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Vince went on a 3 day hiking trip. Each day, he walked (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^("st ") day of the trip? Round your final answer to the nearest kilometer. km

Vince went on a 33 day hiking trip.\newlineEach day, he walked 34 \frac{3}{4} the distance that he walked the day before. He \mathrm{He} walked 8383.2525 kilometers total in the trip.\newlineHow far did Vince walk on the 1st  1^{\text {st }} day of the trip?\newlineRound your final answer to the nearest kilometer.\newlinekm \mathrm{km}

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Full solution

Q. Vince went on a 33 day hiking trip.\newlineEach day, he walked 34 \frac{3}{4} the distance that he walked the day before. He \mathrm{He} walked 8383.2525 kilometers total in the trip.\newlineHow far did Vince walk on the 1st  1^{\text {st }} day of the trip?\newlineRound your final answer to the nearest kilometer.\newlinekm \mathrm{km}
  1. Denote Distance Vince Walked: Let's denote the distance Vince walked on the first day as DD. On the second day, he walked (3/4)D(3/4)D, and on the third day, he walked (3/4)(3/4) of the second day's distance, which is (3/4)×(3/4)D=(9/16)D(3/4) \times (3/4)D = (9/16)D. The total distance walked over the three days is the sum of these distances.\newlineSo, the equation representing the total distance walked is:\newlineD+(3/4)D+(9/16)D=83.25D + (3/4)D + (9/16)D = 83.25
  2. Find Common Denominator: Now, let's find a common denominator to combine the terms on the left side of the equation. The common denominator for 11, 34\frac{3}{4}, and 916\frac{9}{16} is 1616. So we can rewrite the equation as:\newline(1616)D+(1216)D+(916)D=83.25(\frac{16}{16})D + (\frac{12}{16})D + (\frac{9}{16})D = 83.25
  3. Combine Terms: Combine the terms on the left side of the equation:\newline(1616)D+(1216)D+(916)D=(16+12+916)×D=(3716)D(\frac{16}{16})D + (\frac{12}{16})D + (\frac{9}{16})D = (\frac{16+12+9}{16}) \times D = (\frac{37}{16})D\newlineSo, (3716)D=83.25(\frac{37}{16})D = 83.25
  4. Divide by (37/16)(37/16): To find DD, we need to divide both sides of the equation by (37/16)(37/16):
    D=83.25(37/16)D = \frac{83.25}{(37/16)}
    D=83.25×(1637)D = 83.25 \times (\frac{16}{37})
  5. Perform Multiplication: Now, let's perform the multiplication to find DD:
    D=83.25×(1637)D = 83.25 \times (\frac{16}{37})
    D≈83.25×0.4324D \approx 83.25 \times 0.4324
    D≈36D \approx 36
    Since we are asked to round the final answer to the nearest kilometer, Vince walked approximately 3636 kilometers on the first day.

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