Calculating Karl's Walking Distance on Day 5: A Comprehensive Guide

Karl's journey has been anything but straightforward. He started later than intended, walked 12 km on the first day, and then pushed his pace over the next four days, covering 45 km each day. By the end of the fourth day, Karl had covered a total distance of 147 km. The question arises: how far did Karl walk on the fifth day, given that he was exactly halfway through his journey by the end of this day?

Breaking Down the Problem

To determine Karl's walking distance on day 5, we need to break down the problem step-by-step. Let's start by calculating the total distance Karl had walked in the first four days.

Distance Walked in the First Four Days

Day 1: 12 km Day 2: 45 km Day 3: 45 km Day 4: 45 km

Adding these distances together:

Total distance after four days:
12 45 45 45 147 km

Distance after Day 5

Let (x) be the distance Karl walked on day 5. Therefore, the total distance after five days is:

147 (x) km

Halfway Point

According to the problem, after day 5, Karl was exactly halfway through his journey. This means that the total distance he needs to walk is twice the distance he has walked by day 5.

Setting Up the Equation

Let (D) represent the total distance of the journey. Therefore, we can express the total distance in terms of (x):

147 (x) (frac{1}{2})D

Substituting (D) into the equation, we get:

(D 2 times (147 x))

Since 147 (x) (frac{1}{2})D, we can simplify:

D 2 times 147 2x

147 x 147 2x

This equation simplifies to:

x 147 km

Conclusion

Therefore, Karl's walking distance on day 5 is 147 km. This means that after day 5, he will have walked a total of 294 km, making him exactly halfway through his journey.

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