Simplification and Fractions

Overview

Simplification and Fractions appears about 0.8 times per paper across the last four AFCAT solved papers, placing it in the solid add-on band of Numerical Ability.

The Simplification and Fractions bucket is the lightest numerical sub-topic on AFCAT, averaging only 0.75 questions per paper across the last four solved sittings. Despite the low frequency, it is the highest-accuracy mark in the section because the entire item collapses to mechanical rule-following — there is no concept to model, no equation to set up, no trap embedded in the wording. You either apply BODMAS correctly or you do not.

Most candidates who lose this mark do so for one of three reasons: they attempt the chain mentally instead of writing each reduction line, they fumble division-versus-multiplication precedence (both are left-to-right, neither outranks the other), or they invert the wrong fraction during division. Each of these slips is preventable with a written protocol that takes 40–60 seconds. The aim of this module is to install that protocol so that whenever a simplification or fraction-comparison item appears, you bank it without thought.

The other reason this topic deserves a careful read is that the underlying skills — fraction arithmetic, decimal equivalents, recurring-decimal recognition, surd handling — are infrastructure for the rest of Numerical Ability. Percentage problems convert to fractions. Ratio problems demand quick fraction comparison. Compound interest formulas need clean surd and power simplification. So even if only one item in the actual paper carries the label, the skill carries the whole section.

Why simplification is the highest-accuracy mark on the paper

Across the AFCAT papers analysed, simplification items show two patterns:

• The arithmetic chain is long but mechanical — usually 4 to 7 operations with mixed brackets, fractions, and decimals.
• The four options are arithmetically close — designed to catch a single mis-step rather than a conceptual error.

This combination means that a candidate who writes every reduction line will almost always reach the correct option, while a candidate who tries to compress the chain mentally will land on a distractor roughly one time in three. The cost of writing is 15–20 seconds; the cost of guessing wrong under negative marking is −1 against a potential +3. So even at 0.75 questions per paper average, the expected return on writing the steps is positive and large.

A second reason this topic matters: the four options are usually generated by simulating common errors. If you spot which distractor corresponds to which error (forgot the bracket, inverted wrong fraction, mis-ordered division and addition), you can self-check your answer by ruling out the trap options. We will see this in the worked examples.

BODMAS — the only order you need

BODMAS is an acronym that fixes the order in which you must evaluate a mixed arithmetic expression. The letters stand for:

The two pitfalls in this table are the same-rank rules. Division does not outrank multiplication — they share rank D-M and you apply them strictly left to right. Same for addition and subtraction. So 20 ÷ 4 × 5 is 25, not 1, because you do the division first as it appears earlier.

Walkthrough: 12 + 3 × (8 − 2)² ÷ 4

1. Brackets first: (8 − 2) = 6. Expression becomes 12 + 3 × 6² ÷ 4.
2. Orders: 6² = 36. Expression becomes 12 + 3 × 36 ÷ 4.
3. D and M left-to-right: 3 × 36 = 108, then 108 ÷ 4 = 27. Expression becomes 12 + 27.
4. A: 12 + 27 = 39.

Notice that step 3 went multiplication-then-division because that is the left-to-right order in the original expression. If the problem had been 3 ÷ 36 × 4, you would have done division first.

Nested brackets — innermost first

When brackets are nested, work from the deepest layer outward. The conventional ordering is round → square → curly, written symbolically as { [ ( ) ] }. In AFCAT stems you will normally see at most two levels of nesting.

The discipline is identical to non-nested BODMAS: complete the inner layer fully (apply BODMAS inside it), then drop one level of brackets and continue.

Walkthrough: 50 − [30 − {15 − (8 − 3)}]

1. Innermost round: (8 − 3) = 5. Becomes 50 − [30 − {15 − 5}].
2. Curly: {15 − 5} = 10. Becomes 50 − [30 − 10].
3. Square: [30 − 10] = 20. Becomes 50 − 20.
4. Final: 50 − 20 = 30.

The most common trap here is the sign flip: when a minus precedes an opening bracket, students sometimes drop the bracket without flipping signs inside. The protocol above avoids this entirely because you always evaluate the inside fully before stripping the bracket.

Fraction operations — the four basic moves

Every fraction problem reduces to four operations: add, subtract, multiply, divide. Each has a fixed rule.

Two practical points:

• Reduce before you multiply when possible. In 2/3 × 9/4, the 3 and 9 share a factor 3, and the 2 and 4 share a factor 2. Cancel first to get 1/1 × 3/2 = 3/2, avoiding the 18/12 step.
• Invert the divisor, not the dividend. In a ÷ b you flip b, never a. Writing the operation as a fraction-of-a-fraction (the main bar separating the two) makes this visually obvious.

Mixed numbers — convert before you operate

A mixed number such as 2¾ is a shorthand for 2 + 3/4. Before you do any arithmetic, convert it to an improper fraction.

Rule: mixed-to-improper = (whole × denominator + numerator) ÷ denominator.

• 2¾ = (2 × 4 + 3) / 4 = 11/4
• 5⅔ = (5 × 3 + 2) / 3 = 17/3
• 1⅛ = (1 × 8 + 1) / 8 = 9/8

To go the other way (improper-to-mixed), divide the numerator by the denominator. The quotient is the whole part; the remainder over the denominator is the fraction part. So 23/5 = 4 with remainder 3 = 4⅗.

Never try to add or multiply mixed numbers directly. 2¾ × 1⅛ is not 2 × 1 with the fractions multiplied separately — it is 11/4 × 9/8 = 99/32 = 3 and 3/32.

Decimal-to-fraction conversion (with recurring decimals)

Terminating decimals convert by inspection: 0.75 = 75/100 = 3/4. The denominator is a power of ten matching the number of decimal places, then reduce.

Recurring decimals are trickier but follow a fixed pattern. Memorise these standard equivalents — they appear repeatedly in AFCAT distractors:

The general derivation: for a purely recurring decimal with cycle length k, the fraction equals (the repeating block) over (k nines). So 0.27 = 27/99 = 3/11. And 0.142857 = 142857/999999 = 1/7.

Fraction comparison — three methods, pick by situation

Comparing fractions is the most common form this topic takes in AFCAT — the stem lists three or four fractions and asks for largest or smallest. Use whichever method fits the count:

• Two fractions — cross-multiply. For a/b versus c/d, compare the cross-products a × d and b × c. The side with the larger product is the larger fraction. (Assumes positive denominators, which is the AFCAT default.)
• Three or four fractions — common denominator. Find LCM of all denominators, convert each fraction, compare numerators. Slower per fraction but linear in count.
• Mixed denominator sizes — convert to decimals. If the denominators are unrelated (3, 7, 11, 13), the LCM is huge. Faster to compute decimal values to three places using the memorised table.

Heuristic for special forms

If all fractions have the shape n/(n+k) for a fixed gap k, the largest n gives the largest fraction. So 4/5, 5/6, 6/7, 7/8 all have gap 1 between numerator and denominator, and they increase in the order listed (7/8 is largest).

If all fractions have the shape n/(n−k), the same rule applies but inverted — larger n gives a fraction closer to 1 from above. And if the gap k is bigger, the fraction is further from 1.

Memorise this fraction-to-decimal table

This is the single highest-leverage memorisation in the Numerical Ability syllabus. It speeds up not just simplification but also percentage, ratio, and interest items. Commit it to memory until recall is instant.

Powers and roots — quick simplification rules

AFCAT simplification stems sometimes embed a small power or surd. Three rules cover almost every case:

• Same base, add exponents in multiplication: a^m × aⁿ = a^m+n. So 2³ × 2⁴ = 2⁷ = 128.
• Same base, subtract exponents in division: a^m ÷ aⁿ = a^m−n. So 5⁶ ÷ 5⁴ = 5² = 25.
• Power of a power, multiply exponents: (a^m)ⁿ = a^m×n. So (3²)³ = 3⁶ = 729.

For surds, the basic identities are √a × √b = √(ab) and √(a/b) = √a / √b. So √8 = √(4 × 2) = 2√2, and √(50/2) = √25 = 5.

Common values worth remembering: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236. These cover most surd-approximation distractors.

Algebraic simplification — like terms and distribution

A small minority of simplification items wrap the arithmetic inside a tiny algebraic expression. The two skills you need are:

• Combining like terms: 3x + 5x − 2x = 6x. Only terms with the same variable and the same exponent combine.
• Distributing: a(b + c) = ab + ac. So 4(3 + 2) = 4 × 3 + 4 × 2 = 20, which matches 4 × 5 = 20 — a quick self-check.

When the expression has fractions of variables, treat the variable as an opaque token and apply standard fraction rules. So x/3 + x/4 = (4x + 3x)/12 = 7x/12. Then if asked to evaluate at x = 6, substitute at the end: 7(6)/12 = 42/12 = 7/2.

Approximation tactics for AFCAT

If the four options are well-separated (say, 12, 18, 25, 31), you can approximate aggressively. Rules of thumb:

If the options are tight (say, 0.45, 0.48, 0.51, 0.54), do not approximate — compute exactly. The distractors are designed to trap a rounding step.

Speed shortcuts

Two micro-shortcuts that pay back in the simplification slot:

• Split-and-add for 11-based products. n × 11 = n with its digits added in the middle. So 23 × 11 = 2(2+3)3 = 253. And 45 × 11 = 4(4+5)5 = 495.
• Perfect-square recognition. Memorise squares to 25. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. When a √ appears, scan the table first.

Other useful instant facts: 12³ = 1728, 15² = 225, 25² = 625, 35² = 1225, 45² = 2025. The pattern for any n5: take n × (n+1), then append 25. So 75² = 7 × 8 = 56, append 25 → 5625.

Common AFCAT trap patterns

Across the solved papers, the simplification distractors cluster around five errors:

1. Division-before-multiplication when they sit in opposite order. 20 × 5 ÷ 10 is 10, not 0.4. The candidate who does ÷10 first because it is on the right gets 0.4.
2. Sign loss when stripping a bracket preceded by minus. 10 − (3 − 5) = 10 − (−2) = 12, not 10 − 3 − 5 = 2.
3. Inverting the wrong fraction in division. (3/4) ÷ (2/5) = 3/4 × 5/2 = 15/8, not 4/3 × 2/5 = 8/15.
4. Mis-cancelling across a plus. In (2x + 4)/2, you cannot cancel the 2 with just the 2x — both terms in the numerator must be divisible. Correct: (2x + 4)/2 = x + 2.
5. Forgetting the cycle in a recurring decimal. 0.36 is 36/99 = 4/11, not 36/100 = 9/25.

When you pick an answer, glance at the other three options. If one of them matches the result of skipping a step, you have just confirmed that you did not skip it.

Time budget

Across 100 questions in 120 minutes you have 72 seconds per item on average. Simplification deserves less than that — it is a known-quick slot. Aim for:

• Single BODMAS chain (no nesting): 30–40 seconds.
• Nested brackets: 45–60 seconds.
• Fraction comparison (three or four fractions): 50–70 seconds.
• Mixed expression with fractions and decimals: 60–80 seconds.

If you cross 90 seconds on a simplification item, mark and move. Spending two minutes here costs you the chance to attempt a high-yield percentage or work item later.

Worked AFCAT-style examples

Exam-day strategy

1. Write every BODMAS reduction line. Mental compression is the single biggest source of error in this slot.
2. For two-fraction comparison, cross-multiply. For three or more, go to common denominator or memorised decimals.
3. Reduce fractions before multiplying. Smaller numbers, fewer arithmetic slips.
4. Memorise the fraction-to-decimal table up to 1/12. It pays back in every numerical sub-topic, not just here.
5. When options are tight, never approximate. When options are well-separated, approximate aggressively to save 30 seconds.
6. If a recurring decimal appears in the stem or options, convert it using the (repeating block) / (k nines) rule.
7. Time budget: 30–80 seconds depending on the form. Mark-and-skip beyond 90.
8. Self-check: glance at the three distractors. If one matches the result of a step-skip you almost did, that confirms your route.

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