Let's solve the problem step-by-step.
1. Initial Ratios and Variables Setup:
- Let [tex]\( A \)[/tex] be the number of adults and [tex]\( C \)[/tex] be the number of children on the ferry initially.
- The ratio of adults to children is given by:
[tex]\[
\frac{A}{C} = \frac{8}{3}
\][/tex]
- We can express [tex]\( A \)[/tex] in terms of [tex]\( C \)[/tex]:
[tex]\[
A = \frac{8}{3}C
\][/tex]
2. Condition After Adults Get Down:
- When 10 adults get down, the number of adults becomes [tex]\( A - 10 \)[/tex].
- The new ratio of adults to children is given as [tex]\( 2:1 \)[/tex]:
[tex]\[
\frac{A - 10}{C} = \frac{2}{1}
\][/tex]
- This gives us the equation:
[tex]\[
A - 10 = 2C
\][/tex]
3. Substitute [tex]\( A \)[/tex] from the First Equation:
- We already have [tex]\( A = \frac{8}{3}C \)[/tex].
- Substitute [tex]\( A \)[/tex] in the second equation:
[tex]\[
\frac{8}{3}C - 10 = 2C
\][/tex]
4. Solve for [tex]\( C \)[/tex]:
- Clear the fraction by multiplying all terms by 3:
[tex]\[
8C - 30 = 6C
\][/tex]
- Simplify the equation:
[tex]\[
8C - 6C = 30
\][/tex]
[tex]\[
2C = 30
\][/tex]
[tex]\[
C = 15
\][/tex]
Hence, the number of children on the ferry is 15.
