Calculating Traveled Distance: A Comprehensive Guide Using Different Modes of Transportation

When solving problems involving journey calculations, it is often necessary to break down the total trip into different segments and solve for the unknowns. This article will walk you through the process of determining the total distance traveled when using multiple modes of transportation. We will analyze the given data and solve for the unknown distance using various methods.

Introduction

Imagine a scenario where an individual travels using different modes of transportation. In this case, we are given the fractions of the journey traveled by rail and bus, with the remaining distance walked. Let us explore how to find the total distance using the provided information and several methods.

Provided Information and Solution

Let's start with the exact problem statement:

Problem:
You travelled 5/16th of your journey by rail and 7/26th by bus, then you walked the remaining 10 miles. How far did you go?

Let the total distance be (D). The remaining distance walked is 10 miles, which can be represented as:

(frac{5}{16}D frac{7}{26}D 10 D)

To solve for (D), we need to combine the fractions and subtract them from (D):

(D - left(frac{5}{16}D frac{7}{26}Dright) 10)

First, find a common denominator for the fractions, which is the least common multiple (LCM) of 16 and 26, which is 208.

(frac{5}{16}D frac{65}{208}D, frac{7}{26}D frac{56}{208}D)

(D - left(frac{65}{208}D frac{56}{208}Dright) 10)

(D - frac{121}{208}D 10)

(frac{87}{208}D 10)

(D frac{2080}{87})

(D approx 23.9) miles

[Ans]

Further Analysis and Other Methods

There are several other ways to approach this problem. Let's look at a few more methods for solving it:

LCM Method:

The LCM of 16 and 26 is 208. Let the total distance be (208x).

(frac{5}{16} times 208x 65x text{rail}frac{7}{26} times 208x 56x text{bus})

(65x 56x 10 208x)

(121x 10 208x)

(87x 10)

(x frac{10}{87})

(x approx frac{10}{87} approx 23.91) miles

Unit Division Method:

235 of 416 units of the trip is by rail, and 181 of 416 units is by bus and walking 10 miles. Therefore:

(frac{181}{416} frac{10}{text{our total distance traveled}})

(text{our total distance traveled} frac{4160}{181} approx 22.98)

Simplified Visualization:

2/11 or 4/22 of the journey by car, 17/22 by train, and the rest by walking 1 km. The remaining part is 1/22, which is 1 km. Therefore, the total distance is 22 km:

(1 - frac{21}{22} frac{1}{22} 1text{ km})

(frac{1}{frac{1}{22}} 22text{ km})

Conclusion

We have covered three methods to solve the problem of calculating the distance traveled using different modes of transportation. Whether you use the standard method, the LCM method, or the unit division method, the key is to break down the problem into manageable segments and solve for the unknowns step-by-step. This comprehensive guide should help you tackle similar problems with confidence.

Keywords

traveled distance, distance calculation, travel math