4. Ratio, proportion and variation
Free study material · concepts, shortcuts & solved questions
A ratio compares quantities in the same units. If a:b=3:5, total parts are 8 and the shares of T are 3T/8 and 5T/8. In a proportion a/b=c/d, cross multiplication gives ad=bc. Direct variation has y=kx; inverse variation has xy=k.
Method before formula Reduce ratios before using them. If units differ, convert first. In partnership and mixture questions, a ratio often represents capital × time or quantity, not the final answer itself. |
Formula and decision table
Model | Worked setup |
Divide 720 in 5:7 | Parts=12; shares=300 and 420 |
Direct variation | If y=12 when x=3, k=4, so y=4x |
Inverse variation | If 6 workers take 10 days, 12 workers take 5 days, assuming equal efficiency |
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 |
Divide 720 in ratio 5:7. Larger share: | Larger share=720×7/12=420. | C. 420 |
If x:y=4:9 and x=20, y is: | Scale factor is 5, so y=45. | B. 45 |
For inverse variation, which remains constant? | The product remains constant. | C. xy |
What an examiner is testing
The options around Ratio, proportion and variation 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: Divide 720 in ratio 5:7. Larger share: 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 420, because Larger share=720×7/12=420.
Example 2: If x:y=4:9 and x=20, y is: 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 45, because Scale factor is 5, so y=45.
Example 3: For inverse variation, which remains constant? 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 xy, because The product remains constant.
Chapter practice
1. Divide 720 in ratio 5:7. Larger share:
(A) 300 (B) 360 (C) 420 (D) 480
Answer: C. 420 | Explanation: Larger share=720×7/12=420.
2. If x:y=4:9 and x=20, y is:
(A) 36 (B) 45 (C) 54 (D) 60
Answer: B. 45 | Explanation: Scale factor is 5, so y=45.
3. For inverse variation, which remains constant?
(A) x+y (B) x−y (C) xy (D) x/y
Answer: C. xy | Explanation: The product remains constant.
SSC CGL speed and trap clinic
For Ratio, proportion and variation, 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 Ratio, proportion and variation? | 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 Ratio, proportion and variation 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. |