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Hi Fredrick! So I have this summer math packet to complete before school starts, which is on August 7th. I reached a couple of problems which left me stumped because I didn't know what to do. Your help would be so appreciated. Oh, when you help me with these, would you mind telling me the process and everything? Thank you.

78. Working together, Natalie and Paul can pick forty bushels of apples in 4.8 hours. Had he done it alone, it would have taken Paul 12 hours. Find how long it would have taken Natalie.

81. An aircraft carrier left the pier traveling east eight hours before a fishing boat. The fishing boat traveled in the opposite direction for four hours, going 5 mph slower than the aircraft carrier. The ships were then 380 miles apart. What was the aircraft carrier's speed?

Here are the detailed solutions:

78. Paul's rate of picking bushels (alone) = 12 /40 = 0.3 hours per bushel

Let Natalie's rate of picking bushels (alone) = x hours per bushel

If their efforts were combined, then Natalie would have picked 4.8/x bushels while Paul

would have picked 4.8/0.3 = 16 bushels in a span of 4.8 hours.

Since 40 bushels would have been picked in a combined effort scenario,

16 + 4.8/x = 40

4.8/x = 24

x = 4.8/24 = 0.2

Hence, if Natalie was left to pick all 40 bushels by herself, she would have taken

40*0.2 = 8 hours (shown)

81. Let the aircraft carrier's speed be x mph, then the fishing boat's speed would be (x-5)

mph.

After the first eight hours, the aircraft carrier has traveled 8x miles and the distance

between the two ships at this instant would simply be

In the next four hours, the aircraft carrier would have traveled another

east while the fishing boat

Hence, the total distance separating both ships after 12 hours would be

8x + 4x + 4x - 20 = 16x -20

Since this is equivalent to 380 miles,

16x-20 = 380 =====> x =400/16 = 25 mph (shown)

Hope this helps. Peace.

Algebra

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