Business Mathematics - Ratio and Proportion

 

Business Mathematics

RATIO AND PROPORTION

Ratio Formulas

1	If a and b are two quantities of the same kind (in same units), then a/b or a: b are called "ratio of a to b". Here, 'a' is called the first term or antecedent and 'b' is called the second term or consequent.
2	a: b is the inverse ratio of b: a and vice-versa.
3	Compound ratio of a: b and c: d is ac: bd.
4	a^2: b^2 is the duplicate ratio of a: b.
5	a^3: b^3 is the triplicate ratio of a: b.
6	a^ (1/2): b^ (1/2) is the sub-duplicate ratio of a: b.
7	a^ (1/3): b^ (1/3) is the sub-triplicate ratio of  a: b.
8	a: b: c is the Continued Ratio of a, b and c. Here, a, b and c are of the same kind and same unit.

 

Proportion Formulas

1	An equality of the two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion, if a: b = c: d.
2	a, b, c, d are called terms of the proportion; a, b, c, d are called first, second, third and fourth terms of the proportion a: b = c: d respectively.
3	In the proportion a: b = c: d, a and d are called “extremes” and b and c are called “means”.
4	If a, b, c, d are in proportion, i.e. if a: b = c: d, then ad = bc i.e. Product of extremes = Product of means.
5	a, b and c are said to be in continuous proportion if a: b = b: c. Here, a, b and c must be of the same kind and same unit.
6	If a, b and c are in continuous proportion, i.e. if a: b = b: c, then b^2 = ac or, b = (ac) ^ (1/2). Here, b is called the 'mean proportional' between a and c.
7	If a: b = c: d, then b: a = d: c [Invertendo]
8	If a: b = c: d, then a: c = b: d [Alternendo]
9	If a: b = c: d, then (a + b): b = (c + d): d [Componendo]
10	If a: b = c: d, then (a − b): b = (c − d): d [Dividendo]
11	If a: b = c: d, then (a + b): (a − b) = (c + d): (c − d) [Componendo and Dividendo]
12	If a: b = c: d = e: f = ................, then each of these = (a + c + e + ..........): (b + d + f + ..........) [Addendo]
13	If a: b = c: d = e: f = ................, then each of these = (a – c – e − .........): (b – d – f − .........) [Subtrahendo]

 

CMA – Foundation – Ratio and Proportion

Practice Problems

 

1. The ratio of the present age of a father to that of his son is 5:3. Ten years hence the ratio would be 3:2. Find their present ages.

 

2. The monthly salaries of two persons are in the ratio of 3:5. If each receives an increase of Rs 20 in salary, the ratio is altered to 13:21. Find the respective salaries.

 

3. What must be subtracted from each of the numbers 17, 25, 31, 47 so that the remainders may be in proportion?

 

4. If x/ (b + c) = y/(c + a) = z/ (a + b) show that (b – c) x + (c – a) y + (a – b) z = 0

 

5. If (4x – 3z)/4c = (4z – 3y)/3b = (4y – 3x)/2a, show that each ratio = (x + y + z)/ (2a + 3b + 4c)

 

6. If x/(y + z) = y/ (z + x) = z/(x + y) = k prove that k = ½ if (x + y + z) ≠ 0

 

7. If [a^ (1/2) – b^ (1/2)]/ [a^ (1/2) + b^ (1/2)] = ½, prove that (a^2 + ab + b^2)/ (a^2 – ab + b^2) = 91/73

 

8. If a/4 = b/5 = c/9, prove that (a + b + c)/c = 2

 

9. If (b + c)/a = (c + a)/b = (a + b)/c and a + b + c ≠ 0, then show that each of these ratios is equal to 2.

Also prove that a^2 + b^2 + c^2 = ab + bc + ca.

 

10. If a: b = c: d, show that

(xa + yb): (aα – bβ) = (xc + yd): (cα – dβ).

 

11. Monthly incomes of two persons Ram and Rahim are in the ratio 5:7 and their monthly expenditures are in the ratio 7:11. If each of them saves Rs 60 per month find their monthly income.

 

12. There has been increment in the wages of labourers in a factory in the ratio of 22:25, but there has also been a reduction in the number of labourers in the ratio of 15:11. Find out in what ratio the total wage bill of the factory would be increased or decreased.