Seth had some marbles. He gave 2/3 of his marbles to Henry and bought 4 new marbles. After that, he gave 2/7 of his remaining marbles to his cousin and bought another 7 new marbles. In the end, Seth had 47 marbles. How many marbles did Seth give to his cousin? How many marbles did Seth have at first?
Let the initial quantity of marble = X
Then, the number of marbles given to Henry = (2/3) * X
Now,
Quantity of marbles left with Seth = (X - (2/3) * X)
Quantity of marbles left with Seth = (1/3) * X
After adding 4 marbles,
The new level of quantity of marbles left with Seth = (1/3) * X + 4
Now 2/7 of the new level of quantity is given to cousin, then
The remaining quantity of marble with Seth = ((1/3) * X + 4) - (2/7) * ((1/3) * X + 4)
Now 7 new marbles are added, then:
Final quantity = 47
((1/3 * X + 4) - (2/7) * ((1/3) * X + 4) + 7 = 47
((1/3 * X + 4) - (2/7) * ((1/3) * X + 4) = 47 - 7
40 = ((1/3) * X - (2/21) * X + 4 - (2/7) * 4
X = (40 - 4 + (2/7) * 4) / ((1/3) - (2/21))
X = 156
So,
Number of marbles given to cousin =
(2/7) * ((1/3) * 156 + 4)
Number of marbles given to cousin = 16
And,
Number of marbles with Seth at first = 156
Hence the correct answer:
Number of marbles given to cousin = 16
Number of marbles with Seth at first = 156
To solve this problem, let's break it down step by step:
Step 1: Given that Seth gave 2/3 of his marbles to Henry, we can represent the number of marbles he had before as "x". Therefore, Seth gave away (2/3)x marbles to Henry.
Step 2: After giving marbles to Henry, Seth had x - (2/3)x = (1/3)x marbles left.
Step 3: Seth bought 4 new marbles, which means he had (1/3)x + 4 marbles now.
Step 4: Seth gave 2/7 of his remaining marbles to his cousin, so he gave away (2/7)(1/3)x marbles.
Step 5: After giving marbles to his cousin, Seth had (1/3)x + 4 - (2/7)(1/3)x marbles left.
Step 6: Seth bought 7 new marbles, so he had (1/3)x + 4 - (2/7)(1/3)x + 7 = 47 marbles.
Now let's solve the equation:
(1/3)x + 4 - (2/7)(1/3)x + 7 = 47
First, simplify the equation,
(1/3)x + 11/21x + 11 = 47
Combine like terms,
(9/21)x + 11 = 47
Next, convert the fractions to a common denominator,
(9/21)x + (11/21) = 47
Now, add the fractions,
(20/21)x + (11/21) = 47
Subtract (11/21) from both sides,
(20/21)x = 47 - 11/21
Simplify,
(20/21)x = (987/21) - (11/21)
(20/21)x = (976/21)
Divide both sides by (20/21),
x = (976/21)/(20/21)
Simplify,
x = 976/20
x = 48.8
Since we're dealing with marbles, which are whole numbers, we know that Seth had 48 marbles initially.
To find out how many marbles Seth gave to his cousin, substitute x = 48 into the equation we found in Step 5:
(1/3)(48) + 4 - (2/7)(1/3)(48) + 7 = 47
(16/3) + 4 - (32/21) + 7 = 47
Simplify,
(48/3) + 12 - (32/21) + 7 = 47
16 + 12 - (32/21) + 7 = 47
35 - (32/21) + 7 = 47
Now find the common denominator of 21,
(735/21) - (32/21) + (147/21) = 47
Combine like terms,
(735 - 32 + 147)/21 = 47
Simplify,
850/21 = 47
Divide both sides by (850/21),
47 = (850/21)/(47/1)
Simplify,
47 = 850/21 * 1/47
47 = 850/987
So, Seth gave 850/987 of his marbles to his cousin.
Simplified, this is approximately 0.8624.
Therefore, Seth gave approximately 0.8624 * 48 = 41.4 marbles to his cousin.
So, Seth gave approximately 41 marbles to his cousin, and he initially had 48 marbles.
Let's break down the problem step by step:
1. Seth gave 2/3 of his marbles to Henry. Let's represent the number of marbles Seth had at this point as 'x'. After giving away 2/3 of his marbles, Seth had (1 - 2/3) * x = (1/3) * x marbles left.
2. Seth bought 4 new marbles. After buying 4 new marbles, Seth had (1/3) * x + 4 marbles.
3. Seth gave 2/7 of his remaining marbles to his cousin. Let's represent the number of marbles Seth had at this point as 'y'. After giving away 2/7 of his marbles, Seth had (1 - 2/7) * y = (5/7) * y marbles left.
4. Seth bought 7 new marbles. After buying 7 new marbles, Seth had (5/7) * y + 7 marbles.
5. In the end, Seth had 47 marbles. So, we can set up the equation: (5/7) * y + 7 = 47.
Now, let's solve the equation to find the value of y:
(5/7) * y + 7 = 47
(5/7) * y = 47 - 7
(5/7) * y = 40
Multiplying both sides by 7 to get rid of the fraction:
y = 40 * 7 / 5
y = 56
So Seth had 56 marbles at the point where he gave 2/7 of his marbles to his cousin.
To find the number of marbles Seth gave to his cousin, we need to calculate (2/7) * y:
(2/7) * 56
= (2 * 56) / 7
= 112 / 7
= 16
Therefore, Seth gave 16 marbles to his cousin.
To find the number of marbles Seth had at first, we need to add the marbles he gave to Henry, the marbles he bought, the marbles he gave to his cousin, and the marbles he had at the end:
x = (1/3) * x + 4 + 16 + 47
(2/3) * x = 67
x = 67 * (3/2)
x = 100.5
Since we can't have a fraction of a marble, the number of marbles Seth had at first would be 100.
Therefore, Seth gave 16 marbles to his cousin, and he had 100 marbles at first.