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Question
Math
Posted 3 months ago

Vince went on a 3 day hiking trip. Each day, he walked 34\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 1st 1^{\text {st }} day of the trip?
Round your final answer to the nearest kilometer.
km\mathrm{km}
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
Let's denote the distance Vince walked on the first day as dd
step 2
On the second day, Vince walked 34d\frac{3}{4}d
step 3
On the third day, Vince walked 34\frac{3}{4} of the distance of the second day, which is 34×34d=(34)2d\frac{3}{4} \times \frac{3}{4}d = \left(\frac{3}{4}\right)^2 d
step 4
The total distance walked over the three days is the sum of the distances walked each day: d+34d+(34)2dd + \frac{3}{4}d + \left(\frac{3}{4}\right)^2 d
step 5
This sum equals the total distance walked, which is 83.25 km: d+34d+(34)2d=83.25d + \frac{3}{4}d + \left(\frac{3}{4}\right)^2 d = 83.25
step 6
Solving for dd, we combine like terms: d(1+34+(34)2)=83.25d \left(1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2\right) = 83.25
step 7
Calculating the coefficient of dd: 1+34+(34)2=44+34+916=1616+1216+916=37161 + \frac{3}{4} + \left(\frac{3}{4}\right)^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 3716d=83.25\frac{37}{16}d = 83.25
step 9
Dividing both sides by 3716\frac{37}{16} to solve for dd: d=83.253716=83.25×1637d = \frac{83.25}{\frac{37}{16}} = \frac{83.25 \times 16}{37}
step 10
Calculating the value of dd: d=133237=36d = \frac{1332}{37} = 36
step 11
Rounding dd to the nearest whole number, we get d=36d = 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 34\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.

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