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

What are the three types of fractions and how do you identify each type?

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

How do you convert a mixed fraction to an improper fraction?

**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

How do you convert an improper fraction to a mixed fraction?

**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

What are equivalent fractions and how do you create them?

**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!

How do you add fractions with the same denominator?

**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)

How do you add fractions with different denominators?

**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

What is the LCD and how do you find it?

**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)

How do you subtract fractions with different denominators?

**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

How do you multiply fractions?

**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!

How do you multiply a fraction by a whole number?

**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

How do you divide fractions using the reciprocal method?

**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

What is the cross multiply method for dividing fractions?

**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

How do you simplify or reduce fractions to lowest terms?

**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

How do you compare fractions with the same denominator?

**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)

How do you compare fractions with the same numerator?

**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!

How do you compare fractions with different numerators and denominators?

**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

How do you convert fractions to decimals?

**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.

Solve: 2⅓ + 1¾ =

**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**

A recipe calls for ¾ cup of flour. If you want to make 2½ times the recipe, how much flour do you need?

**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 ✓

Which is larger: 5/8 or 7/12? Show your work.

**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**

If 3/4 of a number is 450, what is the number?

**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 ✓

What is the difference between proper and improper fractions? Give examples of each.

**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.

Simplify: 36/48

**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**

Convert 0.375 to a fraction in lowest terms.

**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 ✓