To solve the problem of finding the ratio [tex]\(a : b : c\)[/tex] given [tex]\(a : b = 2 : 3\)[/tex] and [tex]\(b : c = 4 : 9\)[/tex], follow these steps:
1. Express the given ratios:
- [tex]\(a : b\)[/tex] is given as [tex]\(2 : 3\)[/tex]. This means [tex]\( \frac{a}{b} = \frac{2}{3} \)[/tex].
- [tex]\(b : c\)[/tex] is given as [tex]\(4 : 9\)[/tex]. This means [tex]\( \frac{b}{c} = \frac{4}{9} \)[/tex].
2. Find a common term for [tex]\( b \)[/tex]:
We need to find a common value for [tex]\( b \)[/tex] in both ratios to easily compare [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
To do this, we need to find a common multiple of the two [tex]\( b \)[/tex]-values from both ratios. In [tex]\(a : b = 2 : 3\)[/tex], [tex]\(b = 3x\)[/tex], and in [tex]\(b : c = 4 : 9\)[/tex], [tex]\(b = 4y\)[/tex].
3. Determine the least common multiple (LCM) of 3 and 4:
The least common multiple (LCM) of 3 and 4 is [tex]\(12\)[/tex]. This gives us a consistent value for [tex]\( b \)[/tex] across both ratios.
4. Adjust the ratios accordingly:
- For the ratio [tex]\(a : b = 2 : 3\)[/tex]:
[tex]\(\frac{a}{b} = \frac{2}{3}\)[/tex]
If we want [tex]\(b\)[/tex] to be 12, we adjust [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:
[tex]\(\frac{a}{b} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)[/tex]
- For the ratio [tex]\(b : c = 4 : 9\)[/tex]:
[tex]\(\frac{b}{c} = \frac{4}{9}\)[/tex]
If we want [tex]\(b\)[/tex] to be 12, we adjust [tex]\(b\)[/tex] and [tex]\(c\)[/tex] as follows:
[tex]\(\frac{b}{c} = \frac{4 \times 3}{9 \times 3} = \frac{12}{27}\)[/tex]
5. Combine the adjusted ratios:
Now we have:
[tex]\[
a : b = 8 : 12 \\
b : c = 12 : 27
\][/tex]
Combining these, we get [tex]\( a : b : c \)[/tex].
6. Final ratio:
The final ratio is [tex]\( a : b : c = 8 : 12 : 27 \)[/tex].
So, the solution yields the ratio [tex]\( a : b : c = 8 : 12 : 27 \)[/tex].
