Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।
Exam strategyपरीक्षा रणनीतिRarely taughtकम सिखाई जाने वाली
Back-solve from the middle option
Use it when:कब इस्तेमाल करें: The question is easy to check but slow to set up as an equation (ages, mixtures, “find the number”).
Options on this site are in ascending order (as in most printed papers), so start with B or C.
Plug it into the question. Too big? The answer is above it. Too small? Below.
At most two checks find the answer.
Exampleउदाहरण
The father is 3 times his son’s age; in 12 years he will be twice the son’s age. Son’s age? (a) 10 (b) 12 (c) 14 (d) 16
→ Try 12: father 36; in 12 years 48 and 24 → 2 times. Done.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: Checking a candidate is a few multiplications; with sorted options each check halves the choices, so two checks cover four options.
Limits:सीमाएँ: Not for “cannot be determined” options or questions with several unknowns. Write the equation then.
Solved examplesहल किए गए उदाहरण
Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।
Example 1उदाहरण 1 · Mixtures and alligation · Easyआसान
In what ratio must a grocer in Bhopal mix basmati rice costing ₹35/kg with basmati rice costing ₹55/kg so that the mixture costs ₹37/kg?
9 : 1
19 : 2
7 : 11
1 : 9
Answer:उत्तर:9 : 1
Cheaper : dearer = (55 − 37) : (37 − 35)
= 18 : 2 = 9 : 1.
Example 2उदाहरण 2 · Mixtures and alligation · Mediumमध्यम
In what ratio must a grocer in Nagpur mix sugar costing ₹65/kg with sugar costing ₹95/kg so that the mixture costs ₹70/kg?
1 : 5
13 : 19
5 : 1
26 : 5
Answer:उत्तर:5 : 1
Cheaper : dearer = (95 − 70) : (70 − 65)
= 25 : 5 = 5 : 1.
Example 3उदाहरण 3 · Mixtures and alligation · Hardकठिन
A container has 50 litres of milk at a dairy in Anand. 5 litres are taken out and replaced with water. This is done 2 times in all. How many litres of milk are left?
32.5 litres
36.5 litres
40.5 litres
48.5 litres
Answer:उत्तर:40.5 litres
Fraction kept each time = 1 − 5/50 = 0.9.
Milk left = 50 × (0.9)^2 = 40.5 litres.
Example 4उदाहरण 4 · Mixtures and alligation · Mediumमध्यम
In what ratio must a grocer in Bhopal mix sugar costing ₹60/kg with sugar costing ₹70/kg so that the mixture costs ₹62/kg?
6 : 7
4 : 1
1 : 4
9 : 2
Answer:उत्तर:4 : 1
Cheaper : dearer = (70 − 62) : (62 − 60)
= 8 : 2 = 4 : 1.
Example 5उदाहरण 5 · Mixtures and alligation · Mediumमध्यम
In what ratio must a grocer in Patna mix tea costing ₹45/kg with tea costing ₹75/kg so that the mixture costs ₹68/kg?
8 : 23
3 : 5
7 : 23
23 : 7
Answer:उत्तर:7 : 23
Cheaper : dearer = (75 − 68) : (68 − 45)
= 7 : 23.
Example 6उदाहरण 6 · Mixtures and alligation · Mediumमध्यम
In what ratio must a grocer in Pune mix wheat costing ₹55/kg with wheat costing ₹70/kg so that the mixture costs ₹65/kg?
2 : 1
1 : 2
11 : 14
3 : 5
Answer:उत्तर:1 : 2
Cheaper : dearer = (70 − 65) : (65 − 55)
= 5 : 10 = 1 : 2.
Common mistakesआम गलतियाँ
Putting the ratio the wrong way round.
Using simple subtraction for repeated replacement instead of the power formula.
Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न
What is the rule of alligation?
Quantity of cheaper : quantity of dearer = (dearer − mean) : (mean − cheaper).
Does alligation work for averages and percentages too?
Yes, it works for any weighted average: speeds, marks, profit percentages and concentrations.
How do I solve milk-and-water replacement questions?