Question

The continued ratio of \[4:3\] and \[5:6\] is_________
A) \[4:15:6\]
B) \[4:5:6\]
C) \[20:15:12\]
D) \[20:15:18\]

Answer 1

Hint: Here in this question, we have to find the next or continued ratio of given two ratios. to find this, Multiplying ratio 1 by antecedent of ratio 2 and multiply ratio 2 by consequent of ratio 1 and further compare the two resultant ratios and by using the properties of ratios we can write the continued ratios.

Complete step by step solution:
 a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent.
The most common way to write a ratio using a colon i.e., \[a:b\]
Where, a is antecedent and b is the consequent.
Consider the given ratio,
\[ \,\,4:3\] and \[5:6\]
In ratio 1- 4 is antecedent and 3 is consequent.
In ratio 2- 5 is antecedent and 6 is consequent.
Now find the continued ratio of \[4:3\] and \[5:6\].
To find the continued ratio:
First Multiply, ratio 1 by antecedent of ratio 2 and multiply ratio 2 i.e.,
\[ \Rightarrow \,\,4:3\]
It can be written as
\[ \Rightarrow \,\,\dfrac{4}{3}\]
Antecedent of 2 ratio is 5, then
\[ \Rightarrow \,\,\dfrac{{4 \times 5}}{{3 \times 5}}\]
\[ \Rightarrow \,\,\dfrac{{20}}{{15}}\]
Or
\[ \Rightarrow \,\,20:15\]-------(1)
Now multiply, ratio 2 by consequent of ratio 1
\[ \Rightarrow \,\,5:6\]
It can be written as
\[ \Rightarrow \,\,\dfrac{5}{6}\]
Consequent of 1 ratio is 3, then
\[ \Rightarrow \,\,\dfrac{{5 \times 3}}{{6 \times 3}}\]
\[ \Rightarrow \,\,\dfrac{{15}}{{18}}\]
Or
\[ \Rightarrow \,\,15:18\]-------(2)
From the definition, for two ratios \[a:b\] and \[b:c\] then \[a:b:c\] is called the continued ratio. Then
From (1) and (2), we get
\[ \Rightarrow \,\,20:15:18\]

Therefore, \[20:15:18\] is the continued ratio of \[4:3\] and \[5:6\]. So, Option (D) is correct.

Note:
In mathematics, a ratio can also be in 3 quantities as well as 2 quantities. Remember when 3 or more quantities are involved in a ratio, it is called a "continued ratio" but the ratio is continuing beyond 2 quantities it means there may be a situation when we have to compare more than two quantities which are in a continued ratio.