Jack and Jill got an equal number of marbles from their grandpa's collection. Jill gave 50 marbles to Jack because she saw that he was more interested in marbles than she was. Now Jack has 5 times as many marbles as Jill. How many total marbles did grandpa give away and how many did they each get?
This is a logic problem. I will write out the steps below.
Ja= Jack
Ji= Jill
Jill gives 50 to Jack, so she has 50 less and he has 50 more.
And, this quantity of 50 more is equal to 5 times Jill's amount.
So, now Jill has Ji-50 and Jack has Ja + 50
So, now just think how to set this up algebraically: If they both had 100 to start with, and Jill gives away 50, she has 50 now, and Jack has 150. So, the difference between the two will always be 100, no matter what number you choose for the beginning value. So, Jill will have 100 less marbles than Jack now.
Ji= Ja- 100
Now we develop a new relationship:
Ja= 5 Ji
Ja= 5 ( Ja - 100)
Ja= 5Ja-500
Ja + 500= 5Ja
4Ja= 500
Ja= 125
Now solve for Ji using Ja= 5 Ji
125= 5 Ji
J= 25
Now since Jill gave him 50 and he took 50, you subtract 50 from his pile, and add 50 to hers.
So, Jill had 75 to begin with and Jack had 75 to begin with.
Check backwards:
75-50= 25= Ji now
75 + 50= 125= Ja now
Is this statement true: Ja= 5 Ji?
125= 5 * 25
125= 125
Yes! It is. So, this is your answer!
Hope this helps!
This is a question to the answer above
Got it. I was confused with the question because what if they did NOT start with the same amount but did receive same amount from grandpa? Is that a different problem or cannot be done?
Yoyoma, can you please answer the question on what if Jack & Jill did NOT start with equal amount of marbles?
To solve this problem, we can use algebraic equations. Let's call the number of marbles Jack and Jill received from their grandpa "x." According to the information given:
Jack received x marbles from grandpa.
Jill received x marbles from grandpa.
After Jill gave 50 marbles to Jack, Jack now has 5 times as many marbles as Jill. We can express this as an equation:
Jack's marbles = 5 * Jill's marbles
Since Jack received x marbles and Jill gave him 50 marbles, Jack now has x + 50 marbles. Jill, on the other hand, gave away 50 marbles, so she has x - 50 marbles.
Using the equation mentioned earlier, we can write:
x + 50 = 5 * (x - 50)
Now we can solve this equation to find the value of x, which represents the number of marbles initially received from their grandpa.
Expanding the right side of the equation:
x + 50 = 5x - 250
Rearranging the equation:
4x = 300
Dividing both sides by 4:
x = 75
So, each of them initially received 75 marbles from their grandpa. To find out the total number of marbles grandpa gave away, we can sum up the marbles Jack and Jill received:
Total marbles = Jack's marbles + Jill's marbles = 75 + 75 = 150
Therefore, grandpa gave away 150 marbles in total.