"A”, "B”, and "C” spent their afternoon picking bananas and mangoes. "A” had twice as many bananas as "B”, while "B” had twice as many bananas as "C”. "B” had three times as many mangoes as "C”, who had three times as many mangoes as "A”. At the end of the day they had fewer than 250 pieces of fruit in total, and an equal positive number of bananas and mangoes. How many pieces of fruit did they pick?
"A”, "B”, and "C” spent their afternoon picking bananas and mangoes. "A” had twice as many bananas as "B”, while "B” had twice as many bananas as "C”. "B” had three times as many mangoes as "C”, who had three times as many mangoes as "A”. At the end of the day they had fewer than 250 pieces of fruit in total, and an equal positive number of bananas and mangoes. How many pieces of fruit did they pick?
The correct option is B 182
Suppose that "C” had a bananas. Then "B” has 2a bananas, and "A” has 4a bananas, so in total the three have 7a bananas. Now suppose that "A” has b mangoes. Then "C” has 3b mangoes, and "B” has 9b mangoes, so together the three have 13b mangoes. We are told that 7a = 13b. Since 7 and 13 are distinct primes, 7 must divide b, and hence b = 7k for some positive integer k. Therefore 13b = 91k, and the total number of bananas and mangoes the group picked is 182k. Since we are told that this number is less than 250, we have k = 1.
182
Suppose that "C” had a bananas. Then "B” has 2a bananas, and "A” has 4a bananas, so in total the three have 7a bananas. Now suppose that "A” has b mangoes. Then "C” has 3b mangoes, and "B” has 9b mangoes, so together the three have 13b mangoes. We are told that 7a = 13b. Since 7 and 13 are distinct primes, 7 must divide b, and hence b = 7k for some positive integer k. Therefore 13b = 91k, and the total number of bananas and mangoes the group picked is 182k. Since we are told that this number is less than 250, we have k = 1.
