Fractions, Decimals, and Surds

Fractions and decimals seem basic, but in timed exams, the person who has fraction-decimal equivalents memorized crushes the person who has to calculate them every time. Surds (square roots that can’t be simplified to nice numbers) show up in geometry and simplification questions. Let’s build a toolkit we can use instantly.

Fraction-Decimal Equivalents — Memorize This Table

This table is absolutely worth memorizing. It saves us 10-20 seconds per question, and over a full exam, that adds up to 5-10 extra minutes.

Pro tip: Once we know 1/n, we can get any k/n instantly. For example, 3/8 = 3 × 0.125 = 0.375. And 5/6 = 5 × 0.1667 = 0.8333.

Comparing Fractions — Shortcuts

Method 1: Cross Multiplication (Best for 2 fractions)

To compare a/b and c/d: compare a×d with c×b.

Example: Which is larger: 7/11 or 5/8?

• 7 × 8 = 56
• 5 × 11 = 55
• 56 > 55, so 7/11 > 5/8

Method 2: Converting to Same Denominator

When comparing multiple fractions, find LCM of denominators.

Method 3: The Benchmark Method

Compare each fraction to a benchmark like 1/2 or 1.

Example: Arrange 3/7, 4/9, 5/11 in ascending order.

• 3/7 = 0.4286…
• 4/9 = 0.4444…
• 5/11 = 0.4545…
• Ascending: 3/7 < 4/9 < 5/11

Shortcut for fractions of the form (n)/(2n+1): They increase as n increases. So 3/7 < 4/9 < 5/11 < 6/13… This pattern is common in exams.

Recurring Decimals to Fractions

Pure recurring (all digits repeat)

If 0.abcabc… (the block “abc” repeats), the fraction = abc/999

Examples:

• 0.333… = 3/9 = 1/3
• 0.272727… = 27/99 = 3/11
• 0.142857142857… = 142857/999999 = 1/7

The rule: Put the repeating block in the numerator, and as many 9s as digits in the block in the denominator.

Mixed recurring (some digits don’t repeat)

If 0.abc (a doesn’t repeat, bc repeats): fraction = (abc - a) / 990

Example: 0.1666… = 0.16

• Numerator: 16 - 1 = 15
• Denominator: 90 (one 9 for repeating digit, one 0 for non-repeating digit)
• = 15/90 = 1/6 ✓

General rule: Denominator has as many 9s as repeating digits, followed by as many 0s as non-repeating digits after the decimal point.

Surds — The Basics

A surd is a root that can’t be simplified to a rational number. √2, √3, √5, ∛7 are all surds. √4 is NOT a surd because √4 = 2.

Key Values to Memorize

Simplifying Surds

Look for perfect square factors inside the root.

• √18 = √(9 × 2) = 3√2
• √75 = √(25 × 3) = 5√3
• √200 = √(100 × 2) = 10√2

Operations with Surds

We can only add/subtract like surds (same number under the root):

• 3√2 + 5√2 = 8√2 ✓
• 3√2 + 5√3 → can’t simplify ✗

Multiplication works across different surds:

• √2 × √3 = √6
• 3√2 × 4√5 = 12√10

Rationalizing the Denominator

We don’t leave surds in the denominator. To remove them, we multiply by the conjugate.

Simple case: 1/√a

Multiply top and bottom by √a:

• 1/√3 = (1 × √3)/(√3 × √3) = √3/3

Conjugate case: 1/(a + √b)

The conjugate of (a + √b) is (a - √b). Multiply both sides by it.

Example: Rationalize 1/(3 + √2)

• Multiply by (3 - √2)/(3 - √2)
• = (3 - √2)/(9 - 2)
• = (3 - √2)/7

Double surd case: 1/(√a + √b)

Example: Rationalize 1/(√5 + √3)

• Multiply by (√5 - √3)/(√5 - √3)
• = (√5 - √3)/(5 - 3)
• = (√5 - √3)/2

Worked Examples

Example 1: Simplify √50 + √32 - √18

• √50 = √(25×2) = 5√2
• √32 = √(16×2) = 4√2
• √18 = √(9×2) = 3√2
• = 5√2 + 4√2 - 3√2 = 6√2

Example 2: Which is greater: √7 - √6 or √6 - √5?

Trick: Rationalize each by multiplying by the conjugate.

• √7 - √6 = (7-6)/(√7+√6) = 1/(√7+√6) ≈ 1/5.095
• √6 - √5 = (6-5)/(√6+√5) = 1/(√6+√5) ≈ 1/4.685
• Since 1/4.685 > 1/5.095, √6 - √5 is greater

The pattern: for consecutive numbers, the gap between their square roots decreases as numbers get larger.

Example 3: Convert 0.5̄7̄ (0.575757…) to a fraction.

Repeating block = 57, which has 2 digits. Fraction = 57/99 = 19/33

Example 4: Find the value of (√5 + √3)² - (√5 - √3)²

We could expand both, but there’s a shortcut using the identity a² - b² = (a+b)(a-b):

• Let a = √5 + √3 and b = √5 - √3
• a + b = 2√5
• a - b = 2√3
• a² - b² = 2√5 × 2√3 = 4√15

Answer: 4√15

Example 5: Arrange in ascending order: 3/5, 5/8, 7/12, 2/3

Convert to decimals:

• 3/5 = 0.6
• 5/8 = 0.625
• 7/12 = 0.5833…
• 2/3 = 0.6667…

Ascending: 7/12 < 3/5 < 5/8 < 2/3

Common Exam Patterns

1. “Rationalize the denominator” → Multiply by conjugate
2. “Arrange fractions in order” → Cross multiply pairwise, or convert to decimals
3. “Convert recurring decimal to fraction” → Repeating digits/999… formula
4. “Simplify surd expression” → Factor out perfect squares, combine like surds
5. “Which is greater: √a - √b or √c - √d?” → Rationalize and compare

Practice Problems

Q1: Simplify: 3/(√7 - √3)

Q2: Convert 0.23̄ (where only 3 repeats) to a fraction.

Q3: If √3 = 1.732, find the value of 1/(√3 - 1) correct to 3 decimal places.

Answers:

A1: Multiply by (√7 + √3)/(√7 + √3) = 3(√7 + √3)/(7-3) = 3(√7 + √3)/4

A2: 0.2333… Here, 2 doesn’t repeat, 3 repeats. = (23 - 2)/90 = 21/90 = 7/30

A3: 1/(√3 - 1) = (√3 + 1)/((√3)² - 1²) = (√3 + 1)/2 = (1.732 + 1)/2 = 2.732/2 = 1.366