Vince went on a 3 day hiking trip. Each day, he walked $\frac{3}{4}$ the distance that he walked the day before. He walked 83.25 kilometers total in the trip.

How far did Vince walk on the $1^{ ext {st }}$ day of the trip?
Round your final answer to the nearest kilometer.
$\mathrm{km}$

Question

Vince went on a 3 day hiking trip. Each day, he walked $\frac{3}{4}$ the distance that he walked the day before. He walked 83.25 kilometers total in the trip.

How far did Vince walk on the $1^{	ext {st }}$ day of the trip?
Round your final answer to the nearest kilometer.
$\mathrm{km}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Let's denote the distance Vince walked on the first day as $d$. ‖ ‖ step 2 ⋮ On the second day, Vince walked $\frac{3}{4}d$. ‖ ‖ step 3 ⋮ On the third day, Vince walked $\frac{3}{4}$ of the distance of the second day, which is $\frac{3}{4} imes \frac{3}{4}d = \left(\frac{3}{4} ight)^2 d$. ‖ ‖ step 4 ⋮ The total distance walked over the three days is the sum of the distances walked each day: $d + \frac{3}{4}d + \left(\frac{3}{4} ight)^2 d$. ‖ ‖ step 5 ⋮ This sum equals the total distance walked, which is 83.25 km: $d + \frac{3}{4}d + \left(\frac{3}{4} ight)^2 d = 83.25$. ‖ ‖ step 6 ⋮ Solving for $d$, we combine like terms: $d \left(1 + \frac{3}{4} + \left(\frac{3}{4} ight)^2 ight) = 83.25$. ‖ ‖ step 7 ⋮ Calculating the coefficient of $d$: $1 + \frac{3}{4} + \left(\frac{3}{4} ight)^2 = \frac{4}{4} + \frac{3}{4} + \frac{9}{16} = \frac{16}{16} + \frac{12}{16} + \frac{9}{16} = \frac{37}{16}$. ‖ ‖ step 8 ⋮ The equation becomes $\frac{37}{16}d = 83.25$. ‖ ‖ step 9 ⋮ Dividing both sides by $\frac{37}{16}$ to solve for $d$: $d = \frac{83.25}{\frac{37}{16}} = \frac{83.25 imes 16}{37}$. ‖ ‖ step 10 ⋮ Calculating the value of $d$: $d = \frac{1332}{37} = 36$. ‖ ‖ step 11 ⋮ Rounding $d$ to the nearest whole number, we get $d = 36$ km. ‖ ∻Answer∻ ⚹ Vince walked 36 km on the first day of his trip. ⚹ ∻Key Concept∻ ⚹Geometric Series and Solving Equations⚹ ∻Explanation∻ ⚹The distance walked each day forms a geometric series with a common ratio of $\frac{3}{4}$. To find the distance walked on the first day, we set up an equation based on the total distance and solve for the first term of the series.⚹◊What is the formula to determine the distance Vince walked on the first day of his hiking trip, given the information provided?⍭ Generate me a similar question◊

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