2. Fractions, decimals and surds
Free study material · concepts, shortcuts & solved questions
Convert a decimal to a fraction by place value and reduce it. For operations with fractions, use a common denominator; for division, multiply by the reciprocal. A terminating decimal in lowest form has a denominator containing only powers of 2 and/or 5. Surds are irrational roots such as √2; simplify by extracting perfect-square factors.
Method before formula Rationalise only when required. √a × √b = √(ab), but √a + √b cannot be combined unless the surds are like terms. In approximation questions, square nearby values rather than expanding blindly. |
Formula and decision table
Concept | Method |
0.375 | 375/1000 = 3/8 |
√72 | √(36×2)=6√2 |
1/(√a+√b) | Multiply by conjugate √a−√b |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
0.125 equals: | 0.125=125/1000=1/8. | B. 1/8 |
√72 simplifies to: | √72=√36×√2=6√2. | B. 6√2 |
A terminating decimal has reduced denominator made of: | Only factors 2 and 5 permit termination. | B. 2 and/or 5 |
What an examiner is testing
The options around Fractions, decimals and surds usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: 0.125 equals: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 1/8, because 0.125=125/1000=1/8.
Example 2: √72 simplifies to: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 6√2, because √72=√36×√2=6√2.
Example 3: A terminating decimal has reduced denominator made of: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 2 and/or 5, because Only factors 2 and 5 permit termination.
Chapter practice
1. 0.125 equals:
(A) 1/4 (B) 1/8 (C) 3/8 (D) 5/8
Answer: B. 1/8 | Explanation: 0.125=125/1000=1/8.
2. √72 simplifies to:
(A) 3√8 (B) 6√2 (C) 8√2 (D) 9√2
Answer: B. 6√2 | Explanation: √72=√36×√2=6√2.
3. A terminating decimal has reduced denominator made of:
(A) 2 and 3 only (B) 2 and/or 5 (C) 3 and 7 only (D) Any prime
Answer: B. 2 and/or 5 | Explanation: Only factors 2 and 5 permit termination.
SSC CGL speed and trap clinic
For Fractions, decimals and surds, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Fractions, decimals and surds? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Fractions, decimals and surds question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |