# CLASS-8DIVIDING A QUANTITY IN A RATIO

Dividing A Quantity In A Given Ratio

If a number n is divided into two parts in the ratio a : b

a

1st Part would be =  ------------  X  n

a + b

b

2nd Part would be  =  -----------  X  n

a + b

If a number n is divided into three parts in the continued ratio  a : b : c,

a

1st Part would be = -------------  X  n

a + b + c

b

2nd Part would be = -------------  X  n

a + b + c

c

3rd Part would be = -------------  X  n

a + b + c

There are some examples are given below for your understanding –

Example.1)  If 3a = 4b = 5c, then find a : b : c = ?

Ans.) As per the given condition we have  3a = 4b = 5c

Let, 3a = 4b = 5c = k  ( k 0 )

So, a = k / 3 ,   b = k / 4,   c = k / 5

k             k             k

So, a : b : c  = -------- : --------- : ---------

3             4             5

1              1             1

= --------- : --------- : ---------

3              4             5

( Now, we will find the LCM of denominators of obtained fraction and we find  LCM of 3, 4, 5 = 60 )

Now we will multiply by obtained LCM 60 with

1          1            1

-------, --------, -------- and we find -

3          4            5

1                    1                     1

= ( ------- X 60 ) : ( ------- X 60 ) : ( ------- X 60 )

3                    4                     5

=   ( 1 X 20 ) : ( 1 X 15 ) : ( 1 X 12 )

= 20 : 15 : 12

So,  a : b : c  =  20 : 15 : 12        (Ans.)

Example.2) Mr. Richard divided his retirement benefit worth \$ 2,16,800 to his Son, Daughter & Wife in

2                1              3

the ratio 1 -------- :  2 -------- : 1 --------.

5                3              7

Find how much money Richard’s each family member will get ?

Ans.)  As per the given condition, Mr. Richard has distributed his money worth \$ 2,16,800 to his three family members

2              1               3

in the ratio of  =  1 -------- : 2 -------- :  1 --------

5              3               7

First of all, we have to simplify the given ratio –

2               1                3

1 --------- : 2 --------- : 1 ----------

5               3                7

7              7             10

=  --------- : --------- : ----------

5              3              7

Now, we have to find out LCM of the denominator of the obtained fraction is 105, now we multiply each ratio by 105 and we will get –

7                     7                      10

= ( ------- X 105 ) : ( ------- X 105 ) : ( ------- X 105 )

5                      3                      7

=    ( 7 X 21 ) : ( 7 X 35 ) : ( 10 X 15 )

=   147 : 245 : 150

Now we will find the money distributed to his family members in the ratio of  147 : 245 : 150

147

Son will get =  \$ 2,16,800 X  ---------------------

( 147 + 245 + 150 )

147

=   \$ 2,16,800 X  ------------

542

=   400 X 147 =  \$ 58, 800

245

Daughter will get =  \$ 2,16,800 X  ----------------------

( 147 + 245 + 150 )

245

=   \$ 2,16,800 X -------------

542

=   400 X 245  =   \$ 98,000

150

Wife will get =  \$ 2,16,800 X  -----------------------

( 147 + 245 + 150 )

150

=  \$ 2,16,800 X --------------

542

=   400 X 150  =   \$ 60,000

So, Richard’s money will be distributed to his son, daughter & wife by \$ 58,800, \$ 98,000 and \$ 60,000 respectively. (Ans.)

Example.3) A certain quantity of burger is put into two boxes in the ratio 5 : 3, if the bigger box contains 80 pcs of burger then what is the quantity of total burger ?

Ans.)   Let the burger in the smaller box be Y pcs.

Then,  80 pcs : Y pcs = 5 : 3

80              5

So,  ---------- = ----------

Y              3

So,   5 Y  =  80 X 3

80 X 3

So,  Y  =  -------------  = 16 X 3 =  48 pcs

5

So, the total pieces of burger is = 80 + 48 =  128 pieces     (Ans.)

Example.4) Three friends x, y & z share \$ 780 in such that ratio x : y = 3 : 1 & y : z = 3 : 1, then find the each share of distributed money.

Ans.) As per the given condition given ratio are  x : y = 3 : 1 &  y : z = 3 : 1

Here we can observe that, consequent of 1st ratio is y = 1 but in the second ratio antecedent y = 3.

So, 3 ÷ 1 = 3

So, now we will multiply 1st ratio by 3 and find -

x : y = 3 : 1  =  ( 3 X 3 ) : ( 1 X 3 ) =  9 : 3

Now we have,  x : y = 9 : 3  &  y : z = 3 : 1

So, now we can find  x : y : z  =  9 : 3 : 1

9

So, friend x will get the money =  \$ 780 X ---------------

( 9 + 3 + 1 )

9

=  \$ 780 X  ---------  =  \$ 60 X 9 =  \$ 540

13

3

So, friend y will get the money =  \$ 780 X --------------

( 9 + 3 + 1 )

3

=  \$ 780 X --------  = \$ 60 X 3  = \$ 180

13

1

So, friend z will get the money  =  \$ 780 X --------------                                                                                                                                                     ( 9 + 3 + 1 )

1

=  \$ 780 X -------- = \$ 60 X 1  = \$ 60

13

So, distributed money to x, y, & z is \$ 540, \$ 180, & \$ 60 respectively  (Ans.)