Answer:
36 ladybugs on the marigolds.
Explanation:
Let's denote the initial number of ladybugs on the marigolds as
2
�
2x (since the ratio is 2:3) and on the roses as
3
�
3x.
After 6 ladybugs moved from the roses to the marigolds, the new ratio becomes 7:8. So, the new number of ladybugs on the marigolds is
2
�
+
6
2x+6 and on the roses is
3
�
−
6
3x−6.
The new ratio is given as:
2
�
+
6
3
�
−
6
=
7
8
3x−6
2x+6
=
8
7
Now, cross-multiply to solve for
�
x:
8
(
2
�
+
6
)
=
7
(
3
�
−
6
)
8(2x+6)=7(3x−6)
Expand and solve for
�
x:
16
�
+
48
=
21
�
−
42
16x+48=21x−42
48
+
42
=
21
�
−
16
�
48+42=21x−16x
90
=
5
�
90=5x
�
=
18
x=18
Now that we have the value of
�
x, we can find the initial number of ladybugs on the marigolds:
2
�
=
2
×
18
=
36
2x=2×18=36
So, there were initially 36 ladybugs on the marigolds.