FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 1.Proof : Let ab d=c=k,then a = bk, c = dk
L. H. S. =
bk b b(k 1) k 1
bk b b(k 1) k 1+ = + = +
− − −
R. H. S. =
dk d d(k 1) k 1
dk d d(k 1) k 1+ = + = +
− − −. Hence the result, L.H.S. = R.H.S. (Proved)Note. 1.
x y z
a b c= = is sometimes written as x : y : z = a : b : c.
- If x : y = a : b, it does not mean x = a, y = b. It is however to take x = ka, y = kb.
Solved Examples :
Example 5 : 4x 3z 4z 3y 4y 3x4c− = 3b− = 2a,− If , show that each ratio is equal to 2a 3b 4cx y z++ ++.
Each of the given ratio =4x 3z 4z 3y 4y 3x− 4c 3b 2a++−++ − =2a 3b 4cx y z++ ++
Example 6 : The marks obtained by four examinees are as follows :
A : B = 2 : 3, B : C = 4 : 5, C : D = 7 : 9, find the continued ratio.
A : B = 2 : 3
B : C = 4 : 5 = 4 :5×4 43 3× =3 :^154 [for getting same number in B, we are to multiply by^34 ]
C : D = 7 : 9 = 7 ×28 9 28 4 2815 1 15 15 135: × = : [to same term of C, multiply by^1528 ]
A : B : C : D = 2 : 3 :^154 :^13528 =56 : 84 :105:135.
Example 7 : Two numbers are in the ratio of 3 : 5 and if 10 be subtracted from each of them, the
remainders are in the ratio of 1 : 5, find the numbers.
Let the numbers be x and y, so that xy 5=^3 or,5x 3y...(1)=
Again
x 10 1
y 10 5
− =
−
or, 5x–y = 40 ....(ii) , Solving (I) & (ii), x = 12, y =
∴ Required Numbers are 12 and 20.
Example 8 : The ratio of annual incomes of A and B is 4 : 3 and their annual expenditure is 3 : 2. If each of
them saves ` 1000 a year, find their annual income.