Decimal Fractions and Simplification

AFCAT Numerical Ability is at matriculation (Class 10) level, and simplification is its speed core - you get roughly 72 seconds per question overall, so clean mental arithmetic here frees time for tougher items. With -1 negative marking, accuracy matters as much as speed.

BODMAS - the order of operations

Solve in order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Example: 12 + 6 / 3 x 2 = 12 + (6/3) x 2 = 12 + 2 x 2 = 12 + 4 = 16.

Decimal handling

• Adding/subtracting: line up the decimal points. 2.5 + 0.75 = 3.25.
• Multiplying: multiply as whole numbers, then place the decimal by counting total decimal places. 0.3 x 0.2 = 0.06 (two places).
• Dividing by a decimal: shift both points. 4.5 / 0.5 = 45/5 = 9.

Fraction and percentage shortcuts (memorise)

1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 1/5 = 0.2 = 20%; 1/8 = 0.125 = 12.5%; 3/8 = 0.375 = 37.5%. Using these, 37.5% of 480 = 3/8 x 480 = 180 - no long multiplication.

Exam Tricks & Tips

• 🎯 Always follow BODMAS strictly - division/multiplication before addition/subtraction.
• 🎯 Convert repeating decimals and percentages to fractions (0.125 = 1/8, 12.5% = 1/8) to compute faster.
• 🎯 To divide by a decimal, shift the point in BOTH numbers equally (4.5/0.5 = 45/5 = 9).
• 🎯 Count decimal places when multiplying: total places in the answer = sum of places in the factors.
• 🎯 Use approximation when options are far apart - round, compute, pick the closest.
• ❌ Common mistake: doing addition before division - "12 + 6/3" is 14, NOT 6; BODMAS puts division first.

Quick recap

Apply BODMAS, align decimals for add/subtract, count places for multiply, shift points for divide, and lean on the fraction-percentage table. Simplification is pure speed and accuracy - drill it daily.

Cover the answer, recall, then check. 11 cards on decimals and simplification.

Q1. What does BODMAS stand for?
A1. Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.

Q2. Solve 12 + 6 / 3 x 2 using BODMAS.
A2. 16 — do 6/3 = 2, then x2 = 4, then 12 + 4 = 16.

Q3. 0.3 x 0.2 = ?
A3. 0.06 (two decimal places total).

Q4. 4.5 / 0.5 = ?
A4. 9 — shift both points: 45/5 = 9.

Q5. Write 1/8 as a decimal and a percentage.
A5. 0.125 and 12.5%.

Q6. 37.5% of 480 using fractions?
A6. 3/8 x 480 = 180.

Q7. 2.5 + 0.75 = ?
A7. 3.25 (line up the decimal points).

Q8. 3/8 as a percentage?
A8. 37.5%.

Q9. When multiplying decimals, how do you place the point?
A9. Count the total decimal places in both factors; the product has that many.

Q10. 1/5 as a decimal and percentage?
A10. 0.2 and 20%.

Q11. Why is "12 + 6/3" equal to 14, not 6?
A11. BODMAS does division first: 6/3 = 2, then 12 + 2 = 14.

Simplification is where AFCAT Numerical Ability quietly rewards speed. A messy-looking expression collapses in seconds if you apply the order of operations correctly and use fraction-decimal shortcuts. Waste time doing it long-hand and you lose marks you already knew how to get.

Core idea / what this tests: whether you can evaluate arithmetic expressions with fractions, decimals, brackets and mixed operations quickly and correctly, using the BODMAS rule and smart conversions.

Deep explanation

Beginner — BODMAS is law

Evaluate in strict order: Brackets → Orders (powers/roots) → Division and Multiplication (left to right) → Addition and Subtraction (left to right). Division and multiplication share a rank (do whichever comes first, left to right); same for addition and subtraction. Skipping this order is the number-one cause of wrong answers.

Intermediate — decimals and fractions fluently

• Decimal ↔ fraction: 0.5 = 1/2, 0.25 = 1/4, 0.2 = 1/5, 0.125 = 1/8, 0.333... = 1/3. Converting to the easier form speeds calculation.
• Multiplying/dividing decimals: count decimal places. 0.6 × 0.05 = 0.030 (1 + 2 = 3 places). To divide by a decimal, shift both numbers' points equally: 4.5 ÷ 0.9 = 45 ÷ 9 = 5.
• Recurring decimals: 0.\overline{3} = 1/3, 0.\overline{6} = 2/3.

Advanced — algebraic shortcuts and 'of'

Use identities to avoid grinding: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², a²−b² = (a+b)(a−b). So 102² = (100+2)² = 10000+400+4 = 10404 instantly. The word "of" means multiply: "1/2 of 60" = 30, and it binds like multiplication in BODMAS. Simplify fractions before multiplying (cancel common factors) to keep numbers small.

Worked example

Simplify: 12 + 6 ÷ 2 × 3 − (4 + 1)

Step 1 (Brackets): (4 + 1) = 5 → 12 + 6 ÷ 2 × 3 − 5.
Step 2 (÷ and × left to right): 6 ÷ 2 = 3; then 3 × 3 = 9 → 12 + 9 − 5.
Step 3 (+ and − left to right): 12 + 9 = 21; 21 − 5 = 16.

Answer: 16. The trap is doing 2 × 3 first — but division comes first here because it is to the left. BODMAS order settles it.

AFCAT relevance

Simplification questions are among the fastest in AFCAT Numerical Ability and often open the section. Clearing them quickly and accurately banks time for the heavier word problems, and with −1 negative marking the discipline of BODMAS protects against silly, avoidable errors.

Speed tricks and shortcuts

• Memorise common decimal-fraction equivalents (1/8=0.125, 1/3=0.333) to switch to the easier form.
• Cancel before you multiply fractions to keep numbers tiny.
• Use (a±b)² and a²−b² to square/multiply near-round numbers fast.
• Left-to-right for equal-rank operations (÷ vs ×, + vs −) — don't reorder them.

Key formulas

• BODMAS order: Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction.
• a² − b² = (a+b)(a−b); (a+b)² = a²+2ab+b²; (a−b)² = a²−2ab+b².
• (a+b)³ = a³+3a²b+3ab²+b³; a³+b³ = (a+b)(a²−ab+b²).
• Approximation: round to convenient values, track error; percentage ≈ conversions.
• Fractions: LCM of denominators to add/subtract.
• √(ab) = √a·√b; recurring decimal x.ȳ converts via 9s.

Usage: apply algebraic identities to avoid long multiplication in simplification.

Worked Example

Problem: Simplify: 3.2 + 0.8 × 5 − 4 ÷ 2.

Solution:
Follow the BODMAS/order-of-operations rule: do Brackets first, then Orders (powers), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).

Step 1 — Division and multiplication first (there are no brackets or powers):
0.8 × 5 = 4.0
4 ÷ 2 = 2

Step 2 — Rewrite the expression with these results:
3.2 + 4.0 − 2.

Step 3 — Now addition and subtraction, left to right:
3.2 + 4.0 = 7.2
7.2 − 2 = 5.2.

Answer: 5.2.

Solution note: A common error is to add 3.2 + 0.8 first; but multiplication and division must be performed before addition and subtraction.