1. During one year the ratio of milk prices per quart to bread prices per loaf was 3.00,1 answer below »

1. During one year the ratio of milk prices per quart to bread prices per loaf was 3.00, whereas during the next year the ratio was 2.00.

(a) Find the arithmetic mean of these ratios for the 2-year period.

(b) Find the arithmetic mean of the ratios of bread prices to milk prices for the 2-year period.

(c) Discuss the advisability of using the arithmetic mean for averaging ratios.

(d) Discuss the suitability of the geometric mean for averaging ratios.

2. The bacterial count in a certain culture increased from 1000 to 4000 in 3 days. What was the average percentage increase per day?

3. THE HARMONIC MEAN

Find the harmonic mean H of the numbers 3, 5, 6, 6, 7, 10, and 12.

Jan 26 2021 05:01 PM

1 Approved Answer

Sonia B
answered on
January 28, 2021

4
Ratings,(24 Votes)

Solution 1: a) The mean ratio of milk to bread prices is (3+2)/2 = 2.5. b) Since, the ratio of the milk to bread prices for the first year is 3 and the ratio of the bread to milk prices is 1/3 = 0.33. The ratio of the bread to milk prices for the second year is ½ =0.5. Mean ratio of bread to milk prices = (0.33+0.5)/2 = 0.417. c) We would expect the mean ratio of milk to bread prices to be the reciprocal of the mean ratio of bread to milk prices if the mean is an appropriate average. But 1/0.417 = 2.40 ≠ 2.5. So, arithmetic mean is a poor average to use for ratios. d) The geometric mean of the ratios of milk to bread prices = sqrt(3*2) = sqrt(6). The geometric mean of the ratios of bread to milk prices = sqrt(0.33*0.5) = sqrt(0.0167) = 1/sqrt(6) . Since, these averages are reciprocals, so our conclusion is that the geometric mean is...

more suitable than the arithmetic mean for averaging ratios of this type of problem. Solution 2: Let the average percentage increase be r. The total bacterial count after 1 day is 1000+1000r = 1000(1+r). The total bacterial count after 2 days is 1000(1+r) + 1000(1+r)r = 1000(1+r)2. The total bacterial count after 3 days is 1000(1+r)2 + 1000(1+r)2 r = 1000(1+r)3. We have: 1000(1+r)3 = 4000 i.e. 1+r = cube root of(4) . So, r = 1.587-1 = 0.587. So, r = 58.7%. Solution 3: The harmonic mean of 3, 5, 6, 6, 7, 10, and 12 is: (1/3 /15 +1/6+ /16+1/ 7+ 1/10 +1/ 12) = 5.87.

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