Decimal fractions and simplification
Simplify in BODMAS order: brackets, of, division, multiplication, addition, subtraction. Turn recurring decimals into fractions before you calculate.
Rules to remember
| Rule | What to do | Example |
|---|---|---|
| Order (VBODMAS) | Bar, brackets ( ) { } [ ], of, then ÷ and × from left to right, then + and − from left to right | 10 − 2 × 3 = 4, not 24 |
| 'Of' | Multiply, and do it before division | 48 ÷ 4 of 3 = 48 ÷ 12 = 4 |
| Terminating decimal | Write over 10, 100, 1,000 … and reduce | 0.375 = 375/1,000 = 3/8 |
| Pure recurring decimal | Repeating digits over as many 9s | 0.444… = 4/9; 0.3636… = 36/99 = 4/11 |
| Mixed recurring decimal | (All digits − non-repeating digits) over 9s for repeating digits, then 0s for non-repeating digits | 0.1666… = (16 − 1)/90 = 15/90 = 1/6 |
| Multiplying decimals | Multiply as whole numbers; decimal places in the answer = total places in the factors | 0.12 × 0.3 = 0.036 |
| Dividing decimals | Shift the point equally in both numbers until the divisor is whole | 1.44 ÷ 0.12 = 144 ÷ 12 = 12 |
| Comparing fractions | Change each to a decimal | 5/8 = 0.625 is more than 3/5 = 0.6 |
Worked example 1: recurring decimal to fraction
Express 0.74545… (45 repeating) as a fraction in its lowest terms.
- Standard method (algebra): let x = 0.74545… Then 10x = 7.4545… and 1,000x = 745.4545…
- Subtract: 1,000x − 10x = 745.4545… − 7.4545…, so 990x = 738 and x = 738/990.
- Reduce: divide top and bottom by 18 to get 41/55.
- Shortcut (rule): (all digits − non-repeating digits) ÷ (two 9s for '45', one 0 for '7') = (745 − 7)/990 = 738/990 = 41/55.
- Check: 41 ÷ 55 = 0.74545…
41/55.
Worked example 2: decimals with an algebra pattern
Simplify (0.6 × 0.6 × 0.6 + 0.4 × 0.4 × 0.4) ÷ (0.6 × 0.6 − 0.6 × 0.4 + 0.4 × 0.4).
- Standard method: numerator = 0.216 + 0.064 = 0.280; denominator = 0.36 − 0.24 + 0.16 = 0.28.
- 0.28 ÷ 0.28 = 1.
- Shortcut (identity): the pattern is (a³ + b³) ÷ (a² − ab + b²) with a = 0.6 and b = 0.4. Since a³ + b³ = (a + b)(a² − ab + b²), the answer is a + b = 0.6 + 0.4 = 1.
- Check: both parts equal 0.28, so the quotient is exactly 1.
1.
Worked example 3: BODMAS with brackets and 'of'
Simplify: 48 ÷ 4 of 3 + 18 − [6 + {12 − (9 − 5)}].
- Standard method: innermost bracket (9 − 5) = 4; curly bracket {12 − 4} = 8; square bracket [6 + 8] = 14.
- 'Of' before division: 4 of 3 = 12, so 48 ÷ 12 = 4.
- Left to right: 4 + 18 − 14 = 8.
- Shortcut (open all brackets at once, flipping signs inside a bracket that follows a minus): 6 + 12 − 9 + 5 = 14. Then 48 ÷ 12 + 18 − 14 = 8.
- Check: doing 48 ÷ 4 first would give 12 × 3 = 36 and a wrong total of 40. 'Of' comes first.
8.
Worked example 4: approximation (bank-exam style)
Find the approximate value of 24.97 × 16.02 + 299.89 ÷ 5.98.
- Standard method (exact): 24.97 × 16.02 = 400.0194 and 299.89 ÷ 5.98 ≈ 50.149, so the total is about 450.17.
- Shortcut (round each number to a friendly value): 25 × 16 + 300 ÷ 6 = 400 + 50 = 450.
- Check: the exact value, about 450.17, rounds to 450, so the rounding did not mislead.
About 450.
Bank and insurance prelims open with simplification and approximation sets: long decimal expressions where you round each number and pick the nearest option. SSC and railway papers test the BODMAS order and recurring-decimal-to-fraction conversion.
2 marks: “Express 0.363636… as a fraction in its lowest terms.” Two repeating digits go over 99: 36/99 = 4/11.
5 marks: “State the BODMAS rule with an example, and simplify 5 + 4 of 3 ÷ 6 − 2.” Explain the order (bar, brackets, of, ÷ and ×, + and −). Working: 4 of 3 = 12; 12 ÷ 6 = 2; 5 + 2 − 2 = 5.
10 marks: “(a) Convert 0.1666… and 0.2727… into fractions. (b) Simplify 48 ÷ 4 of 3 + 18 − [6 + {12 − (9 − 5)}]. (c) Find the approximate value of 24.97 × 16.02 + 299.89 ÷ 5.98.” (a) 1/6 and 3/11; (b) 8; (c) about 450. Show every step and one check line for each part.
V-B-O-D-M-A-S: bar, brackets, of, then ÷ and × left to right, then + and − left to right. 'Of' beats ÷. Repeating digits go over 9s; non-repeating digits add 0s.