Civil Service Exam (Professional) Numerical Ability — Fractions — Operations, Conversion & ComparisonFlash Cards

Answer

1. Proper Fractions: Numerator < Denominator (less than 1). Examples: 1/2, 3/8, 5/7
2. Improper Fractions: Numerator ≥ Denominator (equal to or greater than 1). Examples: 5/2, 4/3, 9/7
3. Mixed Fractions: Whole number + proper fraction. Examples: 2½, 1⅓, 3⅝

Answer

Steps:

1. Multiply the whole number by the denominator
2. Add the product to the numerator (this becomes the new numerator)
3. Keep the same denominator

Example: 2⅜ = (2×8+3)/8 = 19/8
Formula: a b/c = (a×c+b)/c

Answer

Steps:

1. Divide the numerator by the denominator
2. The quotient becomes the whole number
3. The remainder becomes the new numerator
4. Keep the same denominator

Example: 17/5 = 3 remainder 2 = 3⅖
Check: 17 ÷ 5 = 3 R2

Answer

Definition: Fractions with the same value but different numerators and denominators.

Rule: Multiply or divide both numerator and denominator by the same non-zero number.

Example: ¼ = 2/8 = 3/12 = 4/16
(multiply by 2, 3, 4 respectively)

Key: Whatever you do to the top, do to the bottom!

Answer

Rule: Add the numerators and keep the same denominator.

Steps:

1. Add the numerators
2. Keep the denominator unchanged
3. Simplify if possible

Example: 2/7 + 3/7 = 5/7
Example: 4/8 + 2/8 = 6/8 = ¾ (simplified)

Answer

Steps:

1. Find the LCD (Least Common Denominator)
2. Convert fractions to equivalent fractions with LCD
3. Add the numerators, keep the LCD
4. Simplify if needed

Example: ¾ + ⅕
LCD of 4 and 5 = 20
¾ = 15/20, ⅕ = 4/20
15/20 + 4/20 = 19/20

Answer

LCD = Least Common Denominator (same as LCM - Least Common Multiple)

Method: List multiples of each denominator, find the smallest common one.

Example: LCD of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
LCD = 12 (smallest common multiple)

Answer

Same process as addition, but subtract instead:

1. Find the LCD
2. Convert to equivalent fractions
3. Subtract numerators, keep LCD
4. Simplify

Example: 5/6 - ⅓
LCD = 6
5/6 - 2/6 = 3/6 = ½

Answer

Rule: Multiply numerators together, multiply denominators together.

Steps:

1. Multiply numerator × numerator
2. Multiply denominator × denominator
3. Simplify if possible

Example: ⅔ × ¾ = (2×3)/(3×4) = 6/12 = ½

Note: Denominators don't need to be the same!

Answer

Method: Convert whole number to a fraction with denominator 1, then multiply.

Steps:

1. Write whole number as fraction/1
2. Multiply as usual
3. Simplify

Example: ¾ × 8 = ¾ × 8/1 = 24/4 = 6
Shortcut: Multiply numerator by whole number: ¾ × 8 = (3×8)/4 = 24/4 = 6

Answer

Rule: Multiply by the reciprocal of the divisor.

Steps:

1. Keep the first fraction (dividend)
2. Change ÷ to ×
3. Flip the second fraction (reciprocal of divisor)
4. Multiply normally

Example: ¾ ÷ ⅖ = ¾ × 5/2 = 15/8 = 1⅞

Answer

Method: Cross multiply - multiply diagonally.

Steps:

1. Multiply numerator of dividend × denominator of divisor
2. Multiply denominator of dividend × numerator of divisor
3. Form new fraction and simplify

Example: ¾ ÷ ⅖
Cross multiply: (3×2)/(4×2) = 6/8... Wait! Use (3×5)/(4×2) = 15/8

Answer

Methods:

Method 1 - GCF:

1. Find GCF of numerator and denominator
2. Divide both by GCF

Method 2 - Prime Division:

1. Divide by small primes (2,3,5,7...)
2. Continue until no common factors

Example: 12/18 ÷ 2 = 6/9 ÷ 3 = 2/3
GCF method: GCF(12,18) = 6, so 12/18 ÷ 6/6 = 2/3

Answer

Rule: When denominators are equal, compare numerators.

Larger numerator = larger fraction
Smaller numerator = smaller fraction

Example: Compare 3/8, 5/8, 7/8
5 > 3 and 7 > 5, so: 3/8 < 5/8 < 7/8

Order: 3/8 < 5/8 < 7/8 (least to greatest)

Answer

Rule: When numerators are equal, compare denominators.

Larger denominator = smaller fraction
Smaller denominator = larger fraction

Example: Compare ¼, ½, ⅛
Denominators: 4, 2, 8
2 < 4 < 8, so: ½ > ¼ > ⅛

Think: More pieces (larger denominator) means smaller pieces!

Answer

Method: Convert to equivalent fractions with same denominator (LCD), then compare numerators.

Steps:

1. Find LCD of all denominators
2. Convert each fraction to equivalent fraction with LCD
3. Compare numerators

Example: Compare ⅔, ¾, 5/6
LCD = 12: 8/12, 9/12, 10/12
So: ⅔ < ¾ < 5/6

Answer

Method: Divide numerator by denominator.

For denominators 10, 100, 1000:

• 10: Move decimal 1 place left
• 100: Move decimal 2 places left
• 1000: Move decimal 3 places left

Examples:
3/10 = 0.3
25/100 = 0.25
125/1000 = 0.125

General: Use long division for other denominators.

Answer

Solution:

1. Convert to improper fractions:
   2⅓ = 7/3, 1¾ = 7/4
2. Find LCD of 3 and 4 = 12
3. Convert: 7/3 = 28/12, 7/4 = 21/12
4. Add: 28/12 + 21/12 = 49/12
5. Convert back: 49/12 = 4 1/12

Answer: 4 1/12

Answer

Solution:
¾ × 2½

1. Convert mixed to improper: 2½ = 5/2
2. Multiply: ¾ × 5/2 = (3×5)/(4×2) = 15/8
3. Convert to mixed: 15/8 = 1⅞

Answer: 1⅞ cups of flour

Check: 15 ÷ 8 = 1 remainder 7 = 1⅞ ✓

Answer

Solution:
Compare 5/8 and 7/12 using LCD method.

1. Find LCD of 8 and 12:
   Multiples of 8: 8, 16, 24...
   Multiples of 12: 12, 24...
   LCD = 24
2. Convert:
   5/8 = 15/24
   7/12 = 14/24
3. Compare: 15/24 > 14/24

Answer: 5/8 is larger than 7/12

Answer

Solution:
Let the number be x.
¾ × x = 450

Method: Divide by ¾ (multiply by reciprocal)
x = 450 ÷ ¾
x = 450 × 4/3
x = 1800/3
x = 600

Answer: The number is 600
Check: ¾ × 600 = 450 ✓

Answer

Proper Fractions:

• Numerator < Denominator
• Value < 1
• Examples: ½, ⅓, ⅝, 9/10

Improper Fractions:

• Numerator ≥ Denominator
• Value ≥ 1
• Examples: 5/4, 7/3, 9/9, 11/8

Key Difference: Proper fractions represent parts less than a whole, while improper fractions represent one whole or more.

Answer

Solution using prime factorization:

1. Find GCF of 36 and 48:
   36 = 2² × 3²
   48 = 2⁴ × 3¹
   GCF = 2² × 3¹ = 12
2. Divide both by GCF:
   36/48 ÷ 12/12 = 3/4

Alternative method:
36/48 ÷ 2 = 18/24 ÷ 2 = 9/12 ÷ 3 = 3/4

Answer: 3/4

Answer

Solution:

1. Write as fraction: 0.375 = 375/1000
2. Simplify by dividing by common factors:
   375/1000 ÷ 5 = 75/200
   75/200 ÷ 5 = 15/40
   15/40 ÷ 5 = 3/8

Answer: 3/8

Check: 3 ÷ 8 = 0.375 ✓