Let's break down the given problem into clear, logical steps.
### Part (a)
First, we need to determine how many books Sulin had initially.
1. Initial Ratio:
The initial ratio of the number of Meili's books to Sulin's books is 1:2. Let's denote the number of Meili's books as [tex]\( M \)[/tex] and the number of Sulin's books as [tex]\( S \)[/tex].
Thus, we can express this ratio as:
[tex]\[
\frac{M}{S} = \frac{1}{2} \implies M = \frac{S}{2}
\][/tex]
2. Ratio After Meili Buys More Books:
After Meili buys 12 more books, the new ratio of Meili's books to Sulin's books becomes 2:1.
This can be expressed as:
[tex]\[
\frac{M + 12}{S} = 2 \implies M + 12 = 2S
\][/tex]
3. Solving the Equations:
- First, substitute [tex]\( M \)[/tex] from the initial ratio into the second equation:
[tex]\[
\left(\frac{S}{2}\right) + 12 = 2S
\][/tex]
- Next, clear the fraction by multiplying every term by 2:
[tex]\[
S + 24 = 4S
\][/tex]
- Finally, solve for [tex]\( S \)[/tex]:
[tex]\[
24 = 3S \implies S = 8
\][/tex]
So, Sulin initially had 8 books.
### Part (b)
Next, we need to determine the new ratio of the number of Meili's books to Sulin's books if Sulin buys another 5 books.
1. Initial Books:
- Meili initially had:
[tex]\[
M = \frac{S}{2} = \frac{8}{2} = 4
\][/tex]
2. Books After Buying More:
- After buying 12 more books, Meili had:
[tex]\[
4 + 12 = 16
\][/tex]
- If Sulin buys another 5 books, he will have:
[tex]\[
8 + 5 = 13
\][/tex]
3. New Ratio:
The new ratio of Meili's books to Sulin's books will be:
[tex]\[
\frac{16}{13}
\][/tex]
Hence, after Sulin buys 5 more books, the new ratio of the number of Meili's books to Sulin's books will be 16:13.
