The ratio of boys to girls in a group of children was 3:5.Then 24 girls left the group and 24 more boys joined the group.The ratio of boys to girls became 5:3.How many boys and girls were in the original group?
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6 Answers
This is a system of two mathematical linear equations:
x - boys, y - girls.
x / y = 3 / 5
(x + 24) / (y - 24) = 5 / 3
x = (3 * y ) / 5
((3 * y) / 5 + 24) / (y - 24) = 5 / 3
here you will get number of girls, and then you will get number of boys.
Ok.* is multiplication symbol. / is division symbol.
We stop here:
((3*y)/5 + 24)/(y-24) = 5 / 3
Then:
(3 * y) / 5 + 24 = (5 / 3) * ( y - 24)
3/5 * y - 5/3 * y = - 5/3 * 24 - 24
y * ( 3/5 - 5/3) = (-5 * 24 - 24 * 3) / 3
y * ( (3 * 3 - 5 * 5) / 15) = (-120 - 72) / 3
y * (9 - 25) = (-120 - 72) * 5
- 16 * y = - 960
y = 960 / 16
y = 60, x = 36
36 boys and 60 girls was the original group
We stop here:
((3*y)/5 + 24)/(y-24) = 5 / 3
Then:
(3 * y) / 5 + 24 = (5 / 3) * ( y - 24)
3/5 * y - 5/3 * y = - 5/3 * 24 - 24
y * ( 3/5 - 5/3) = (-5 * 24 - 24 * 3) / 3
y * ( (3 * 3 - 5 * 5) / 15) = (-120 - 72) / 3
y * (9 - 25) = (-120 - 72) * 5
- 16 * y = - 960
y = 960 / 16
y = 60, x = 36
36 boys and 60 girls was the original group
B:G
3:5
5:3. After 24 girls left and 24 boys joined
So changes occur in both boys and girls, now check the difference between the ratios
Boys becomes 3 to 5 I.e difference of 2
Girls becomes 5 to 3 I.e difference if 2
So this 2 is equals to 24 original difference is of 24
2=24
1=12
Therefore originally
Boys = 3*12. ( by taking original ratio,)
=36
Girls = 5* 12
60
3:5
5:3. After 24 girls left and 24 boys joined
So changes occur in both boys and girls, now check the difference between the ratios
Boys becomes 3 to 5 I.e difference of 2
Girls becomes 5 to 3 I.e difference if 2
So this 2 is equals to 24 original difference is of 24
2=24
1=12
Therefore originally
Boys = 3*12. ( by taking original ratio,)
=36
Girls = 5* 12
60
Let the number of boys be x
Let the no. Of girls be y.
According to statement 1, the ratio of boys to girls is 3:5, thus, the number of boys initially is 3x
And girls is 5y.
On adding 24 boys, the ratio becomes 5x.
Therefore
24+3x=5x
24=5x-3x
24=2x
12= x
Similarly
5y-24=3y
2y=24
y=12
Thus, based on condition one, the initial number
Of boys: 3(12) =36
Girls: 5(12)=60
Let the number of boys and girls in the original group be 3x and 5x respectively.
After 24 girls left the group, the number of girls became 5x-25.
After 24 more boys joined the group, the number of boys became 3x+24. At the point the ratio of boys to girls became 5:3. So we can write:
(3x+24)/(5x-24)=5/3
Cross-multiplying and simplifying, we get:
9x+72 =25x-120
after simplifying, we get
x=12, then we have 36 boys and 60 girls as final answer
After 24 girls left the group, the number of girls became 5x-25.
After 24 more boys joined the group, the number of boys became 3x+24. At the point the ratio of boys to girls became 5:3. So we can write:
(3x+24)/(5x-24)=5/3
Cross-multiplying and simplifying, we get:
9x+72 =25x-120
after simplifying, we get
x=12, then we have 36 boys and 60 girls as final answer