**Question 1: **Find the AM, GM and HM of the four values 3, 6, 24 and 48. Verify HM < GM < AM.

** Solution: **

Arithmetic Mean = $\frac{3+6+24+48}{4}$ = $\frac{81}{4}$ = 20.25

Geometric Mean = $\sqrt[4]{3\times 6\times 24\times 48}$ = 12

Harmonic Mean = $\frac{4}{\frac{1}{3}+\frac{1}{6}+\frac{1}{24}+\frac{1}{48}}$

= $\frac{64}{9}$ ≈ 7.11

The computed values verify HM < GM < AM.

**Question 2: **James, Janet, Jenny, Jack and Jerome work together and produce wooden
toys. If they can correspondingly complete making one toy in 4.5, 6, 6,
5.5 and 5 hrs, find the average number of toys completed in one hour. If
they work 8 hrs per day for 5 days in a week find the average number of
toys produced by a person in one week?

** Solution: **

Average rate of work is $\frac{1}{HM}$ of the five times given.

HM of 4.5, 6, 6, 5.5 and 5 = $\frac{5}{\frac{1}{4.5}+\frac{1}{6}+\frac{1}{6}+\frac{1}{5.5}+\frac{1}{5}}$

= $\frac{5}{\frac{2}{9}+\frac{1}{6}+\frac{1}{6}+\frac{2}{11}+\frac{1}{5}}$

= $\frac{5\times 990}{2\times 110+165+165+2\times 90+198}$

= $\frac{4950}{928}$ = $\frac{2475}{464}$

Average Number of units produce in one hr = $\frac{1}{HM}$ = $\frac{464}{2475}$ ≈ 0.1875

Average number of toys produced by a person in one week = 5 x 8 x 0.1875 = 7.5

**Question 3: **An airplane covers a distance of 1000 Km in two phases. During the
first phase it averaged a speed of 400 Km/hr and during the second phase
the average speed was 600 Km/hr. Find the average speed for the entire
trip.

** Solution: **

Average speed of the trip is the Harmonic Mean of the average speeds of the two phases.

Harmonic Mean of the two average speeds = $\frac{2}{\frac{1}{400}+\frac{1}{600}}$

= $\frac{2\times 1200}{3+2}$ = $\frac{2400}{5}$ = 480 Km/hr

The average speed of the plane for the entire trip = 480 Km/hr.