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If a group of $12$ boys with different heights: $5'$ ; $5'2''$ ; $5'4''$ ; $5'6''$ ; $5'8''$ ; $5'10''$ ; $6'$ ; $6'2''$ ; $6'4''$ ; $6'6''$ ; $5'2''$ $0$ ; $5'2''$ $1$ , which of the following is correct? $\newline$ A) No boy in the group represents the median height of the group. $\newline$ B) There is a boy in the group who represents the arithmetic mean (average) height of the group. $\newline$ C) The sum of the number of boys with less than the median height and the number of boys with greater than the median height is equal to $5'2''$ $2$ . $\newline$ D) The arithmetic mean (average) height is greater than the median height of the group.

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One-step inequalities: word problems

Full solution

Q. If a group of

12

boys with different heights:

5'

;

5'2''

;

5'4''

;

5'6''

;

5'8''

;

5'10''

;

6'

;

6'2''

;

6'4''

;

6'6''

;

5'2''

0

;

5'2''

1

, which of the following is correct?

\newline

A) No boy in the group represents the median height of the group.

\newline

B) There is a boy in the group who represents the arithmetic mean (average) height of the group.

\newline

C) The sum of the number of boys with less than the median height and the number of boys with greater than the median height is equal to

5'2''

2

.

\newline

D) The arithmetic mean (average) height is greater than the median height of the group.

Calculate Median Height: Step $1$ : Calculate the median height of the group. $\newline$ Since there are $12$ boys, the median will be the average of the heights of the $6$ th and $7$ th tallest boys. $\newline$ $6$ th tallest: $5′10′′$ $\newline$ $7$ th tallest: $6′$ $\newline$ Median $= (5′10′′ + 6′) / 2$ $\newline$ Median $= (70 \text{ inches} + 72 \text{ inches}) / 2$ $\newline$ Median $= 142 \text{ inches} / 2$ $\newline$ Median $6$ $0$ or $6$ $1$
Calculate Mean Height: Step $2$ : Calculate the arithmetic mean (average) height of the group. $\newline$ Sum of heights = $5′ + 5′2′′ + 5′4′′ + 5′6′′ + 5′8′′ + 5′10′′ + 6′ + 6′2′′ + 6′4′′ + 6′6′′ + 6′8′′ + 6′10′′$ $\newline$ Convert all to inches: $60 + 62 + 64 + 66 + 68 + 70 + 72 + 74 + 76 + 78 + 80 + 82 = 852$ inches $\newline$ Average = $852$ inches $/$ $12$ $\newline$ Average = $71$ inches or $5′11′′$
Check Each Statement: Step $3$ : Check each statement. $\newline$ A) Incorrect. The median height is $5′11′′$ , and there is no boy exactly at $5′11′′$ . $\newline$ B) Incorrect. The average height is $5′11′′$ , and there is no boy exactly at $5′11′′$ . $\newline$ C) Incorrect. There are $6$ boys below and $6$ boys above the median height, summing to $12$ , not $11$ . $\newline$ D) Incorrect. The arithmetic mean and the median are equal, both are $5′11′′$ .

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