Question

Can someone please help me understand thank you

Answer 1

(6): The current is 2 miles per hour

(7): 20/3 hours

For (7), if your software allows you to use mixed numbers, you can write 6 2/3 instead of 20/3 if you prefer.

Explanation:

(6) Finding the speed of the river's current:

Step 1: Calculate the effective speed upstream

When paddling upstream, Jeanette's speed relative to the ground is reduced by the speed of the river's current. So her effective speed upstream is (5 - c) miles per hour.

Step 2: Calculate the effective speed downstream

When paddling downstream, Jeanette's speed relative to the ground is increased by the speed of the river's current. So her effective speed downstream is (5 + c) miles per hour.

Step 3: Calculate the time taken to travel 18 miles upstream

Using the formula time = distance / speed, the time taken to travel 18 miles upstream is:

Time upstream = 18 / (5 - c) hours

Step 4: Calculate the time taken to travel 42 miles downstream

Using the same formula, the time taken to travel 42 miles downstream is:

Time downstream = 42 / (5 + c) hours

Step 5: Set up the equation based on the given information

According to the problem, the time taken upstream is equal to the time taken downstream. So we can set up the equation:

Time upstream = Time downstream

Step 6: Solve the equation

Substituting the expressions for time upstream and time downstream from steps 3 and 4 respectively, we have:

18 / (5 - c) = 42 / (5 + c)

Step 7: Cross-multiply and solve for 'c'

18(5 + c) = 42(5 - c)

90 + 18c = 210 - 42c

60c = 120

c = 2

Therefore, the speed of the river's current is 2 miles per hour.

(7) Finding the time it takes for Hannah and Destiny to paint the room simultaneously:

Step 1:

• Let x be the number of hours it takes for both Hannah and Destiny to paint the room together.

Step 2:

• In one hour, Hannah can paint 1/15 of the room, and Destiny can paint 1/12 of the room.
• The sum of their fractions will tell how much of the room they can paint in an hour:

(1/15) + (1/12) = (4/60) + (5/60) = 9/60 = 3/20

Thus, they can paint 3/20 of the room in one hour

Step 3:

• Since they can paint 3/20 of the room in one hour, 1 represents the entire room being painted (technically 20/20 which reduces to 1/1 or just 1).

Thus, we can create a proportion to determine how many hours it will take for them to paint the entire room:

3/20 painted / 1 hour = 1 painted / x hours

(3/20) / 1 = 1 / x

(3/20 = 1/x) * x

(3/20x = 1) / 3/20

x = 1 * 20/3

x = 20/3

Thus, it would 20/3 hours (about 6.67 hours) for the pair to paint the room while working together.

Answer 2

6. 2 mph

7. 6 2/3 hours

Explanation:

You want the speed of a river's current if it takes a kayak paddling 5 mph the same time for 18 miles upstream as for 42 miles downstream. You want to know the "working together" time for Hannah and Destiny if they can paint a room in 15 and 12 hours, respectively, working alone.

Rate

These are rate problems. The general relation involved is ...

rate = quantity / time

This equation can be rearranged to give you quantity or time, as you may need.

For the river current problem, the quantity is miles, and the time is hours. The rate (no surprise here) is miles per hour.

For the painting problem, the quantity is painted rooms, and the time is hours. The rate is given by its inverse: hours per painted room. You can only add rates expressed as quantity/time, so the given hours per room will need to be inverted to give rooms per hour.

6. Current

The speed upstream is 5-c, and the distance is 18 miles, so the time is ...

18/(5 -c)

The speed downstream is 5+c, and the distance is 42 miles, so the time is ...

42/(5 +c)

These times are said to be the same, so we have ...

18/(5 -c) = 42/(5 +c)

Multiplying by (5-c)(5+c)/6, we have ...

3(5 +c) = 7(5 -c)

15 +3c = 35 -7c . . . . . eliminate parentheses

10c = 20 . . . . . . . . . . . add 7c-15

c = 2 . . . . . . . . . divide by 2

The current is 2 miles per hour.

7. Painting

Hannah's rate of painting rooms is ...

(1 room)/(15 hours) = 1/15 room/hour

Destiny's rate of painting rooms is ...

(1 room)/(12 hours) = 1/12 room/hour

These rates add when they work together:

(1/15 room/hour) +(1/12 room/hour) = (12 +15)/(12×15) room/hour

= 3/20 room/hour

Then the time it takes to paint 1 room is ...

time = quantity/rate

(1 room)/(3/20 room/hour) = 20/3 hour = 6 2/3 hour

Working together it take the two painters 6 2/3 hours to paint the room.

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Additional comment

It helps to be comfortable with arithmetic using fractions and rational expressions (same thing, but with variables).

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