Question

Find the compound ratios of the following:
3:4 and 2:3

Answer 1

Hint: First we will look at the definition of compound ratio. It is the ratio of the product of antecedents to consequents of the ratios. Then, we will apply it to the given ratios. To do so, we will consider antecedent as 3 and 2 respectively. Then, we will consider consequent as 4 and 3 respectively. The ratio of the product of these will get us the compound ratio. The value of this compound ratio is the required result in this question.

Complete step-by-step answer:
Compound ratio: For two or more ratios, if we take antecedent as the product of antecedents of ratios and consequent as the product of consequences of ratio, then the ratio thus formed is called the compound ratio. It is also known as the mixed ratio (generally). It can also be defined as the ratio obtained when two or more ratios we multiplied term wise.
The antecedent of a ratio: The term which is before the ratio symbol (given by “:”) is called as the antecedent of that ratio.
The consequence of a ratio: The term which is after the ratio symbol (given by “:”) is called as the consequence of that ratio.
We can write the given ratios in the question as
3:4 and 2:3
Let us assume the first ratio to be denoted by X.
Let us assume the second ratio to be dented by Y.
By writing the value of X, we can write it in the form of:
X=3:4
By definition, we can say the value of antecedent of X as:
consequent of X = 3 …… (i)
By definition, we can say the value of consequent of X as:
consequent of X =4 …… (ii)
By definition, we can say the value of antecedent of Y as:
antecedent of Y = 2 …… (iii)
By definition, we can say the value of consequent of Y as:
consequent of Y = 3 …… (iv)
By equations (i), (iii), we can say the value of the product as:
product of us antecedents = $3\times 2$
By simplifying the above equation, we get the equation as:
product of antecedents = 6 --- (v)
By equations (ii), (iv), we can say the value of the product as:
product of consequents = $4\times 3$
By simplifying the above equation, we get the equation as:
product of consequents = 12 --- (vi)
By the equations (v), (vi), we can say the value of compound ratio:
Compound ratio = 6:12 = 1:2.
Therefore 1:2 is the compound ratio for the given ratios.

Note: Be careful while finding the value of antecedents, consequents, because this forms the base for our solution. Students get confused and find the product of antecedents, consequent instead of the antecedent, antecedent. This way you will get the wrong result, so do the product of similar terms carefully. At last, don't forget to write the simplest form possible. We can also solve such questions using another technique. We have ratios as a:b and c:d. To find their compound ratio, we write them as ac:bd. This is the same as $\dfrac{ac}{bd}\Rightarrow \dfrac{a}{b}\times \dfrac{c}{d}$ . So, we just have to multiply the ratios in fraction form. For this question, we can get the compound ratio as $\dfrac{3}{4}\times \dfrac{2}{3}\Rightarrow \dfrac{2}{4}\Rightarrow \dfrac{1}{2}$ . This is a faster and simpler method.