7. One night the King could not sleep. He went to the royal kitchen, where he found a bowl full of mangoes. Being hungry, he took 1/6 of the mangoes in the bowl. Later that same night, the Queen could not sleep, and she was hungry. She found the mangoes and took 1/5 of what the King had left in the bowl. Still later, the youngest Prince awoke, went to the kitchen, and ate ¼ of the remaining mangoes. Even later, the second Prince ate 1/3 of what his younger brother had left. Finally, the third Prince, the heir to the throne, ate ½ of what his younger brothers had left, and then there were only three mangoes left in the bowl. How many mangoes were in the bowl when the King found them?
each person took 1/6, so 3 is 16 as well.
There were 18 mangoes to begin
Original number of mangoes = x
after king, amount left:
x - 1 / 6 x = 6 / 6 x - 1 / 6 x = 5 / 6 x
after queen, amount left :
4 / 5 ∙ 5 / 6 x = 4 / 6 x = 2 / 3 x
after youngest prince, amount left:
3 / 4 ∙ 2 / 3 x = 2 / 4 x = 1 / 2 x
after 2nd prince, amount left:
2 / 3 ∙ 1 / 2 x = 1 / 3 x
after 3rd prince, amount left:
1 / 2 ∙ 1 / 3 x = 1 / 6 x
1 / 6 x = 3
Multiply both sides by 6
x = 18
Check Result:
King ate 1 / 6 of 18 = 3
leaving 18 - 3 = 15
Queen ate 1 / 5 of 15 = 3
leaving 15 - 3 = 12
1st prince ate 1 / 4 of 12 = 3
leaving 12 - 3 = 9
2nd prince ate 1 / 3 of 9 = 3
leaving 9 - 3 = 6
3rd prince ate 1 / 2 of 6 = 3
leaving 6 - 3 = 3
Let's solve this step by step.
Let's assume the number of mangoes in the bowl when the King found them is "x".
After the King takes 1/6 of the mangoes, there are (x - x/6) = (5x/6) mangoes left.
Next, the Queen takes 1/5 of what the King had left. So, she takes (1/5) * (5x/6) = (x/6) mangoes.
Now, there are (5x/6 - x/6) = (4x/6) = (2x/3) mangoes left.
Then, the youngest Prince takes 1/4 of the remaining mangoes. So, he takes (1/4) * (2x/3) = (x/6) mangoes.
Now, there are (2x/3 - x/6) = (4x/6 - x/6) = (3x/6) = (x/2) mangoes left.
The second Prince takes 1/3 of what his younger brother had left. So, he takes (1/3) * (x/2) = (x/6) mangoes.
Now, there are (x/2 - x/6) = (3x/6 - x/6) = (2x/6) = (x/3) mangoes left.
Finally, the third Prince eats 1/2 of what his younger brothers had left. So, he eats (1/2) * (x/3) = (x/6) mangoes.
After the third Prince eats (x/6) mangoes, there are 3 mangoes left.
Therefore, (x/3 - x/6) = (2x/6 - x/6) = (x/6) = 3.
To find the value of x, we can multiply both sides of the equation by 6:
x = 18.
So, there were 18 mangoes in the bowl when the King found them.
To find the number of mangoes that were originally in the bowl when the King found them, we can work backwards and reverse the process of the mangoes being taken.
Let's start from the point where there were three mangoes left in the bowl.
The third Prince ate 1/2 of what his younger brothers had left, and there were three mangoes left. This means that before the third Prince ate, there were 2 * 3 = 6 mangoes.
Similarly, the second Prince ate 1/3 of what his younger brother had left (6 mangoes), which means that before the second Prince ate, there were 3 * 6 = 18 mangoes.
Continuing backwards, the youngest Prince ate 1/4 of the remaining mangoes (18 mangoes), which means that before the youngest Prince ate, there were 4 * 18 = 72 mangoes.
Next, the Queen took 1/5 of what the King had left. If we let the number of mangoes the King had left be x, then we can write this equation: x - (1/5)*x = 72.
Multiplying both sides of the equation by 5, we get: 5x - x = 360.
Simplifying the equation, we find: 4x = 360.
Dividing both sides of the equation by 4, we find: x = 90.
Therefore, when the King found the mangoes, there were 90 mangoes in the bowl.