Simplification and Approximation — AAI ATC GK Lesson

Before you start

Learning objectives and outcomes

Objective 1Apply BODMAS correctly, including the left-to-right rule within each priority pair.

Objective 2Convert common percentages to fractions instantly.

Objective 3Recall squares to 30, cubes to 15, and common roots without calculating.

Outcome ASolve simplification questions for speed, not just correctness.

Subtopic 1

BODMAS, the order of operations

Every simplification problem depends on applying operations in the correct order. Getting this order wrong is the most common source of errors in this entire topic.

The BODMAS order

Brackets → Orders (powers/roots) → Division/Multiplication (left to right) → Addition/Subtraction (left to right)

Division and multiplication share equal priority and are done left to right as they appear, not division always before multiplication. The same applies to addition and subtraction.

✈ AAI ATC relevance — a BODMAS question, or a longer expression relying on it, appears in nearly every attempt

Memory trick — "BODMAS is a queue, not a hierarchy of two"

Students often wrongly assume multiplication always beats division, or addition always beats subtraction. Within each pair, it's whichever comes first, left to right, that goes first.

Easy

Simplify: 12 + 8 ÷ 4 × 2 − 3

Following BODMAS: 8÷4=2, then 2×2=4, so the expression becomes 12+4−3 = 13.

Moderate

Simplify: 36 ÷ (6 − 2) × 2

(6−2)=4, 36÷4=9, 9×2 = 18.

Subtopic 2

Fractions, percentages and quick conversions

Converting familiar percentages to their fraction form, and back, is far faster than long decimal multiplication — and this exam rewards speed as much as accuracy.

Percentages worth knowing as fractions

12.5% = 1/8 · 25% = 1/4 · 20% = 1/5 · 331/3% = 1/3 · 50% = 1/2 · 75% = 3/4

Once these are automatic, a question like "12.5% of 640" becomes a one-step division (640÷8=80) instead of a multiplication with decimals.

Figure 2.1 — Common percentage-fraction pairs A simple reference table pairing each common percentage with its fraction equivalent, from 121/2% up to 75%. Upload image here

Memory trick — "Eighths, quarters, fifths, thirds"

Group the common percentages by their denominator family: eighths (12.5%, 37.5%...), quarters (25%, 75%), fifths (20%, 40%...), and thirds (331/3%, 662/3%) — remembering one family at a time is easier than memorising them as a flat list.

Easy

What is 12.5% of 640?

12.5% = 1/8, so 1/8 of 640 = 80.

Moderate

What is 25% of 400?

25% of 400 = 0.25 × 400 = 100.

Subtopic 3 · highest yield

Squares, cubes and roots

Knowing squares up to 30, cubes up to 15, and the corresponding square/cube roots by heart eliminates calculation time on a large share of simplification questions.

Worth memorising outright

Squares: 11²=121 … 25²=625 · Cubes: 11³=1331, 12³=1728
√144=12, √0.0009=0.03 · ∛27=3, ∛64=4

For decimal square roots like √0.0009, rewrite the decimal as a power of 10 first (0.0009 = 9×10⁻⁴), take the root of each part separately, then recombine.

✈ AAI ATC relevance — square/cube recall questions are pure memorisation, making them the fastest possible points on this exam once learned

Memory trick — "Match the decimal places to the root"

A square root halves the number of decimal places moved in the power of 10; a cube root divides it by three. This shortcut avoids long division entirely for decimal roots.

Moderate

What is the value of 11³ (11 cubed)?

11³ = 11×11×11 = 1331.

Hard

Simplify: √0.0009

√0.0009 = √(9×10⁻⁴) = 3×10⁻² = 0.03.

Exam strategy

Facts worth having automatic

BODMAS orderBrackets, Orders, then Division/Multiplication and Addition/Subtraction left to right.

Division = Multiplication priorityNeither always comes first — whichever appears first, left to right, goes first.

12.5% = 1/8One of several percentages worth knowing as fractions instantly.

Squares to 30, cubes to 15Memorise outright — pure recall, fastest points available.

Decimal rootsRewrite as a power of 10 first, then take the root of each part.

Quick revision

Chapter summary sheet

BODMASBrackets, Orders, Division/Multiplication (L-R), Addition/Subtraction (L-R).

12.5%= 1/8

25%= 1/4

20%= 1/5

331/3%= 1/3

11³= 1331

√144= 12

∛27= 3

↓ Free download — Simplification and Approximation question PDF

All 6 questions with a full answer key.

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End of chapter

Comprehensive assessment

Full solutions are in the downloadable PDF above.

Multiple choice questions

1. State the full order of operations under BODMAS.

2. Do division and multiplication have a fixed priority over each other?

3. What fraction is equivalent to 12.5%?

4. What fraction is equivalent to 331/3%?

5. What is the value of 11 cubed?

6. What is √144 + √64?

7. How do you simplify a decimal square root like √0.0009 quickly?

8. Simplify: (2.5)²

9. Simplify: 7 ÷ 0.7

10. What is 45% of 80 minus 20% of 50?

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