Question

If a:b = 2:1, b:c = 1:3, c:d = 2:3 and d:e = 1:2, find a:b:c:d:e.

Answer 1

Hint: This a question of ratio and proportions. The concept of ratio is similar to division. The first number is taken as numerator and second as denominator. For example-
X:Y = 1:2 is the same as $\dfrac{\mathrm X}{\mathrm Y}=\dfrac12$

Complete step by step answer:
To solve this problem, find all the variables in terms of one variable(say a), and then write them in a ratio so that the common variable gets cancelled.

Now, we are given the ratios, so we will first convert them into fractions-
$\dfrac{\mathrm a}{\mathrm b}=2\\\mathrm a=2\mathrm b....\left(1\right)\\\dfrac{\mathrm b}{\mathrm c}=\dfrac13\\\mathrm c=3\mathrm b....\left(2\right)\\\dfrac{\mathrm c}{\mathrm d}=\dfrac23\\\mathrm d=\dfrac32\mathrm c\\\mathrm{Using}\;\mathrm{equation}\left(2\right),\;\\\mathrm d=\dfrac32\times3\mathrm b=\dfrac92\mathrm b....\left(3\right)\\\dfrac{\mathrm d}{\mathrm e}=\dfrac12\\\mathrm e=2\mathrm d\\\mathrm{Using}\;\mathrm{equation}\left(3\right),\;\\\mathrm e=2\times\dfrac92\mathrm b=9\mathrm b....\left(4\right)\\\;$
We have to find a:b:c:d:e. Using equations (1), (2), (3) and (4), we can write that-
$\mathrm a:\mathrm b:\mathrm c:\mathrm d:\mathrm e=2\mathrm b:\mathrm b:3\mathrm b:\frac92\mathrm b:9\mathrm b\\\mathrm{Since}\;\mathrm b\;\mathrm{is}\;\mathrm{common},\;\mathrm{it}\;\mathrm{can}\;\mathrm{be}\;\mathrm{elIminated}.\\\mathrm a:\mathrm b:\mathrm c:\mathrm d:\mathrm e=2:1:3:\frac92:9\\\mathrm{To}\;\mathrm{simplify}\;\mathrm{the}\;\mathrm{answer}\;\mathrm{we}\;\mathrm{can}\;\mathrm{multiply}\;\mathrm{all}\;\mathrm{the}\;\mathrm{terms}\;\mathrm{by}\;2\\\mathrm a:\mathrm b:\mathrm c:\mathrm d:\mathrm e=4:2:6:9:18$
This is the required answer.

Note: Instead of b, any other variable can be taken as reference. A common mistake is that the students don't simplify the final answer and leave it in fractional form, which is incorrect. To simplify the ratio, one can multiply the whole ratio by the LCM of the denominators.