Ali, Ben and Calvin divided some stickers among themselves.
Ali took 20 stickers and 1/5 of the remaining stickers.
Next, Ben took 1/6 of the remaining stickers and 30 more.
Finally Calvin took the 240 stickers that were left over.
(a) How many stickers did Ben take?
(b) How many stickers were there in total at first?
starting with x stickers,
a = 20 + 1/5 (x-20) = x/5 + 16
b = 1/6 (x-a) + 30
c = x-(a+b) = 240
Now finish it off
the all → x (set an unknown)
Ali 20+1/5(x -20)
Ben 1/6[x-20-1/5(x-20)]+30
Calvin 240
20+1/5(x-20)+1/6[x-20-1/5(x-20)]+30+240=x
20x=8500 (solve equation)
x=425
(a) 425 - 20 = 405
1/5 × 405 = 82
Ali: 20 + 81 =1 01
Ben: 1/6 × (425 - 101) + 30 = 84
(b) 425
(a) 84
(b) 425
To find the answers, let's go step by step:
(a) How many stickers did Ben take?
Let's start with the total number of stickers left after Ali took some stickers.
Since Ali took 20 stickers and 1/5 of the remaining stickers, we can calculate the number of stickers left as follows:
Total stickers left = Total stickers - Ali's stickers
Total stickers left = Total stickers - 20
Next, let's calculate the number of stickers Ben took:
Ben took 1/6 of the remaining stickers and 30 more. Therefore:
Ben's stickers = (1/6) * Total stickers left + 30
Substituting the value of Total stickers left, we get:
Ben's stickers = (1/6) * (Total stickers - 20) + 30
Now, we need to simplify this equation to find the value of Ben's stickers.
(b) How many stickers were there in total at first?
We'll calculate the total number of stickers based on the information given at the end.
Calvin took the 240 stickers that were left over. So:
Total stickers = Calvin's stickers + Leftover stickers
Total stickers = 240 + Leftover stickers
Now, we just need to find the value of the Leftover stickers to get the total number of stickers at first.
To summarize:
(a) Ben's stickers = (1/6) * (Total stickers - 20) + 30
(b) Total stickers = 240 + Leftover stickers
Let's solve these step by step.
To solve this problem, we'll work step by step and keep track of the number of stickers at each stage.
(a) To find how many stickers Ben took, we need to calculate the number of remaining stickers after Ali and then after Ben took their portions.
Step 1: Ali's share
Ali took 20 stickers and 1/5 of the remaining stickers. Let's call the number of remaining stickers after Ali took his portion "x."
After Ali took 20 stickers, there were x - 20 stickers left.
Ali's share is 1/5 of x - 20 stickers, which can be written as (1/5)(x - 20).
Step 2: Ben's share
Ben took 1/6 of the remaining stickers and 30 more. Let's call the number of remaining stickers after Ben took his portion "y."
After Ben took his share, there were y - (1/6)(x - 20) - 30 stickers left.
Ben's share is (1/6)(y - (1/6)(x - 20) - 30).
We know that Calvin took the remaining stickers, so y - (1/6)(x - 20) - 30 = 240.
To solve this equation, we first need to rewrite it in a simplified form:
y - (1/6)(x - 20) = 270.
Now, we have two equations:
(1) Ali's share: (1/5)(x - 20)
(2) Ben's share: (1/6)(y - (1/6)(x - 20) - 30)
Let's solve these equations simultaneously to find the values of x and y.
From equation (2), we can rewrite it in terms of y:
y = (6/5)(y - (1/6)(x - 20) - 30).
Now, substitute equation (1) into equation (2):
y = (6/5)([(1/5)(x - 20)] - (1/6)(x - 20) - 30).
Simplifying further, we get:
y = (6/5)[(1/5)(x - 20) - (1/6)(x - 20) - 30].
Next, simplify again:
y = (6/5)[(6(x - 20) - 5(x - 20) - 30(5)) / 30].
Continuing to simplify:
y = (6/5)(x - 20 - x + 120 - 150).
y = (6/5)(-30).
Finally, solve for y:
y = -36.
Now that we have the value of y, we can substitute it back into equation (1) to find x:
(1/5)(x - 20) = 36.
x - 20 = 180.
x = 200.
Now, substituting the values of x and y into the equation y - (1/6)(x - 20) - 30 = 240, we can find the number of stickers Ben took.
-36 - (1/6)(200 - 20) - 30 = 240.
-6 - (1/6)(180) - 30 = 240.
-6 - (30) - 30 = 240.
-6 - 30 - 30 = 240.
-66 - 30 = 240.
-96 = 240.
Since -96 does not equal 240, there must be a mistake in our calculations. Let's go back and find the error.
From equation (2), we can rewrite it in terms of y:
y = (6/5)(y - (1/6)(x - 20) - 30).
Now, substitute equation (1) into equation (2):
y = (6/5)([(1/5)(x - 20)] - (1/6)(x - 20) - 30).
Simplifying further, we get:
y = (6/5)[(1/5)(x - 20) - (1/6)(x - 20) - 30].
Next, simplify again:
y = (6/5)[(6(x - 20) - 5(x - 20) - 30(5)) / 30].
Continuing to simplify:
y = (6/5)(x - 20 - x + 120 - 150).
y = (6/5)(-30).
Finally, solve for y again:
y = -36.
Now, substituting the values of x and y into the equation y - (1/6)(x - 20) - 30 = 240, we can find the number of stickers Ben took.
-36 - (1/6)(200 - 20) - 30 = 240.
-36 - (1/6)(180) - 30 = 240.
-36 - (30) - 30 = 240.
-36 - 30 - 30 = 240.
-66 - 30 = 240.
-96 = 240.
Since -96 does not equal 240, there seems to be an error in the problem statement or in our calculations.