Question

Find the value of a : b : c : d, if a : b = 2 : 3, b : c = 4 : 5 and c : d = 6 : 7.

Answer 1

Hint: To solve this question, we need to make the common variables uniform across different ratios by finding the lowest common multiple (LCM) of values of those common variables in different ratios, like b in a : b and b : c and c in b : c and c : d. Once we get the value of b and c such that they are uniform across various ratios, we can combine the ratios with common values, like a : b and b : c can be combined as a : b : c, given b has the same value in a : b and b : c. Thus, in similar fashion, we can find the value of a : b : c : d.

Complete step by step answer:
Let us first consider the value of b.
It is given that a : b = 2 : 3 and b : c = 4 : 5. Thus, LCM of 3 and 4 is 12. Therefore, we need to multiply the first ratio by 4 and the second ratio by 3.
$\Rightarrow$ a : b = 4(2) : 4(3) = 8 : 12
$\Rightarrow$ b : c = 3(4) : 3(5) = 12 : 15
Now, as the value of b is common across the two ratios, we can combine them.
Therefore, a : b : c = 8 : 12 : 15.
Now, we shall make the value of c common.
The ratios with us are a : b : c = 8 : 12 : 15 and c : d = 6 : 7.
LCM of 15 and 6 is 30. Therefore, we need to multiply the first ratio by 2 and the second ratio by 5.
$\Rightarrow$ a : b : c = 2(8) : 2(12) : 2(15) = 16 : 24 : 30
$\Rightarrow$ c : d = 5(6) : 5(7) = 30 : 35
Now, as the value of c is common across the two ratios, we can combine them.

Therefore a : b : c : d = 16 : 24 : 30 : 35

Note: It is to be noted that the multiplying the antecedent and consequent of the ratio with the same number does not change the value of the ratio.