Understanding the Journey and Calculating Distance

This problem asks us to find the total distance of a journey that is completed in two separate parts. We are given the speed and time for each part. To find the total distance, we need to calculate the distance covered in each part and then add them together. The key is to make sure our units are consistent, usually using kilometers for distance and hours for time when speed is given in km/h.

Step-by-Step Distance Calculation

The journey has two distinct parts. Let's calculate the distance covered in each part separately.

Part 1: The First Part of the Journey

• Time taken: 37.5 minutes
• Speed: 90 km/h

First, we need to convert the time from minutes to hours, because the speed is given in km/h. There are 60 minutes in an hour.

\( \text{Time in hours} = \frac{\text{Time in minutes}}{60} \)

\( \text{Time for Part 1} = \frac{37.5}{60} \text{ hours} \)

To simplify the fraction, we can multiply the numerator and denominator by 10 to remove the decimal:

\( \frac{37.5}{60} = \frac{375}{600} \)

Now, we can simplify this fraction. Both 375 and 600 are divisible by 25:

\( 375 \div 25 = 15 \)

\( 600 \div 25 = 24 \)

So the fraction becomes \( \frac{15}{24} \). This can be further simplified by dividing both by 3:

\( 15 \div 3 = 5 \)

\( 24 \div 3 = 8 \)

So, the time taken for Part 1 is \( \frac{5}{8} \) hours.

Now, we can calculate the distance for Part 1 using the formula: Distance = Speed \( \times \) Time.

\( \text{Distance for Part 1} = 90 \text{ km/h} \times \frac{5}{8} \text{ hours} \)

\( \text{Distance for Part 1} = \frac{90 \times 5}{8} \text{ km} \)

\( \text{Distance for Part 1} = \frac{450}{8} \text{ km} \)

We can simplify this fraction by dividing both numerator and denominator by 2:

\( \frac{450}{8} = \frac{225}{4} \text{ km} \)

Part 2: The Remaining Part of the Journey

• Time taken: 14 minutes
• Speed: 80 km/h

Again, convert the time from minutes to hours:

\( \text{Time for Part 2} = \frac{14}{60} \text{ hours} \)

Simplify the fraction by dividing both numerator and denominator by 2:

\( \frac{14}{60} = \frac{7}{30} \text{ hours} \)

Now, calculate the distance for Part 2 using the formula: Distance = Speed \( \times \) Time.

\( \text{Distance for Part 2} = 80 \text{ km/h} \times \frac{7}{30} \text{ hours} \)

\( \text{Distance for Part 2} = \frac{80 \times 7}{30} \text{ km} \)

We can cancel out a 10 from the numerator and denominator:

\( \text{Distance for Part 2} = \frac{8 \times 7}{3} \text{ km} \)

\( \text{Distance for Part 2} = \frac{56}{3} \text{ km} \)

Calculating the Total Distance

The total distance of the journey is the sum of the distances covered in Part 1 and Part 2.

\( \text{Total Distance} = \text{Distance for Part 1} + \text{Distance for Part 2} \)

\( \text{Total Distance} = \frac{225}{4} \text{ km} + \frac{56}{3} \text{ km} \)

To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.

Convert \( \frac{225}{4} \) to a fraction with denominator 12:

\( \frac{225}{4} = \frac{225 \times 3}{4 \times 3} = \frac{675}{12} \)

Convert \( \frac{56}{3} \) to a fraction with denominator 12:

\( \frac{56}{3} = \frac{56 \times 4}{3 \times 4} = \frac{224}{12} \)

Now, add the fractions:

\( \text{Total Distance} = \frac{675}{12} + \frac{224}{12} \)

\( \text{Total Distance} = \frac{675 + 224}{12} \)

\( \text{Total Distance} = \frac{899}{12} \text{ km} \)

Converting the Improper Fraction to a Mixed Number

The total distance is \( \frac{899}{12} \) km. We can convert this improper fraction into a mixed number by dividing the numerator (899) by the denominator (12).

\( 899 \div 12 \)

Divide 899 by 12:

\( 899 = 12 \times 74 + 11 \)

So, 899 divided by 12 is 74 with a remainder of 11. This means \( \frac{899}{12} \) is equal to the mixed number \( 74\frac{11}{12} \).

\( \text{Total Distance} = 74\frac{11}{12} \text{ km} \)

Therefore, the total distance of the journey is \( 74\frac{11}{12} \) km.

Revision Table: Journey Calculations

Additional Information on Distance, Speed, and Time

The relationship between distance, speed, and time is fundamental in physics and everyday life. Here's a quick look at the core concepts:

• Speed: This is how fast an object is moving. It is defined as the distance covered per unit of time. The standard formula is \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). Common units include kilometers per hour (km/h), meters per second (m/s), or miles per hour (mph).
• Distance: This is the total length of the path traveled by an object. It is calculated by rearranging the speed formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). Units usually include kilometers (km), meters (m), or miles.
• Time: This is the duration for which an object is in motion. It can be calculated as \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). Units usually include hours (h), minutes (min), or seconds (s).

It is crucial to ensure that the units are consistent when using these formulas. If speed is in km/h, time must be in hours to get distance in km. If speed is in m/s, time must be in seconds to get distance in meters. Converting units is a common step in solving such problems.