# CLASS-10RATIO & PROPORTION - PROBLEM & SOLUTION ON PROPORTION

PROBLEM & SOLUTION ON PROPORTION -

Example.1) What least number must be added to each of the numbers 6, 15, 20, & 43, so that the resulting numbers are proportional ?

Ans.) Let the required number to be added be x. Then,

(6 + x) : (15 + x) : : (20 + x) : (43 + x)

=> (6 + x) (43 + x) = (20 + x) (15 + x) [so, Product of extremes = Product of means]

=> x² + 49x + 258 = x² + 35x + 300

=> 49x – 35x = 300 – 258

=> 14x = 42

=> x = 3

Hence, the required number is 3 (Ans.)

Example.2) Find the fourth proportional to 7, 13, and 35

Ans.) Let the fourth proportional to 7, 13, and 35 be ‘x

Then, 7 : 13 : : 35 : x

7           35

=> ------- = --------   [so, Product of extremes = Product of means]

13           x

35 X 13

=> x = ------------ = 65

7

Hence the fourth proportional to 7, 13, & 35 is 65. (Ans.)

Example.3) Find the third proportional to 9, and 15

Ans.) Let the third proportional to 9 & 15 be ‘x

Then, 9 : 15 : : 15 : x

9            15

=> -------- = -------- [so, Product of extremes = Product of means]

15            x

15 X 15

=> x = ----------- = 25

9

Hence, the third proportional to 9 and 15 is 25. (Ans.)

Example.4) If b is the mean proportion between a & c, then prove that

a² - b² + c²

---------------- = b⁴

a¯² - b¯² + c¯²

Ans.) Since b is the mean proportion between a & c, so as per rules we have b = √ac

=> b² =  ac

a² - b² + c²

------------------

a¯² - b¯² + c¯²

a² - b² + c²

=> ----------------------------

1          1          1

------ - ------ + ------

a²         b²        c²

(a²- b² + c²)

=> ------------------------

b²c² - a²c² + a²b²

--------------------

a²b²c²

a²b²c² (a² - b² + c²)

=> ------------------------

b²c² - a²c² + a²b²

a²(ac)c² (a² - ac + c²)

=> ------------------------         [where ac = b²]

(ac) c² - a²c² + a² (ac)

a³c³ (a² - ac + c²)

=> ----------------------

ac³- a²c² + a³c

a³c³ (a² - ac + c²)

=> ----------------------

ac (a² - ac + c²)

=> a²c² = (b²)² = b⁴       [where ac = b²]                  (Proven)