Two shelves contain 55 books. If half of the books from the second shelf were relocated to the first shelf, then the first shelf would contain 4 times more books than the second one. How many books are there on each shelf?
number of books on first shelf ---- x
number of books on 2nd shelf ---- 55-x
after re-location
first shelf --- (55-x)/2 + x
2nd shelf ---- (55-x)/2
(55-x)/2 + x = 4(55-x)/2
times 2
55-x + 2x = 220 - 4x
5x = 165
x = 33
before the shift:
1st shelf ---- 33 books
2nd shelf ---- 22 books
Well, it seems like those shelves are having quite the book-keeping dilemma! Let's try to solve it with a touch of humor.
Let's call the number of books on the first shelf 'x' and the number of books on the second shelf 'y'. According to the given information, we have two equations.
Equation 1: x + y = 55 (since the total number of books on both shelves is 55)
Equation 2: x + y/2 = 4y (since if half of the books from the second shelf were moved to the first, the first shelf would have 4 times more books than the second)
Now, let's put on our math hats and solve this puzzle!
From Equation 1, we can rearrange it as x = 55 - y.
Substituting this value for x in Equation 2, we get (55 - y) + y/2 = 4y.
Simplifying, we have: 55 - y + y/2 = 4y.
To get rid of that pesky fractional y/2, let's multiply the whole equation by 2. That gives us: 110 - 2y + y = 8y.
Bringing the y terms to one side, we have: 110 = 8y + 2y - y.
Simplifying further, we get: 110 = 9y.
Now we can solve for y by dividing both sides by 9. So: y = 110/9.
Now, since that result is not a whole number, let's get those clown noses on and make a silly assumption. Since we can't have a fraction of a book, let's say that y = 12 (I'll keep my clown fingers crossed it works!).
Now, substituting that value of y back into Equation 1, we have: x + 12 = 55.
Rearranging this, we find: x = 55 - 12.
So, x = 43.
Therefore, we have 43 books on the first shelf and 12 books on the second shelf.
Let's assume the number of books on the first shelf is x and the number of books on the second shelf is y.
According to the problem, we know that:
x + y = 55 ---(1) (The total number of books on both shelves is 55)
After relocating half of the books from the second shelf to the first shelf, the number of books on the first shelf would be x + y/2, and the number of books on the second shelf would be y/2.
According to the problem, the first shelf would contain 4 times more books than the second one. Mathematically, this can be expressed as:
x + y/2 = 4 * (y/2) ---(2)
We can simplify equation (2) by multiplying both sides by 2:
2x + y = 4y ---(3)
Now, we have a system of equations (equations (1) and (3)) that we can solve to find the values of x and y.
First, let's multiply equation (1) by 2:
2x + 2y = 110 ---(4)
Now, we can subtract equation (3) from equation (4) to eliminate the variable y:
(2x + 2y) - (2x + y) = 110 - 4y
Simplifying:
2y - y = 110 - 4y
y = 110 - 4y
Combining like terms:
5y = 110
Dividing both sides by 5:
y = 22
Now, we can substitute the value of y back into equation (1) to find x:
x + 22 = 55
x = 55 - 22
x = 33
So, there are 33 books on the first shelf and 22 books on the second shelf.
Let's assume that the number of books on the first shelf is represented by x, and the number of books on the second shelf is represented by y.
According to the problem, we know that the total number of books on both shelves is 55. Therefore, we can create the equation:
x + y = 55 -- Equation 1
We are also given that if half of the books from the second shelf were relocated to the first shelf, then the first shelf would contain 4 times more books than the second shelf. Mathematically, this can be represented as:
x + (y/2) = 4y -- Equation 2
To solve this system of equations, we can substitute Equation 1 into Equation 2:
55 - y + (y/2) = 4y
Multiplying through by 2 to clear the fraction:
110 - 2y + y = 8y
Combining like terms:
9y = 110
Dividing both sides by 9:
y = 110/9 = 12.22 (approximately)
Since the number of books must be a whole number, we know that y must be 12.
Substituting y=12 back into Equation 1:
x + 12 = 55
x = 55 - 12
x = 43
Therefore, there are 43 books on the first shelf and 12 books on the second shelf.