In a library, if 14 girls leave the library, the ratio of the number al boys 1o
the number of giris that remain in the library will be 2:1. If 14 boys
leave the library, the ratio of the number of boys to the number of girs
that remain in the library will be 3: 5.
How many children were there in the library altogether?
2x-14=1x+14 = 3:5
To solve this problem, let's assume the initial number of girls in the library is G and the initial number of boys is B.
According to the first statement, if 14 girls leave the library, the ratio of boys to the remaining girls is 2:1. This can be written as:
(B)/(G-14) = 2/1
Cross-multiplying, we get:
2(G - 14) = B
Expanding the equation, we have:
2G - 28 = B ----(Equation 1)
According to the second statement, if 14 boys leave the library, the ratio of boys to the remaining girls is 3:5. This can be written as:
(B - 14)/(G) = 3/5
Cross-multiplying, we get:
5(B - 14) = 3G
Expanding the equation, we have:
5B - 70 = 3G ----(Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with two unknowns (G and B). We can solve these equations to find the values of G and B.
Using Equation 1, we can rewrite it as:
B = 2G - 28
Substituting this value of B into Equation 2:
5(2G - 28) - 70 = 3G
Expanding and simplifying:
10G - 140 - 70 = 3G
Combining like terms:
7G - 210 = 0
Adding 210 to both sides:
7G = 210
Dividing both sides by 7:
G = 30
Substituting the value of G back into Equation 1:
B = 2(30) - 28
B = 32
Therefore, there were initially 30 girls and 32 boys in the library. To find the total number of children, we add the number of girls and boys:
Total number of children = 30 + 32 = 62
So, there were 62 children in the library altogether.
To find the total number of children in the library, we need to solve this problem using algebra.
Let's assume the initial number of boys in the library is B, and the initial number of girls is G.
According to the first condition, when 14 girls leave the library, the ratio of the number of boys to the number of girls that remain is 2:1. So, we can write an equation for this condition:
(B) / (G - 14) = 2/1
Simplifying this equation, we get:
B = 2(G - 14) ----(Equation 1)
According to the second condition, when 14 boys leave the library, the ratio of the number of boys to the number of girls that remain is 3:5. So, we can write another equation for this condition:
(B - 14) / (G) = 3/5
Simplifying this equation, we get:
5(B - 14) = 3G ----(Equation 2)
Now, we can solve these two equations simultaneously to find the values of B and G.
Let's start by expanding Equation 2:
5B - 70 = 3G ----(Equation 3)
Now, we can multiply Equation 1 by 5:
5B = 4G - 112 ----(Equation 4)
Substituting Equation 3 into Equation 4, we get:
5(4G - 70) = 4G - 112
Simplifying this equation, we get:
20G - 350 = 4G - 112
Bringing like terms to one side, we get:
20G - 4G = 350 - 112
16G = 238
Dividing both sides by 16, we get:
G = 14.875
Since the number of children cannot be in decimal form, we need to round up the value of G to the nearest whole number:
G = 15 (approximate)
Now, substitute the value of G back into Equation 1 to find the number of boys:
B = 2(G - 14)
B = 2(15 - 14)
B = 2(1)
B = 2
So, there were approximately 2 boys and 15 girls in the library initially.
To find the total number of children, we add the number of boys and girls together:
Total number of children = 2 + 15 = 17
Therefore, there were approximately 17 children in the library altogether.
b/(g-14) = 2/1
(b-14)/(g-14) = 3/5
b=20 and g=24
b+g = 44