Fractions and Decimals
"Fractions and decimals are two sides of the same coin — every fraction can be written as a decimal, and every terminating decimal can be written as a fraction."
1. Multiplication of Fractions
Multiplying a Fraction by a Whole Number
'Multiply the numerator by the whole number. Keep the denominator the same.' (a/b) × c = (a × c)/b.
Example: (3/7) × 5 = (3 × 5)/7 = 15/7 = 2¹/₇.
Multiplying a Fraction by Another Fraction
'Multiply numerator × numerator and denominator × denominator. Simplify if possible.' (a/b) × (c/d) = (a × c)/(b × d).
Example: (2/3) × (4/5) = (2 × 4)/(3 × 5) = 8/15.
Example: (5/8) × (16/25) = (5 × 16)/(8 × 25) = 80/200 = 2/5 (after simplification).
Multiplying Three or More Fractions
Multiply all numerators together and all denominators together.
Example: (1/2) × (3/4) × (2/5) = (1 × 3 × 2)/(2 × 4 × 5) = 6/40 = 3/20.
Fraction as an Operator 'OF'
'OF means MULTIPLY.' (1/2) of 20 = (1/2) × 20 = 10. (3/4) of 60 = (3/4) × 60 = 45.
Real-world example: 'A recipe needs (2/3) cup of sugar. You want to make 3 batches.' Sugar needed = 3 × (2/3) = 2 cups. 'AP farmers: if 3/5 of a 25-acre plot is used for paddy, what area is under paddy?' (3/5) × 25 = 15 acres.
2. Division of Fractions
Reciprocal of a Fraction
The reciprocal of a fraction (a/b) is (b/a). Their product is always 1. (2/3) × (3/2) = 1. 'Zero has NO reciprocal because 1/0 is undefined.'
Dividing a Fraction by a Whole Number
'Multiply the fraction by the reciprocal of the whole number.' (a/b) ÷ c = (a/b) × (1/c) = a/(b × c).
Example: (4/9) ÷ 2 = (4/9) × (1/2) = 4/18 = 2/9.
Dividing a Fraction by Another Fraction
'Multiply the first fraction by the reciprocal of the second fraction.' (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d)/(b × c).
| Example | Steps | Result |
|---|---|---|
| (2/3) ÷ (5/7) | (2/3) × (7/5) | 14/15 |
| (9/4) ÷ (3/8) | (9/4) × (8/3) = 72/12 | 6 |
| (5/12) ÷ (10/3) | (5/12) × (3/10) = 15/120 | 1/8 |
Dividing a Whole Number by a Fraction
c ÷ (a/b) = c × (b/a) = (c × b)/a.
Example: 8 ÷ (2/5) = 8 × (5/2) = 40/2 = 20. 'How many 1/2 kg packets can you make from 12 kg of rice?' 12 ÷ (1/2) = 12 × 2 = 24 packets.
3. Multiplication of Decimals
Rule for Multiplying Decimals
'Multiply the numbers as if they were whole numbers. Count the TOTAL number of decimal places in both factors. Place the decimal point that many places from the RIGHT in the product.'
| Problem | Multiply ignoring decimals | Total decimal places | Final answer |
|---|---|---|---|
| 2.5 × 3.4 | 25 × 34 = 850 | 1 + 1 = 2 | 8.50 = 8.5 |
| 0.06 × 0.4 | 6 × 4 = 24 | 2 + 1 = 3 | 0.024 |
| 1.25 × 2.4 | 125 × 24 = 3000 | 2 + 1 = 3 | 3.000 = 3 |
Real-world example: 'Cost of 1 kg of AP tomatoes is Rs 24.50. Cost of 2.5 kg = 24.50 × 2.5.' 2450 × 25 = 61250. Decimal places: 2 + 1 = 3. Answer: Rs 61.250 = Rs 61.25.
Multiplication by 10, 100, 1000
'Shift the decimal point to the RIGHT by the number of zeros.' 3.456 × 10 = 34.56. 3.456 × 100 = 345.6. 3.456 × 1000 = 3456.
4. Division of Decimals
Dividing a Decimal by a Whole Number
'Perform normal division. Place the decimal point in the quotient directly above the decimal point in the dividend.'
Example: 24.6 ÷ 6 = 4.1. 48.24 ÷ 8 = 6.03.
Dividing a Decimal by Another Decimal
'Shift the decimal point in the divisor to the RIGHT to make it a whole number. Shift the decimal point in the dividend by the SAME number of places. Then divide.'
| Problem | Step 1: Shift divisor | Step 2: Shift dividend | Perform division | Answer |
|---|---|---|---|---|
| 6.25 ÷ 2.5 | 2.5 → 25 (×10) | 6.25 → 62.5 (×10) | 62.5 ÷ 25 | 2.5 |
| 0.084 ÷ 0.07 | 0.07 → 7 (×100) | 0.084 → 8.4 (×100) | 8.4 ÷ 7 | 1.2 |
| 15.6 ÷ 0.12 | 0.12 → 12 (×100) | 15.6 → 1560 (×100) | 1560 ÷ 12 | 130 |
Division by 10, 100, 1000
'Shift the decimal point to the LEFT by the number of zeros.' 345.6 ÷ 10 = 34.56. 345.6 ÷ 100 = 3.456. 345.6 ÷ 1000 = 0.3456.
5. Converting Between Fractions and Decimals
Fraction → Decimal
'Divide the numerator by the denominator.'
| Fraction | Division | Decimal | Type |
|---|---|---|---|
| 3/4 | 3 ÷ 4 = 0.75 | 0.75 | Terminating |
| 1/3 | 1 ÷ 3 = 0.333... | 0.3̅ | Repeating (non-terminating) |
| 7/8 | 7 ÷ 8 = 0.875 | 0.875 | Terminating |
| 2/7 | 2 ÷ 7 = 0.285714... | 0.2̅8̅5̅7̅1̅4̅ | Repeating |
Decimal → Fraction
'Write the decimal as a fraction with denominator as a power of 10. Simplify.'
| Decimal | Fraction | Simplified |
|---|---|---|
| 0.5 | 5/10 | 1/2 |
| 0.75 | 75/100 | 3/4 |
| 0.125 | 125/1000 | 1/8 |
| 2.35 | 235/100 | 47/20 = 2⁷/₂₀ |
6. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| (2/5) × 3 = 2/15 | Forgot to multiply numerator | (2 × 3)/5 = 6/5 |
| (3/4) ÷ (1/2) = 3/2 × NOT taking reciprocal | Must use reciprocal of second fraction | (3/4) × (2/1) = 6/4 = 3/2 |
| 0.3 × 0.2 = 0.6 | Wrong decimal placement | 3 × 2 = 6. Two decimal places → 0.06 |
| 2.5 ÷ 0.5 = 0.5 | Didn't shift decimals properly | 2.5 ÷ 0.5 = 25 ÷ 5 = 5 |
7. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Fraction multiplication | 2-3 | Direct computation |
| Fraction division | 2-3 | Direct computation |
| Decimal multiplication | 2-3 | Word problems |
| Decimal division | 2-3 | Word problems |
| Fraction-decimal conversion | 1-2 | Fill in the blanks |
Quick Self-Test
Q1. Multiply: (7/12) × (4/21). A1. (7 × 4)/(12 × 21) = 28/252 = 1/9.
Q2. Divide: (15/8) ÷ (5/4). A2. (15/8) × (4/5) = 60/40 = 3/2 = 1¹/₂.
Q3. A rope of length 15.6 m is cut into 12 equal pieces. Find the length of each piece. A3. 15.6 ÷ 12 = 1.3 m.
Q4. Convert 7/20 to a decimal. A4. 7 ÷ 20 = 0.35.
Q5. Convert 2.625 to a fraction. A5. 2625/1000 = 21/8 = 2⁵/₈.
Q6. Find the product: 0.75 × 0.04. A6. 75 × 4 = 300. Decimal places: 2 + 2 = 4. Answer: 0.03.
Q7. How many pieces of length 0.25 m can be cut from a 6 m long ribbon? A7. 6 ÷ 0.25 = 600 ÷ 25 = 24 pieces.
