Certainly! Let's solve this step by step to find the ratio of the number of Mathematics books to the number of English books.
1. Identify Given Ratios:
- The ratio of the number of Mathematics books (M) to Science books (S) is 2:3.
- The ratio of the number of Science books (S) to English books (E) is 2:5.
2. Express the Ratios Mathematically:
- The first ratio tells us [tex]\( \frac{M}{S} = \frac{2}{3} \)[/tex].
- The second ratio tells us [tex]\( \frac{S}{E} = \frac{2}{5} \)[/tex].
3. Expressing [tex]\( S \)[/tex] in Terms of [tex]\( M \)[/tex]:
From the first ratio, we can express [tex]\( S \)[/tex] in terms of [tex]\( M \)[/tex]:
[tex]\[
S = \frac{3}{2} M
\][/tex]
4. Expressing [tex]\( S \)[/tex] in Terms of [tex]\( E \)[/tex]:
From the second ratio, we can express [tex]\( S \)[/tex] in terms of [tex]\( E \)[/tex]:
[tex]\[
S = \frac{2}{5} E
\][/tex]
5. Equating the Two Expressions for [tex]\( S \)[/tex]:
Since both expressions equal [tex]\( S \)[/tex], we can set them equal to each other:
[tex]\[
\frac{3}{2} M = \frac{2}{5} E
\][/tex]
6. Solving for [tex]\( \frac{M}{E} \)[/tex]:
To find the ratio [tex]\( \frac{M}{E} \)[/tex], we will solve the equation for [tex]\( M \)[/tex] in terms of [tex]\( E \)[/tex]:
[tex]\[
\frac{3}{2} M = \frac{2}{5} E
\][/tex]
Multiply both sides by 10 to clear the denominators:
[tex]\[
15M = 4E
\][/tex]
7. Isolating [tex]\( M \)[/tex] over [tex]\( E \)[/tex]:
To find the ratio [tex]\( \frac{M}{E} \)[/tex]:
[tex]\[
\frac{M}{E} = \frac{4}{15}
\][/tex]
Thus, the ratio of the number of Mathematics books to the number of English books is [tex]\( \frac{4}{15} \)[/tex].
