A peacekeeping force of 24000 consists of Americans and Russians. The average height of an American is 6 feet 2 inches and that of a Russian is 5 feet 11 inches. The whole peacekee... A peacekeeping force of 24000 consists of Americans and Russians. The average height of an American is 6 feet 2 inches and that of a Russian is 5 feet 11 inches. The whole peacekeeping force's average height is 6 feet. Find the number of Russians in the peacekeeping force.
Understand the Problem
The question involves calculating the number of Russians in a peacekeeping force based on given average heights and total personnel. It challenges the reader to apply reasoning and calculations related to averages and distributions.
Answer
The number of Russians in the peacekeeping force is $R = 16000$.
Answer for screen readers
The number of Russians in the peacekeeping force is ( R = 16000 ).
Steps to Solve
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Assign Variables for Personnel Let ( A ) be the number of Americans and ( R ) be the number of Russians in the peacekeeping force. We have: $$ A + R = 24000 $$
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Convert Heights to Inches Convert the average heights to inches for easier calculations:
- Average height of Americans: ( 6 , \text{feet} , 2 , \text{inches} = 74 , \text{inches} )
- Average height of Russians: ( 5 , \text{feet} , 11 , \text{inches} = 71 , \text{inches} )
- Average height of the whole force: ( 6 , \text{feet} = 72 , \text{inches} )
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Set Up the Average Height Equation Calculate the total height contributed by both groups: The total height of the Americans is ( 74A ) and for Russians is ( 71R ). The equation for the average height is: $$ \frac{74A + 71R}{24000} = 72 $$
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Rearrange the Average Height Equation Multiply both sides by ( 24000 ) to eliminate the fraction: $$ 74A + 71R = 72 \times 24000 $$
Calculating ( 72 \times 24000 ): $$ 72 \times 24000 = 1728000 $$
Thus, the equation becomes: $$ 74A + 71R = 1728000 $$
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Substitute for A From the first equation, we can express ( A ) in terms of ( R ): $$ A = 24000 - R $$
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Substitute A into Height Equation Substituting ( A ) in the height equation: $$ 74(24000 - R) + 71R = 1728000 $$
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Expand and Solve for R Distribute ( 74 ): $$ 1776000 - 74R + 71R = 1728000 $$
Combine like terms: $$ 1776000 - 3R = 1728000 $$
Subtract ( 1776000 ) from both sides: $$ -3R = 1728000 - 1776000 $$
Calculating the right side: $$ -3R = -48000 $$
Divide by -3 to get ( R ): $$ R = \frac{-48000}{-3} = 16000 $$
The number of Russians in the peacekeeping force is ( R = 16000 ).
More Information
This problem illustrates how averages can provide useful insights when the total number and partial averages are known. It involves basic algebra and understanding of average calculations.
Tips
- Forgetting to convert heights to the same units.
- Not multiplying the total personnel by the average height properly.
- Failing to simplify equations correctly during substitution.
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