Two groups of people need to get across a lake. The first group has 36 people. Its boats can carry 5 passengers. The second group has 24 people. Its boats can carry only 4 passengers. Which group needs more boats?
Follow the steps to solve the problem.

1. Group 1 is represented by the equation 5x = 36. Solve this equation for x. Show your work.

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2. How many boats does group 1 need? Explain your answer.

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3. Write and solve an equation to find how many boats group 2 needs. Show your work.

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4. Which group needs more boats?

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The first group needs 8 boats, the second needs 6

Step-by-step explanation:

1: x = 8, divide both sides of the equal sign by 5 and you get x = 7.2

2: 8, 5 goes into 36 7 times evenly with a single person remaining and they would need the 8th boat

3: 4x = 24, divide both sides of the equal sign by 4 and you get x = 6

4: the first group

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt