Question

Math

Posted 3 months ago

```
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 $1^{\text {st }}$ day of the trip?
Round your final answer to the nearest kilometer.
$\mathrm{km}$
```

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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 $d$

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 $\frac{3}{4} \times \frac{3}{4}d = \left(\frac{3}{4}\right)^2 d$

step 4

The total distance walked over the three days is the sum of the distances walked each day: $d + \frac{3}{4}d + \left(\frac{3}{4}\right)^2 d$

step 5

This sum equals the total distance walked, which is 83.25 km: $d + \frac{3}{4}d + \left(\frac{3}{4}\right)^2 d = 83.25$

step 6

Solving for $d$, we combine like terms: $d \left(1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2\right) = 83.25$

step 7

Calculating the coefficient of $d$: $1 + \frac{3}{4} + \left(\frac{3}{4}\right)^2 = \frac{4}{4} + \frac{3}{4} + \frac{9}{16} = \frac{16}{16} + \frac{12}{16} + \frac{9}{16} = \frac{37}{16}$

step 8

The equation becomes $\frac{37}{16}d = 83.25$

step 9

Dividing both sides by $\frac{37}{16}$ to solve for $d$: $d = \frac{83.25}{\frac{37}{16}} = \frac{83.25 \times 16}{37}$

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.

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