5.2 Fractions & Decimals

5.2 Fractions & Decimals

Quick Answer: Treat a fraction as parts of a whole, simplify early, use a common denominator for +/−, multiply across for ×, and multiply by the reciprocal for ÷. Decimals follow place value around the point; convert between fractions and decimals by dividing or by known equivalents (1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2).

Fractions and decimals appear throughout civil-service Mathematics items: shared costs, partial quantities, money in dollars and cents, and “what portion remains” questions. Stay in general arithmetic—do not import firefighting technical formulas.

Fraction vocabulary

• Numerator (top): how many parts you have.
• Denominator (bottom): how many equal parts make one whole. Denominator cannot be 0.
• Proper fraction: numerator < denominator (3/5).
• Improper fraction: numerator ≥ denominator (9/4).
• Mixed number: whole number plus proper fraction (2 1/4 = 9/4).
• Equivalent fractions: same value, different writing (1/2 = 2/4 = 3/6).

Simplifying. Divide numerator and denominator by the same positive whole number (ideally the greatest common factor). 18/24 ÷ 6/6 = 3/4.

Example — class attendance. In a class of 28, 21 students are present. Fraction present: 21/28 = 3/4 after dividing by 7. Fraction absent: 7/28 = 1/4. Check: 3/4 + 1/4 = 1 (the whole class).

Comparing fractions

1. Same denominator → larger numerator is larger (5/9 > 2/9).
2. Same numerator → larger denominator is smaller each piece (1/3 > 1/5).
3. Otherwise convert to a common denominator or to decimals.

Example. Compare 2/3 and 3/5. Common denominator 15: 2/3 = 10/15, 3/5 = 9/15 → 2/3 is larger.

Adding and subtracting fractions

Same denominator: add or subtract numerators; keep the denominator; simplify. 5/12 + 1/12 = 6/12 = 1/2.

Different denominators: rewrite with a least common denominator (LCD), then +/−.

Worked example A — different denominators

Compute 2/3 + 1/4.

1. LCD of 3 and 4 is 12.
2. 2/3 = 8/12; 1/4 = 3/12.
3. 8/12 + 3/12 = 11/12.

Trap: adding denominators (2/3 + 1/4 → 3/7) is never correct for fraction addition.

Worked example B — mixed numbers

A recipe uses 1 1/2 cups of flour in the morning and 2 1/3 cups in the afternoon. Total flour:

1. 1 1/2 = 3/2; 2 1/3 = 7/3.
2. LCD of 2 and 3 is 6: 3/2 = 9/6; 7/3 = 14/6.
3. 9/6 + 14/6 = 23/6 = 3 5/6 cups.

Worked example C — subtraction with borrowing

5 − 2 3/4 = 5/1 − 11/4. LCD 4: 20/4 − 11/4 = 9/4 = 2 1/4.

Or: 5 = 4 4/4; 4 4/4 − 2 3/4 = 2 1/4. Same result.

Multiplying fractions

Multiply numerators; multiply denominators; simplify (before or after).

Example. (2/5) × (3/4) = 6/20 = 3/10.

Whole number × fraction: 6 × (2/3) = 12/3 = 4. Meaning: six groups of two-thirds, or two-thirds of six—same product.

Of means multiply in fraction language: “three-quarters of 40” → (3/4) × 40 = 30.

Worked example D — discount-style wording without percent symbols

A shop states that 3/5 of a $50 voucher remains. Remaining value: (3/5) × 50 = $30. Used amount: 50 − 30 = $20, which is also (2/5) × 50.

Dividing fractions

To divide by a fraction, multiply by its reciprocal (flip numerator and denominator).

Rule: a/b ÷ c/d = a/b × d/c.

Worked example E — division

3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2.

Sense check: how many halves fit into three-quarters? One and a half halves—yes.

Worked example F — sharing a strip of ribbon

You have 2/3 metre of ribbon and cut pieces of 1/6 metre each. Number of pieces: (2/3) ÷ (1/6) = (2/3) × 6 = 4 pieces.

Decimal place value

Money link. In TT dollars, $12.45 means 12 dollars and 45 cents; the 4 is tenths of a dollar (10-cent units) only when you think carefully in decimal dollars—cents are hundredths of a dollar. Practise: $0.05 is five cents; $1.05 is one dollar five cents.

Adding and subtracting decimals

Line up the decimal points. Fill empty places with zeros if helpful.

  12.50
+  3.75
  -----
  16.25

Example. Fuel receipts: $84.90 + $16.25 = $101.15. Subtraction: $50.00 − $27.35 = $22.65.

Multiplying and dividing decimals

Multiply as whole numbers, then place the decimal point so the total number of decimal digits in the factors equals the count in the product.

Example. 1.2 × 0.4 → 12 × 4 = 48; two decimal digits total → 0.48.

Divide by moving the decimal in the divisor to make it a whole number, and move the dividend’s decimal the same number of places.

Example. 4.5 ÷ 0.9 → multiply both by 10 → 45 ÷ 9 = 5.

Converting among fractions, decimals, and percents

Fraction → decimal: divide numerator by denominator. 3/8 = 0.375.

Decimal → fraction: 0.6 = 6/10 = 3/5; 0.25 = 25/100 = 1/4.

Example — water bill portion. A household used 0.6 of its monthly water allocation. As a fraction: 0.6 = 3/5. Remaining: 1 − 0.6 = 0.4 = 2/5.

Common trap table (fractions & decimals)

Multi-step civilian example

A group of 8 friends shares a restaurant bill of $360 equally, then each person also pays 1/4 of a $40 tip pool.

1. Food share each: 360 ÷ 8 = $45.
2. Tip pool: $40; each pays (1/4) only if the stem says each pays one-quarter of the tip—that would be $10 each and total tip paid 8 × 10 = $80, which contradicts a $40 pool. Read carefully.
3. Correct equal tip split of $40 among 8: 40 ÷ 8 = $5 each. Total each: 45 + 5 = $50.

If the stem instead says “the tip is 1/4 of the food bill,” tip = (1/4) × 360 = $90; each tip share 90 ÷ 8 = $11.25; total each 45 + 11.25 = $56.25. Words control the expression.

Practice habits

• Simplify fractions before multiplying when possible to keep numbers small.
• After +/− of fractions, always check whether the answer simplifies.
• For decimals, estimate first: 9.8 × 2.1 ≈ 10 × 2 = 20; exact is 20.58—reject 2.058 or 205.8 as place errors.
• Convert to a convenient form: comparing 0.66 and 2/3 is easier as 0.66 vs 0.666… or 66/100 vs 2/3.

Mastering fractions and decimals unlocks the next section on percentages, which are simply hundredths written with a % sign.

Section checkpoint

You should simplify fractions, perform all four operations, align decimals, convert among forms, and spot traps like adding denominators or misplacing decimal points. Keep contexts civilian—budgets, recipes, class shares—not technical fireground arithmetic.