number is 12. When the digits are reversed, the new number is 18 less than the original. Find the number.
Explanation: Let a = tens digit, b = units digit. Original = 10a + b, reversed = 10b + a.
- • a + b = 12
- • 10b + a = (10a + b) − 18 → a − b = 2
Answer: a = 7, b = 5, number = 75
Check: reversed = 57, and 75 − 57 = 18 ✓
Worked example — Three-digit example
In a three-digit number, the hundreds digit is twice the units digit, and the tens digit is 3 more than the units digit. The sum of the digits is 11. Find the number.
Explanation: Let c = units digit. Then a = 2c and b = c + 3.
- • a + b + c = 11 → 2c + (c + 3) + c = 11 → 4c = 8
Answer: c = 2, a = 4, b = 5, number = 452
Check: 4 + 5 + 2 = 11 ✓
2. Money Problem
P = SP − C
Discount
NSP = SP(1 − D)
NP = NSP − C
Raise
NSP = SP(1 + R)
NP = NSP − C
- P = profit
- SP = selling price
- C = cost
- NSP = new selling price
- NP = new profit
- D = discount
- R = raise
Worked example — Discount
A boutique owner spends ₱450 to produce a dress and sets the original selling price at ₱750. During a flash sale, she offers a 20% discount. Find the new selling price and new profit.
- • NSP = 750(1 − 0.20) = ₱600
- • NP = 600 − 450 = ₱150
Answer: ₱150
Check: original profit was 750 − 450 = ₱300, so the discount cut profit in half
Worked example — Raise
A hardware store buys a tool for ₱200 and originally sells it for ₱280. The owner raises the price by 15%. Find the new selling price and new profit.
- • NSP = 280(1 + 0.15) = ₱322
- • NP = 322 − 200 = ₱122
Answer: ₱122
Check: original profit was 280 − 200 = ₱80, so the raise added ₱42
3. Motion Problem
Object to object relationship
same direction → subtract
opposite direction → add
Object to environment relationship
same direction → add
opposite direction → subtract
Worked example — Object to object — chase problem
Train A is 150 km ahead of Train B on the same track, both moving in the same direction. Train B travels at 90 km/h, Train A at 60 km/h. How long will it take Train B to catch up?
Explanation: Same direction → subtract the speeds to get relative speed. Train B must be the faster one to actually close the gap.
- • relative speed = 90 − 60 = 30 km/h
- • time = 150 ÷ 30
Answer: 5 hours
Worked example — Object to environment — river current
A boat's speed in still water is 20 km/h, on a river with a 5 km/h current. Find downstream and upstream speed, then the total time for a 60 km round trip.
Explanation: Same direction (downstream) → add. Opposite direction (upstream) → subtract.
- • downstream = 20 + 5 = 25 km/h → time = 60 ÷ 25 = 2.4 h
- • upstream = 20 − 5 = 15 km/h → time = 60 ÷ 15 = 4 h
Answer: 6.4 hours total
4. Age Problem
| Past (m) | Present | Future (n) | |
|---|---|---|---|
| Boy | x − m | x | x + n |
| Girl | y − m | y | y + n |
Worked example — Past to present
Six years ago, a father was four times as old as his son. If the father is now 34, how old is the son now?
Explanation: x = father's present age = 34. Past column: x − m and y − m, with m = 6.
- • x − 6 = 4(y − 6)
- • 28 = 4y − 24 → 4y = 52
Answer: y = 13
Check: 6 years ago: father 28, son 7, and 28 = 4 × 7 ✓
Worked example — Present to future
A mother is currently three times as old as her daughter. In 10 years, she will be twice as old as her daughter. Find their present ages.
Explanation: Present: x = 3y. Future column with n = 10: x + n = 2(y + n).
- • 3y + 10 = 2(y + 10)
- • 3y + 10 = 2y + 20
Answer: y = 10, x = 30
Check: in 10 years: mother 40, daughter 20, and 40 = 2 × 20 ✓
5. Percentage Problem
Think of percentage problems as:
PART = RATE × WHOLE (P = R × B)
- P = Part
- R = Rate (decimal)
- B = Base / Whole
Worked example — Solving for Part
A store has 240 items in stock. If 35% are sold in one day, how many are sold?
- • P = 0.35 × 240
Answer: 84 items
Worked example — Solving for Base
45 students represent 15% of all students in a school. How many students in total?
- • 45 = 0.15 × B
- • B = 45 ÷ 0.15
Answer: 300 students
Worked example — Solving for Rate
Out of 60 questions, a student answered 18 correctly. What percent is that?
- • 18 = R × 60
- • R = 18 ÷ 60 = 0.30
Answer: 30%
6. Work Problem
Rate = Work output ÷ time
Workers have different rate
1/A + 1/B = 1/t
Workers have the same rate
Use the concept of manpower (man-days, man-hours) to represent the plan and the done.
Plan = Done
Rate = Work output ÷ (Plan / Done)
Worked example — Different rate
Pedro can paint a fence alone in 6 hours, Juan in 4 hours. Working together, how long will it take?
- • 1/6 + 1/4 = 1/t
- • 2/12 + 3/12 = 5/12 = 1/t
Answer: t = 12/5 = 2.4 hours
Worked example — Same rate (manpower)
It takes 8 workers 10 days to build a wall. How many days for 5 workers at the same rate?
- • Plan = 8 × 10 = 80 man-days
- • Done = 5 × d
- • 80 = 5d
Answer: 16 days
7. Interest Problem
Simple interest
I = P × r × t
Compound interest
A = P(1 + r)n
Interest = A − P
- P = principal (starting amount)
- r = annual interest rate, as a decimal
- t = time in years
- n = number of years (compound interest)
- A = final amount
- I = interest earned
Worked example — Simple interest
₱50,000 is invested at 9% simple annual interest. How much total interest is earned after 2 years and 6 months?
- • I = 50,000 × 0.09 × 2.5
Answer: ₱11,250
Worked example — Compound interest
₱40,000 is invested at 8% annual interest, compounded annually, for 2 years. What is the total amount after 2 years?
- • A = 40,000 × (1.08)²
- • A = 40,000 × 1.1664
Answer: ₱46,656
Check: interest earned = 46,656 − 40,000 = ₱6,656
8. Fractions Cheat Sheet
Worked example
2/3 + 1/4
- • (2×4 + 1×3) / (3×4) = (8+3)/12
Answer: 11/12
Worked example
3/5 − 1/4
- • (3×4 − 1×5) / (5×4) = (12−5)/20
Answer: 7/20
Worked example
2/5 × 3/7
- • (2×3)/(5×7)
Answer: 6/35
Worked example
3/4 ÷ 2/9
- • 3/4 × 9/2 = (3×9)/(4×2)
Answer: 27/8
Worked example
LCD of 4, 6, 9
- • 4=2², 6=2×3, 9=3² → take highest powers: 2²×3²
Answer: 36
Worked example — two-number formula
LCD of 6 and 8
- • GCF(6,8) = 2
- • (6×8) ÷ 2
Answer: 24
Check: multiples of 6: 6,12,18,24... multiples of 8: 8,16,24... first common = 24 ✓
9. Ratio and Proportion
Worked example — Ratio
3 cups flour to 2 cups sugar, as a ratio.
Answer: 3:2 = 3/2
Worked example — Proportion → cross multiplication
Boys to girls is 3:5. There are 12 boys — how many girls?
- • 3/5 = 12/g
- • 3g = 5 × 12 = 60
Answer: g = 20
Worked example — Direct proportion
A car travels 150 km in 3 hours at constant speed. How far in 5 hours?
- • 150/3 = x/5
- • 3x = 750
Answer: 250 km
Worked example — Inverse proportion
4 workers build a wall in 12 days. How many days for 6 workers, same rate?
- • 4 × 12 = 6 × d
- • 48 = 6d
Answer: 8 days
Worked example — Ratio sharing
₱4,500 split between two siblings in ratio 2:3.
- • total ratio = 5
- • first: (2/5)×4500 = 1800
- • second: (3/5)×4500 = 2700
Answer: ₱1,800 and ₱2,700
Check: 1800 + 2700 = 4500 ✓
10. Basic Arithmetic — What to Memorize
Worked example — Addition
47 + 68
Answer: 115
Worked example — Subtraction
152 − 87
Answer: 65
Worked example — Multiplication
23 × 14
Answer: 322
Worked example — Division
144 ÷ 12
Answer: 12
Worked example — Order (PEMDAS)
6 + 2 × (5 − 3)² ÷ 4
Explanation: Parentheses, then Exponents, then Multiplication/Division left to right, then Addition/Subtraction left to right.
- • (5−3) = 2 → 2² = 4
- • 2 × 4 = 8 → 8 ÷ 4 = 2
- • 6 + 2
Answer: 8
Worked example — Average
Find the average of 12, 18, 25, 9, and 16.
- • Sum = 12+18+25+9+16 = 80
- • Number of values = 5
Answer: 80 ÷ 5 = 16
Worked example — Fraction addition
5/6 + 1/4
Explanation: Find the LCD first — LCD(6,4) = 12.
- • 10/12 + 3/12
Answer: 13/12 = 1 1/12
Worked example — Fraction multiplication
3/8 × 4/9
Explanation: Multiply across — numerators together, denominators together.
- • (3×4)/(8×9) = 12/72
Answer: 1/6
Worked example — Fraction division
5/6 ÷ 2/3
Explanation: Keep the first fraction, Change ÷ to ×, Flip the second fraction.
- • 5/6 × 3/2 = 15/12
Answer: 5/4 (or 1 1/4)
Worked example — Same signs
(−8) × (−5)
Explanation: Multiplying two numbers with the same sign gives a positive result.
Answer: 40
Worked example — Different signs
(−8) × 5
Explanation: Multiplying two numbers with different signs gives a negative result.
Answer: −40
Want to go deeper?
- Test-Taking Techniques & Patterns — how to translate word problems, spot trap answers, and pace yourself, using these same formulas under real exam conditions
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