Paul, Kate and Sarah's mother left a bowl of jellybeans on their kitchen counter. Paul took one-third of the jellybeans, but then threw back 4 because he didn't like the black ones. Sarah ate 6 jellybeans and then took one-quater of what remained. Kate took one-third of what remained, and then picked out and ate the last 4 black ones because she really liked them. If after all this there were 6 jellybeans left, how many were there to begin with?
please explain
number at the beginning ----- x
Paul took (1/3)x - 4
number left after Paul's turn = x - (x/3 - 4) = 2x/3 + 4
after Sarah ate 6, number left = 2x/3 + 4 - 6 = 2x/3 - 2
took 1/4 of that, so remaining = (3/4)(2x/3 - 2) = x/2 - 3/2
Kate took 1/3 of that plus 4 = (1/3)(x/2 - 3/2) + 4
= x/6 - 1/2 + 4 = x/6 + 7/2
number left = (x/2 - 3/2) - (x/6 + 7/2)
= x/3 - 5
x/3 - 5 = 6
x/3 = 11
x = 33
check:
Paul took 1/3, which is 11, then put 4 back, so he took 7 , leaving 26
Sarah ate 6, leaving 20, then took 1/4 of that leaving 15
Kate took 1/3 of that, which is 5, leaving 10
Of the remaining, Kate then ate 4, leaving a final count of 6, as given
my answer is correct
Well, let's see if I can do some math while juggling jellybeans in my clown shoes.
Let's call the total number of jellybeans we started with "X."
Now, here's what happened:
Paul took one-third of X, which leaves us with (2/3)X jellybeans.
But then, he threw back 4 black ones. So, we subtract 4 from (2/3)X and we have (2/3)X - 4 jellybeans.
Next up, Sarah ate 6 jellybeans. So, we subtract 6 from (2/3)X - 4, which gives us (2/3)X - 10 jellybeans.
After that, Sarah took one-fourth of what remained. So, we need to multiply (2/3)X - 10 by 1/4. That leaves us with (1/6)X - 5/2 jellybeans.
Last but not least, Kate took one-third of what remained. So, we need to multiply (1/6)X - 5/2 by 1/3. Which gives us (1/18)X - 5/6 jellybeans.
Kate also picked out and ate the last 4 black ones, which means we need to subtract 4 from (1/18)X - 5/6.
So, (1/18)X - 5/6 - 4 = 6.
Let me get my juggling pins ready... solving for X... "X marks the spot!"
(1/18)X - 17/6 = 6.
Alrighty, let's bring the X out of hiding.
(1/18)X = 23/6.
Multiplying both sides by 18 to get rid of that pesky fraction...
X = 23×18/6.
Which gives us...
X = 69 jellybeans!
So, it looks like we started with 69 jellybeans. Time to throw a jellybean party! 🎉
Let's work through the problem step-by-step to determine the initial number of jellybeans.
1. After Paul took one-third of the jellybeans, there were (2/3) remaining.
Let's represent the initial number of jellybeans as 'x'.
So, after Paul took (1/3)x, there were (2/3)x jellybeans left.
2. After Paul threw back 4 jellybeans, the number became (2/3)x - 4.
3. Sarah ate 6 jellybeans, so the number reduced to (2/3)x - 4 - 6.
4. Sarah then took one-quarter of what remained, leaving (3/4) of the jellybeans.
Therefore, (3/4)[(2/3)x - 4 - 6] = (3/4) * [(2/3)x - 10].
5. Kate took one-third of what remained, leaving (2/3) of the jellybeans.
Therefore, (2/3)[(3/4) * ((2/3)x - 10)] = (2/3) * [(3/4) * ((2/3)x - 10)].
6. Kate also ate the last 4 black jellybeans, so the number reduced to 6 jellybeans.
Therefore, (2/3) * [(3/4) * ((2/3)x - 10)] - 4 = 6.
Now, let's solve this equation step-by-step to find the value of 'x'.
Step 1: Distribute the fractions within the equation.
(2/3) * [(3/4) * ((2/3)x - 10)] - 4 = 6
(2/3) * [(3/4) * (2/3)x - (3/4) * 10)] - 4 = 6
Step 2: Simplify the fractions within the equation.
(2/3) * [(1/2) * (2/3)x - 30/4)] - 4 = 6
(2/3) * [(1/2) * (2/3)x - 15/2)] - 4 = 6
Step 3: Distribute the remaining fractions within the equation.
(2/3) * [(2/6)x - 15/2] - 4 = 6
(2/3) * [x/3 - 15/2] - 4 = 6
Step 4: Simplify the expression within the brackets.
(2/3) * [x/3 - 15/2] - 4 = 6
(2/3) * (x/3 - 15/2) - 4 = 6
(2/3) * (x/3) - (2/3) * (15/2) - 4 = 6
(2/9)x - (10/3) - 4 = 6
Step 5: Combine like terms.
(2/9)x - (10/3) - 4 = 6
(2/9)x - 22/3 = 6
Step 6: Add (22/3) to both sides of the equation.
(2/9)x - 22/3 + 22/3 = 6 + 22/3
(2/9)x = 20/3
Step 7: Multiply both sides by 9 to isolate 'x'.
9 * [(2/9)x] = 9 * (20/3)
2x = 60/3
2x = 20
Step 8: Divide both sides by 2.
(2x)/2 = 20/2
x = 10
Therefore, the initial number of jellybeans was 10.
To solve this problem, we can work backward. We know that there were 6 jellybeans left at the end, so let's imagine we're at that point.
Step 1: Kate ate the last 4 black jellybeans because she liked them. That means these 4 jellybeans were originally in the bowl before Kate ate them.
Step 2: Before Kate ate the last 4 jellybeans, she took one-third of what remained. If these 4 jellybeans were one-third of what remained, then there were 4 x 3 = 12 jellybeans before Kate took her share.
Step 3: Sarah ate 6 jellybeans before Kate took her share. So, before Sarah ate her jellybeans, there were 12 + 6 = 18 jellybeans.
Step 4: Paul threw back 4 jellybeans because he didn't like the black ones. These 4 jellybeans were originally part of the 18 jellybeans before Paul threw them back. So, there were initially 18 + 4 = 22 jellybeans.
Therefore, there were originally 22 jellybeans in the bowl.