Understanding the Ratio Problem

This problem involves understanding and working with ratios related to transportation fares and the number of passengers. We are given the ratios of fares for train, bus, and air travel, and the ratios of the number of passengers who used each mode. We also know the total fare collected from all three modes combined on a particular day. Our goal is to find the specific amount of fare collected from air passengers.

Setting up the Ratios

Let's denote the train fare, bus fare, and air fare per passenger by \(F_T\), \(F_B\), and \(F_A\) respectively. Similarly, let the number of passengers who travelled by train, bus, and air be \(N_T\), \(N_B\), and \(N_A\).

We are given the following ratios:

• Fare Ratio: \(F_T : F_B : F_A = 5 : 8 : 12\)
• Passenger Ratio: \(N_T : N_B : N_A = 3 : 4 : 5\)

We can represent the individual fares and number of passengers using proportionality constants. Let the unit fare be \(x\) and the unit number of passengers be \(y\). Then:

• \(F_T = 5x\), \(F_B = 8x\), \(F_A = 12x\)
• \(N_T = 3y\), \(N_B = 4y\), \(N_A = 5y\)

Here, \(x\) and \(y\) are positive constants.

Calculating Total Fare Collected for Each Mode

The total fare collected from each mode of transportation is the product of the fare per passenger and the number of passengers for that mode.

• Total fare from train (\(C_T\)): \(C_T = F_T \times N_T = (5x)(3y) = 15xy\)
• Total fare from bus (\(C_B\)): \(C_B = F_B \times N_B = (8x)(4y) = 32xy\)
• Total fare from air (\(C_A\)): \(C_A = F_A \times N_A = (12x)(5y) = 60xy\)

Finding the Ratio of Collected Fares

The total fare collected from train, bus, and air are in the ratio \(C_T : C_B : C_A\). Substituting the expressions we found:

\(C_T : C_B : C_A = 15xy : 32xy : 60xy\)

Since \(x\) and \(y\) are positive, we can divide by \(xy\) to simplify the ratio:

\(C_T : C_B : C_A = 15 : 32 : 60\)

This ratio tells us the proportion of the total collected fare that comes from each mode.

Using the Total Collected Fare to Find the Value

The total fare collected from all three modes is given as Rs. 1,07,000.

Total Collected Fare = \(C_T + C_B + C_A\)

Using the ratio \(15 : 32 : 60\), we can represent the total collected fare as the sum of the parts of the ratio multiplied by a common factor, let's call it \(k\). So, the total collected fare is \(15k + 32k + 60k\).

We are given that this sum is Rs. 1,07,000.

\(15k + 32k + 60k = 1,07,000\)

\(107k = 1,07,000\)

Now, we can solve for \(k\):

\(k = \frac{1,07,000}{107}\)

\(k = 1,000\)

Calculating Fare Collected from Air Passengers

The fare collected from air passengers corresponds to the "air" part of the collected fare ratio, which is 60. Using the value of \(k\) we just found:

Fare collected from air passengers \( = C_A = 60k\)

\(C_A = 60 \times 1,000\)

\(C_A = 60,000\)

So, the fare collected from air passengers is Rs. 60,000.

Summary of Calculations

Ratio of collected fares: \(15 : 32 : 60\)

Sum of ratio parts = \(15 + 32 + 60 = 107\)

Total collected fare = Rs. 1,07,000

Value of one ratio unit (\(k\)) = \(\frac{1,07,000}{107} = 1,000\)

Fare collected from air passengers = Air ratio part \(\times k = 60 \times 1,000 = 60,000\)

Conclusion

Based on the given fare and passenger ratios and the total collected fare, the fare collected from air passengers is Rs. 60,000.

Ratio Problem Revision Table

Additional Information on Ratios and Proportions

Ratios are fundamental in mathematics and are used to compare quantities. A proportion is an equation that states that two ratios are equal. Problems like this one often involve using ratios to represent unknown quantities and then setting up an equation based on a given total or difference.

• Understanding Compound Ratios: When dealing with ratios of ratios (like fare per person ratio and number of people ratio), multiplying corresponding terms gives a new ratio representing the product of those quantities. Here, (Fare per person ratio) \(\times\) (Number of people ratio) gives (Total Fare Collected ratio for each mode).
• Units in Ratios: While individual quantities in a ratio must be of the same type (e.g., all fares, all passenger counts), the resulting quantities when ratios are combined can represent something new (like total fare collected). The units of the proportionality constants (\(x\) and \(y\)) effectively combine to give the unit for \(xy\), which is the unit of currency per unit squared, but when multiplied by the ratio parts (which are dimensionless), the result is proportional to the currency unit. The final constant \(k\) directly relates the ratio parts of the collected fare to the currency amount.
• Real-World Applications: Ratios are used in many real-world scenarios, including scaling recipes, mixing solutions, interpreting maps, financial analysis, and understanding proportions in populations or distributions, similar to how we analyzed transportation data here.