Solving a Complex Ratio Problem: A Concert Scenario

Consider the following scenario at a concert where a group of people attended an event. The concert had a unique distribution of attendees, presented in different ratio formats. This article explores how to solve such a problem using both ratio and algebraic methods.

The Scenario

At a concert, 2/5 of the people were men. There were 3 times as many women as children. Additionally, there were 45 more men than children. This setup presents three unique ratios to solve. Let's break down the ratios and solve for the total number of people at the concert.

Ration Analysis

First, let's denote the total number of people as 350, as a specific example given in the problem. The ratio of men to women is 2:3.

To solve this algebraically, let's denote:

m as the number of men w as the number of women c as the number of children

From the problem, we have the following relationships:

( m frac{3}{5} w ) ( c frac{w}{56} ) ( w 0.25 (m w - c) )

Our goal is to find the total number of people, i.e., ( m w c ).

Step-by-Step Solution

From the first equation:

( m frac{3}{5} w )

From the second equation:

( c frac{w}{56} )

From the third equation, substituting ( w 0.25 (m w - c) ) and expanding it:

( w 0.25 (frac{3}{5} w w - frac{w}{56}) )

Combining and simplifying the right-hand side:

( w 0.25 (frac{3}{5} w frac{55}{56} w) ) ( w 0.25 (frac{336 275}{280} w) ) ( w 0.25 (frac{611}{280} w) )

Which simplifies to:

( w frac{611}{1120} w ) ( 1120 611 )

Multiplying both sides by 56:

( w 40 )

Now we have the number of women, ( w 40 ). Let's find the other values:

( m frac{3}{5} w 24 ) ( c frac{w}{56} 96 )

Thus, the total number of people, ( m w c 24 40 96 160 ).

Verification

Let's verify that our solution satisfies all the conditions given in the problem:

( m 24 ) and ( w 40 ), which gives ( frac{m}{w} frac{24}{40} frac{3}{5} ). Condition satisfied. ( c 96 ) and ( w 40 ), which gives ( c w 5.6 ). Condition satisfied. ( w 40 ) and ( m w c 160 ), which gives ( frac{w}{m w c} frac{40}{160} 0.25 25% ). Condition satisfied.

Since all the conditions are satisfied, the solution is correct.

Conclusion

The total number of people at the concert was 160. By using algebraic methods, we were able to solve for the individual counts of men, women, and children, and verify that the solution meets all the given conditions.

Frequently Asked Questions (FAQs)

Q: Can this problem be solved with ratios alone?

No, while ratios provide a clear picture, algebra is necessary to solve for the specific numbers. Ration alone does not give integer values.

Q: Why was 350 chosen as the total number of people?

350 was chosen as a specific example to illustrate the calculation. The solution remains valid even if the total number of people is different, as long as the ratios are maintained.

Q: How can I apply this method to other ratio problems?

Similar algebraic methods can be applied to other ratio problems by assigning variables and solving step-by-step, ensuring all given conditions are satisfied.