This is a classic weighted average (alligation) problem. The average age of doctors and lawyers together is 40 years. If the average age of doctors alone is 35 years and that of lawyers alone is 50 years, we need to find the ratio of doctors to lawyers.
Given: Combined average = 40, Doctors avg = 35, Lawyers avg = 50.
Ratio of doctors to lawyers = 2 : 1.
Method 1: Equation → 35d + 50l = 40(d+l) → d:l = 2:1.
Method 2: Alligation → (50−40):(40−35) = 10:5 = 2:1.
Verification: 2 doctors + 1 lawyer → (70+50)/3 = 120/3 = 40 ✓.
Alligation rule: ratio = (B's avg − combined) : (combined − A's avg).
The average age of doctors and lawyers together is 40 years. Average age of doctors alone = 35 years Average age of lawyers alone = 50 years
Find: The ratio of the number of doctors to the number of lawyers.
Let number of doctors = d, number of lawyers = l
Total age of doctors = 35d Total age of lawyers = 50l
Combined average = 40: (35d + 50l) / (d + l) = 40
35d + 50l = 40(d + l) 35d + 50l = 40d + 40l 50l − 40l = 40d − 35d 10l = 5d d/l = 10/5 = 2/1
Ratio of doctors to lawyers = 2 : 1
Alligation is a shortcut for finding ratios in mixture/average problems:
Doctors (35) Lawyers (50)
\ /
\ /
Combined avg (40)
/ \
/ \
50 − 40 = 10 40 − 35 = 5
Ratio of doctors : lawyers = (50 − 40) : (40 − 35) = 10 : 5 = 2 : 1
Ratio = 2 : 1 For every 2 doctors, there is 1 lawyer.
Let doctors = 2, lawyers = 1
Total age = (2 × 35) + (1 × 50) = 70 + 50 = 120 Total people = 2 + 1 = 3 Combined average = 120 / 3 = 40 ✓
The answer is verified: Ratio of doctors to lawyers = 2 : 1
The Alligation rule states:
Quantity A / Quantity B = (Average of B − Combined Average) / (Combined Average − Average of A)
Here: Doctors / Lawyers = (Avg of Lawyers − Combined) / (Combined − Avg of Doctors) = (50 − 40) / (40 − 35) = 10 / 5 = 2 / 1
This method works for any mixture/average problem where two groups combine to form a combined average.
Using alligation: Ratio of doctors to lawyers = (50−40) : (40−35) = 10 : 5 = 2 : 1. There are twice as many doctors as lawyers.
Alligation is a method to find the ratio in which two quantities with different averages are mixed to get a combined average. Formula: Ratio of A to B = (B's value − mean) : (mean − A's value). It is faster than forming equations for mixture and average problems.
Combined average = (2×35 + 1×50) / (2+1) = (70+50)/3 = 120/3 = 40 years.
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