Certainly! Let's solve the problem step by step.
Given the ratios:
[tex]\[
\frac{a}{b} = \frac{2}{5}
\][/tex]
and
[tex]\[
\frac{b}{c} = \frac{3}{4}
\][/tex]
1. Express [tex]\(a\)[/tex] in terms of [tex]\(b\)[/tex]:
[tex]\[
a = \frac{2}{5} b
\][/tex]
2. Express [tex]\(c\)[/tex] in terms of [tex]\(b\)[/tex]:
[tex]\[
c = \frac{4}{3} b
\][/tex]
3. Choose a common multiple to scale [tex]\(b\)[/tex] for consistent comparison:
We can use the least common multiple (LCM) of the denominators involved in the ratios. Here, the denominators are 5 and 4.
The LCM of 5 and 4 is 20.
4. Scale [tex]\(b\)[/tex] to the common multiple:
Let [tex]\(b\)[/tex] be scaled to the common multiple, which is 20.
[tex]\[
b = 20
\][/tex]
5. Calculate [tex]\(a\)[/tex] using the scaled [tex]\(b\)[/tex]:
[tex]\[
a = \frac{2}{5} \times 20 = 8.0
\][/tex]
6. Calculate [tex]\(c\)[/tex] using the scaled [tex]\(b\)[/tex]:
[tex]\[
c = \frac{3}{4} \times 20 = 26.666666666666664
\][/tex]
7. Write the final ratios:
The values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are 8.0, 20, and 26.666666666666664 respectively. Hence, the ratio [tex]\(a : b : c\)[/tex] is:
[tex]\[
8.0 : 20 : 26.666666666666664
\][/tex]
So, the ratio [tex]\(a : b : c\)[/tex] is [tex]\(8.0 : 20 : 26.666666666666664\)[/tex].
