A tree is losing its trees. Two thirds fall off in the first week, two thirds of those remaining fall off in the second week, and two thirds off those remaining fall off in the third week. Now there are 37 leaves. How many leaves did the tree have originally?
original : x
at end of 1st week we have 1/3 x left
at end of 2nd week we have (1/3)(1/3 x) left = (1/3)^2 x
at end of 3rd week we have (1/3)^3 x left
so (1/27)x = 37
x = 999
So there were 999 leaves
check:
number falling off 1st week = (2/3)(999) = 666
number left = 333
number falling off 2nd week - (2/3)(666) = 222
number left = 333 - 222 = 111
number falling off 3rd week = (2/3)(111) = 74
number left at end of 3rd week = 111 - 74 = 37
All is good!
After first week remains 1-\frac{2}{3}=\frac{1}{3}1−
3
2
=
3
1
leaves
After second week remains \frac{1}{3}-\frac{1}{3}\frac{2}{3}=\frac{1}{9}
3
1
−
3
1
3
2
=
9
1
leaves
After third week remains \frac{1}{9}-\frac{1}{9}\frac{2}{3}=\frac{1}{27}
9
1
−
9
1
3
2
=
27
1
leaves
Let x be the number of the leaves, the tree starts with.
We have: \frac{x}{27}=37
27
x
=37
So, the tree starts with x=27*37=999x=27∗37=999 leaves.
Well, it seems like that tree is shedding leaves faster than a squirrel with an irrational fear of foliage! Let's do some math to figure it out.
Let's start with the final count of 37 leaves. According to the problem, in the third week, two-thirds of the leaves fell off. So, before the third week, there were three-thirds of the total leaves.
If we reverse this process, we can calculate the number of leaves in the second week. At that point, two-thirds of the remaining leaves fell off. So in the second week, we'd have four-thirds of the final count.
Continuing backwards, we can figure out the number of leaves in the first week. Again, two-thirds fall off, leaving us with six-thirds of the second-week count.
Now, here's the fun (or should I say 'leafy') part! If you multiply 37 by 3/2, you'll get 55.5. Now, if we multiply 55.5 by 3/4, we get 41.625. Finally, multiplying 41.625 by 3/2, we find that the tree originally had approximately 62.4375 leaves.
Now, I'm not sure if trees can have decimal leaves, but we've done the math! So, let's just round it up to an even number and say the tree had roughly 62 leaves originally.
To solve this problem, we'll work backwards from the given information. Let's start with the current number of leaves, which is 37.
In the third week, two thirds of the remaining leaves fell off. Let's represent the number of leaves after the third week as 'x'. Given that two thirds fell off, we can calculate the number of leaves before the third week using the formula: x = (3/2) * 37.
Simplifying this equation gives us x = 55.5. Since the number of leaves must be a whole number, we can conclude that the tree originally had 56 leaves.
In the second week, two thirds of the remaining leaves fell off. Let's represent the number of leaves after the second week as 'y'. Using the same formula, we can calculate y = (3/2) * 55.5 = 83.25. Again, since the number of leaves must be a whole number, we round up to 84.
Finally, in the first week, two thirds of the remaining leaves fell off. Let's represent the number of leaves after the first week as 'z'. Using the formula, z = (3/2) * 84 = 126.
Therefore, the tree originally had 126 leaves.
kajdhkajhdd
There are (35)2 ⋅ 30 leaves on a tree. What is the total number of leaves on the tree? (5 points)
33
37
310
325