A king couldn't sleep so, he devoured 1/6 of a bowl of mangoes. Later, the queen got hungry and took 1/5 of the bowl of mangoes. The prince awoke and ate 1/4 of the bowl of mangoes. After, he came back and ate another 1/3. Obviously, he was hungry and ate another 1/2. 3 mangoes were left for the servants. How many mangoes were in the bowl before the king, queen, and prince ate mangoes?
I need help solving this. Explain. Thank you! (;
let the number of original mangoes be x
King ate 1/6, so (5/6)x are left
Queen ate 1/5 of that so (4/5)(5/6)x are left
prince ate 1/4 of that so (3/4)(4/5)(5/6)x are left
prince then ate 1/3 of that leaving (2/3)(3/4)(4/5)(5/6)x
prince then ate 1/2 of that, leaving
(1/2)(2/3)(3/4)(4/5)(5/6)x
this cancels very nicely to (1/6)x
but we know (1/6)x = 3
so x = 18
To solve this problem, we need to work backward and find the number of mangoes left in the bowl after each person ate. Then, we can add the number of mangoes consumed by each person to find the total number of mangoes in the bowl originally.
Let's start from the end:
We know that there were 3 mangoes left for the servants. This means that after the prince's last round of eating (1/2), there were 3 mangoes left in the bowl.
Before that, the prince ate 1/3 of the remaining mangoes, which means there were (1 - 1/3 = 2/3) of the bowl's mangoes left.
Before that, the prince ate 1/4 of the remaining mangoes. So, there were (2/3 - 1/4 = 5/12) of the bowl's mangoes left.
Before that, the queen ate 1/5 of the remaining mangoes. So, there were (5/12 - 1/5 = 23/60) of the bowl's mangoes left.
Finally, before the queen, the king ate 1/6 of the mangoes. Therefore, (23/60 - 1/6 = 17/60) of the bowl's mangoes were left.
Since there were 17/60 of the initial bowl remaining, the ratio is 17:60. To find the total number of mangoes, we need to find what quantity represents 17 out of 60. Since the denominator (60) represents the total number of mangoes, we can set up a proportion:
17/60 = x/60
By cross-multiplying, we get:
60 * 17 = x * 60
Simplifying:
x = 17
So, there were 17 mangoes initially in the bowl before the king, queen, and prince ate any mangoes.