Study Guides/Maths/Average Age of Doctors and Lawyers Problem
Study Guide ¡ Maths

Average Age of Doctors and Lawyers is 40 — Solution

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.

Question (Click to Flip)

The average age of doctors and lawyers together is 40. Average of doctors alone is 35, lawyers alone is 50. Find the ratio.

Answer

Using alligation: Ratio of doctors to lawyers = (50−40) : (40−35) = 10 : 5 = 2 : 1. There are twice as many doctors as lawyers.

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Key Facts

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).

Problem Statement

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.

Solution — Using Weighted Average Formula

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

Solution — Using Alligation Method

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.

Verification

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

Key Concept — Alligation Rule

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.

Questions and Answers

The average age of doctors and lawyers together is 40. Average of doctors alone is 35, lawyers alone is 50. Find the ratio.+

Using alligation: Ratio of doctors to lawyers = (50−40) : (40−35) = 10 : 5 = 2 : 1. There are twice as many doctors as lawyers.

What is the alligation method?+

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.

If the ratio of doctors to lawyers is 2:1 and average age of doctors is 35 and lawyers is 50, what is the combined average?+

Combined average = (2×35 + 1×50) / (2+1) = (70+50)/3 = 120/3 = 40 years.

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