# CLASS-10RATIO & PROPORTION - AN IMPORTANT PROPERTY

An Important Property

a           c            e

If, ------ = ------- = ------- = k,

b           d            f                                                                                                                     a + c + e

prove that each ratio is equal to ------------

b + d + f

sum of antecedent

i.e., each ratio = --------------------

sum of consequent

a            c             e

Proof – Let, ------- = -------- = -------- = k.

b            d             f

Then, a = bk, c = dk, and e = fk

a + c + e          bk + dk + fk

So, ------------ = ---------------

b + d + f           b + d + f

k (b + d + f)

= --------------- = k

(b + d + f)

a + c + e

Hence, each ratio = ------------

b + d + f

x + y          y + z         z + x

Example.1) If, --------- = --------- = ---------, prove that each of

ax + by       ay + bz       az + bx

2

these ratios is equal to -------- unless x + y + z = 0

(a + b)

Sum of Antecedents

Ans.) As we know, each Ratio = --------------------

Sum of Consequent

x + y         y + z         z + x          (x + y) + (y + z) + (z + x)

-------- = --------- = -------- = ---------------------------

ax + by       ay + bz      az + bx     (ax + by) + (ay + bz) + (az + bx)

(2x + 2y + 2z)

= -----------------------------

(ax + ay + az) + (by + bz + bx)

2 (x + y + z)

= ---------------------------

a (x + y + z) + b (x + y + z)

2 (x + y + z)

= --------------------

(x + y + z) (a + b)

2

= ---------                (proven)

(a + b)

a            b             c

Example.2) If ------- = -------- = --------,

(b + c)      (c + a)       (a + b)

Sum of Antecedents

Ans.) As we know that, each ratio = ----------------------

Sum of Consequents

a             b             c                   a + b + c

So, -------- = -------- = --------- = ------------------------

(b + c)       (c + a)       (a + b)        (b + c) + (c + a) + (a + b)

a + b + c

= ---------------

(2a + 2b + 2c)

(a + b + c)          1

= --------------- = ------

2 (a + b + c)          2

1

= -------     [where (a + b + c) ≠ 0]

2

Now, when a + b + c = 0, we have

b + c = - a, a + b = - c, and a + c = - b

a           a

so, -------- = ------ = - 1

b + c       - a

b            b

now, -------- = ------- = - 1

a + c        - b

c            c

and, -------- = ------- = - 1

a + b       - c

Hence, in this case, each ratio is – 1        (proven)