Ratio, Proportion, Average, and Mixture
Key Takeaways
- Ratio questions become simple when the total parts are converted into one part before finding individual shares.
- Direct proportion changes in the same direction, while inverse proportion changes in opposite directions.
- Average is a balance point; total sum, count, and weighted contribution must stay consistent.
- Mixture questions can often be solved through weighted average or alligation instead of lengthy equations.
Ratios Are Compressed Comparisons
A ratio compares quantities in the same unit. If two amounts are in the ratio 3:5, they are not necessarily 3 and 5; they are 3x and 5x. This single idea solves most RRB NTPC ratio questions. Convert the ratio into parts, find the value of one part, then build the required value.
Always check units before forming a ratio. Minutes and hours, rupees and paise, kilograms and grams, or metres and centimetres must be converted to the same unit. Many wrong answers come from comparing numbers that look compatible but are not.
Ratio Method
| Question clue | Method |
|---|---|
| Divide total in a ratio | Add parts, find one part, multiply |
| Given difference | Subtract parts, match difference, find one part |
| Given one share | Match that share to its ratio part |
| Change in ratio | Create old and new expressions with same variable |
| Three-way ratio | Use common middle term if needed |
For three ratios such as A:B = 2:3 and B:C = 4:5, make B common. The first ratio has B as 3, the second has B as 4. LCM of 3 and 4 is 12. So A:B becomes 8:12 and B:C becomes 12:15, giving A:B:C = 8:12:15.
Proportion and Variation
A proportion says two ratios are equal. Use cross multiplication: a/b = c/d means ad = bc. This is useful for scale maps, unit rates, wages, and sharing problems.
Direct proportion means both quantities move together. More items cost more money at the same rate. More distance takes more time at the same speed. Inverse proportion means one rises as the other falls. More workers need fewer days for the same work. More speed needs less time for the same distance.
Before calculating, ask whether the answer should be bigger or smaller. If 12 workers take 15 days, then 20 workers should take fewer days, not more. This size check catches inverse-proportion mistakes.
Average as Total Divided by Count
Average is simple only when every item has the same weight. The formula is total sum divided by number of items. Rearranged, total sum equals average x count. Most exam questions use this rearrangement.
If the average of a group changes when a new member enters, work with totals. If 10 people have average 42, their total is 420. If one person joins, add that value and divide by 11. If one person leaves, subtract that value and divide by 9. Avoid guessing from the difference in averages unless you know the shortcut exactly.
Average Traps
| Trap | Correction |
|---|---|
| Averaging averages directly | Use weighted totals if group sizes differ |
| Forgetting count change | Add or remove the person before dividing |
| Treating zero as no item | Zero is a value and counts if included |
| Mixing units | Convert units before averaging |
| Ignoring replacement | Subtract outgoing value and add incoming value |
Weighted average appears in marks, wages, prices, and mixture. If one group of 20 students averages 60 and another group of 30 averages 70, the combined average is not 65. It is (20 x 60 + 30 x 70) / 50 = 66.
Mixture and Alligation
Mixture questions are weighted-average questions with ingredients. They may involve milk and water, two qualities of rice, two interest rates, or two price grades. The final concentration or price must lie between the two original values.
Alligation gives a fast ratio. If cheaper value is C, dearer value is D, and mean is M, then quantity ratio of dearer to cheaper is (M - C):(D - M). The distances from the mean decide the opposite quantities. This opposite placement is the part candidates often reverse.
Example structure: one liquid has 20 percent acid and another has 50 percent acid. To get 35 percent, the mean is exactly midway, so the quantities are equal. If the target were closer to 20 percent, the mixture would need more of the 20 percent liquid.
Replacement Mixtures
A tougher mixture type removes some mixture and replaces it with another liquid. If a container has pure milk and one-fourth is removed and replaced with water, three-fourths of the milk remains. If the same operation is repeated, milk becomes (3/4) x (3/4) = 9/16 of the original. Use the remaining fraction repeatedly.
The replacement formula is: final quantity of original liquid = initial quantity x (1 - removed/total)^n. Use it only when the mixture is well mixed before each removal and the same fraction is removed each time.
Exam Strategy
For ratio and average, write the total relationship first. For mixture, check whether the answer lies between the two given values. For alligation, label cheaper, dearer, and mean before writing the ratio. For inverse proportion, do a size check.
Do not overuse algebra. Many NTPC questions can be solved with assumed totals, parts, or weighted sums. The best method is the one that keeps the total visible and reduces copying errors.
Drill Routine
Practice ten ratio-share questions, ten average-change questions, and ten mixture questions in one sitting. Then review which tool solved each one: parts, cross multiplication, total sum, weighted average, or alligation. The purpose is to recognize the structure quickly in the CBT, not to force one method on every problem.
Two grades of grain cost Rs 36 per kg and Rs 48 per kg. In what ratio should they be mixed to obtain a mixture worth Rs 40 per kg?