CLASS-7
DIVISION OF QUANTITIES OR VALUES ACCORDING TO THEIR RATIOS

DIVISION OF QUANTITIES OR VALUES ACCORDING TO THEIR RATIOS -

Dividing quantities or values according to their ratios involves splitting a given total into parts that maintain a specific ratio. This is a common problem in various applications, such as dividing an inheritance among heirs or distributing resources according to predetermined shares.

Steps to Divide Quantities According to a Given Ratio:-

1. Understand the Ratio:- Let’s say you need to divide a quantity Q into parts based on the ratio a : b : c. Here, a, b, and c are the parts of the ratio.
2. Calculate the Total Number of Parts:- Add the parts of the ratio together: a + b + c. This sum represents the total number of parts into which the quantity will be divided.
3. Determine the Value of Each Part:- To find the value of one part, divide the total quantity Q by the total number of parts:-

                                  Q

      Value of one part = -----------

                               a + b + c

    4. Distribute the Quantity:-  Multiply the value of one part by each ratio number to find the corresponding quantity for each part.

        For the ratio a : b : c, the amounts will be:-

                                 Q

         First part = a × ------------

                             (a + b + c)

                                   Q

        Second part = b × ------------

                               (a + b + c)

                                  Q

         Third part = c × ------------

                              (a + b + c)

Example:-

Problem -

Suppose you have $300 that you want to divide among three friends in the ratio 2 : 3 : 5. 

Solution -

    1. Understand the Ratio:-

            The ratio is 2 : 3 : 5.

    2. Calculate the total number of Parts:-

           Total parts = 2 + 3 + 5 = 10

    3. Determine the Value of Each Part:-

           Value of one part = 300/10.

      4. Distribute the Quantity:-

           First friend’s share = 2 × 30 = 60.

           Second friend’s share = 3 × 30 = 90.

           Third friend’s share = 5 × 30 = 150.

The $300 is divided as $60, $90, and $150 according to the ratio 2 : 3 : 5.    (Ans.)

                                                                                 1      1       1

Example.1) Divide $260 in three children in the ratio ---- : ---- : ----

                                                             2      3       4

Ans.)

At first, we need to convert the ratios to simplest whole number form.

LCM of 2,3,4 = 12.

Multiplying each ratio by 12 we get the whole number ratio as 6 : 4 : 3.

Sum of the ratios = 6 + 4 + 3 = 13

                               6

So,   1st child will get = ------ × 260 = $ 120

                              13

                               4

      2nd child will get = ------ × 260 = $ 80

                              13

                               3

      3rd child will get = ------ × 260 = $ 60

                              13

Ans: The 3 children get $ 120, $ 80 and $ 60 respectively.

Example.2)  A line segment of length 1m is divided into 2 parts such that the first part is 2/3 of the second part. Find the lengths of the 2 parts in cm.

Solution:-

1m = 100cm.

Let the second part be x cm.

                          2

So, the first part is ------ x cm

                          3

                                      2

As per the given condition, x + ------ x cm = 100 cm

                                      3

                              (3x + 2x)

                             Or, ------------ = 100

                                  3

                              5x

                      Or, ------- = 100

                              3

                       Or, 5x = (3 × 100)

                       Or, x = 300/5 = 60

                                  And

                             2              

                     x = ------ × 60 = 40

                             3              

Ans: The 2 parts are 40 cm and 60 cm.     (Ans.)

Example.3) A certain sum of money has been divided into 2 parts in the ratio 5 : 8. If the first part is Rs 250, find the total amount.

Solution:-

Let the 2 parts be $ 5x and $ 8x. As per sum,

   5x = 250.

Or, x = 250/5,

Or, x = 50

So, 8x = 8 × 50

           = 400

So, total amount = 250 + 400 = $ 650.

The total amount is $ 650.  (Ans.)

This method ensures that the quantities are distributed proportionally according to the specified ratio.