Question

Please clearly label parts A and B and explain so I can understand:

Part A: A certain farmer sold one-third of his crop to Miss Liberty, one-fourth to Freewheeling Franklin, one-sixth to Daffy Duck, one-tenth to Tony Soprano, and one-twelfth to JR Dobbs; he gave one-twentieth to his lady friend, and he consumed the last ounce himself. How much did he grow, and how much did each person get?

Part B: The next year, he sold one-third of his crop to Emperor Norton, one-fifth to Roseanne Barr, one-seventh to Uncle Sam, one-tenth to Xena, one-twelfth to Bugs Bunny, one-fourteenth to Betty Boop, one-fifteenth to Gandalf, and he consumed the last ounce himself. How much did he grow, and how much did each customer buy?

Answer 1

Part A

Total fraction of the crops the farmer sold

+ + + + + 3 4 6 10 12 20
  
+ + + + + 20 09

$= \frac{59}{60}$

So, fraction of the total crop the farmer consumed himself

$= 1 - \frac{59}{60} = \frac{1}{60}$

But we are given that the farmer consumed one ounce. So one ounce is one-sixtieth of the total crop the farmer had grown. Let be the total ounces of crop grown by the farmer.

$\\\frac{1}{60} \times x = 1\\ \\\Rightarrow x = 60$

Therefore, we can conclude that the farmer has grown 60 ounces of crops in total.

Amount of crop Miss Liberty got

$= \frac{1}{3} \times 60$   ounces = 20 ounces

By calculating similarly as above, we can see that Freewheeling Franklin got 15 ounces, Daffy Duck got 10 ounces, Tony Soprano got 6 ounces, JR Dobbs got 5 ounces and his lady friend got 3 ounces from the total crop. This completes Part A.

Part B

Part B can be solved in just the same way as Part A.

Total fraction of the crops the farmer sold

$= \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{15}$

$= \frac{140}{420} + \frac{84}{420} + \frac{60}{420} + \frac{42}{420} + \frac{35}{420} + \frac{30}{420} + \frac{28}{420}$

$= \frac{419}{420}$

Thus, the farmer consumed 1/420 of the total crops himself, which equals one ounce. Therefore, the farmer grew 420 ounces of crops.

By calculating similarly as above in Part A, we find out that Emperor Norton got 140 ounces, Roseanne Barr got 84 ounces, Uncle Sam got 60 ounces, Xena got 42 ounces, Bugs Bunny got 35 ounces, Betty Boop got 30 ounces and Gandalf got 28 ounces.