Ratio and Proportion
A Ratio is the relation between two quantities of the same kind. This relation indicates how many times one quantity is equal to the other. In other words, ratio is a number, which expresses one quantity as a fraction of the other.
Example: Ratio of 12 to 13 is 12/13 or 12 : 13.
The numbers forming the ratio are called terms. The numerator, i.e. "12", is known as the antecedent and the denominator, i.e. "13", in this case, is known as the consequent.
The ratio between two quantities a and b if expressed as a/b, is called fractional form and, a : b is called linear form.
If two different ratios, a : b and c : d are expressed in different units, then the two are compounded to obtain a combined ratio.
Compounding of a : b and c : d yields (a*c)/(b*d).
- If a/b=c/d=e/f, then each of these ratios is equal to (a+c+e) ⁄(b+d+f)
- If a/b=c/d, then b/a=d/c (Invertendo)
- If a/b=c/d, then a/c=b/d (Alterendo)
- If a/b=c/d, then (a+b)/b=(c+d)/d (Componendo)
- If a/b=c/d, then (a-b)/b=(c-d)/d (Dividendo)
- If a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d) (Componendo et dividendo)
- Four numbers a, b, c and d are said to be in proportion if a : b = c : d. If on the other hand, a : b = b : c = c : d, then the four numbers are said to be in continued proportion.
- Let us consider the ratios, a : b = b : c. Here b is called the mean proportional and is equal to the square root of the product of a and c i.e. b2 = a *c ⇒ b = √ac
- If a/b=b/c=c/d etc., then a, b, c, d are in geometric progression.
- let a/b=b/c=c/d=k, then, c=dk;b=ck and a=bk
- Since c = dk, b = dk * k = dk2 and a = bk = dk2 * k = dk3, implying that they are in geometric progression.
- If the three ratios, a : b, b : c, c : d are known, we can find a : d by the multiplying these three ratios a/d = a/b * b/c * c/d
- If a, b, c and d are four terms and the ratios a : b, b : c, c : d are known, then one can find the ratio a : b : c : d.
An association of two or more persons who invest money together in order to carry out a certain business is termed as partnership.
Partnerships are of two types:
- Simple Partnership: When all partners invest in the business at the same time i.e. their capital remains in the business for the same duration it is called simple partnership. In this kind of partnership, the profit is simply distributed amongst the partners, in the ratio of their respective invested capital.
- Compound Partnership: When capital of the partners is invested in the business for different time periods, the partnership is known as compound partnership. In this, the profit sharing ratio is calculated by multiplying the capital invested with the unit of time (mostly months).
The sharing of profit and loss can be better understood with the help of the following illustrations:
Rule 1: In a simple partnership, the loss or profit is distributed amongst the partners in the ratio of their respective investments.
Example: Say, P and Q invested Rs. a and b for one year in a business. Then, the share of profit or loss will be,
P's profit (or loss) : Q's profit(or loss) = a : b
Rule 2: In a compound partnership, the profit or loss ratio is calculated as capital multiplied by the duration of investment.
Example: P's profit(or loss) : Q's profit(or loss) = a* t1 : b* t2
where, t1 = P's duration of investment and, t2 = Q's duration of investment