CLASS-7
RELATION BETWEEN AN ORIGINAL RATIO & A NEW RATIO

RELATION BETWEEN AN ORIGINAL RATIO & A NEW RATIO -

Establishing a relationship between an original ratio and a new ratio involves understanding how the ratios are related to each other through multiplication or division. This often means scaling up or down the original ratio to form the new ratio, or determining a common factor that links the two ratios.

Steps to Establish the Relationship Between Original and New Ratios:-

1. Identify the Original and New Ratios:-  Suppose the original ratio is a : b and the new ratio is c : d.
2. Compare the Ratios:-  You need to determine if the new ratio is a scaled version of the original ratio. This can be done by checking if c = k × a and d = k × b, where k is a constant (scaling factor).
3. Find the Scaling Factor (if applicable):-   If you know that the ratios are equivalent or proportional, you can find the scaling factor k by dividing one term of the new ratio by the corresponding term in the original ratio:-

                     c            d

            k  = ------- = -------

                     a            b

       If k is consistent across both terms, then the new ratio is simply a scaled version of the original ratio.

    4. Express the Relationship:-  Once the scaling factor k is identified, you can express the new ratio in terms of the original ratio:-

                             c : d = k × a : k × b

      If the ratios are not directly proportional, other mathematical or contextual relationships may be used to link the original ratio to the new ratio.

Example -

1. Scaling Up:-

• Original Ratio:- 3 : 4
• New Ratio:- 6 : 8
• Here, k = 6/3 = 2 and k = 8/4 = 2. Since k is the same for both terms, the new ratio 6 : 8 is simply the original ratio 3 : 4 scaled by a factor of 2.

2. Scaling Down:-

• Original Ratio:- 5 : 10
• New Ratio:-  1 : 2
• Here, k = 1/5 = 2/10 = 1/5​. 
• The new ratio 1 : 2 is the original ratio 5 : 10 scaled down by a factor of 1/5.

3. Non-Proportional Relationship:-

• Original Ratio: 4 : 5
• New Ratio: 7 : 9
• In this case, there is no simple scaling factor k because the new ratio is not a proportional scaling of the original ratio. To establish a relationship, you would need to explore further context, such as whether these ratios apply to different but related scenarios (e.g., different datasets or conditions).

Summary:-

• Direct Proportionality:-  If the new ratio is a scaled version of the original ratio, find the scaling factor k and establish the relationship.
• Indirect Relationships:-  If the ratios are not directly proportional, look for other mathematical relationships or contextual connections to understand how the new ratio relates to the original ratio.

Some Examples:-

Example.1) The present ages of Richard and David are in the ratio 4:3. Six years hence, their ages will be in the ratio 6:5. Find their present ages.

Ans.)

Ages of Richard : David = 4 : 3.

Let their ages be 4x and 3x respectively.

6 years later, their ages will be (4x+6) and (3x+6) respectively.

As per sum, Richard : David = (4x+6) : (3x+6) = 6 : 5

Or,

                4𝑥 + 6           6

         ----------- = -------

             3𝑥 + 6           5

Cross multiplying, 5 (4x+6) = 6 (3x+6)

                      Or,  (20x + 30) = (18x + 36)

                      Or,  (20x - 18x) = (36 - 30)

                      Or,        2x = 6

                                       6

                      Or,     x = -------

                                       2

                      Or,     x = 3

        So, Richard’s present age = 4x = (4 × 3) = 12 years

And David’s present age = 3x = (3 × 3) = 9 years

So, Richard’s age 12 years and David’s age 9 years.       (Ans.)

Example.2) The ratio of girls to boys in a school of 720 students is 3:5. If 18 new boys join the school, find how many new girls may be admitted so that the ratio of girls to boys may change to 2:3.

Ans.)

Total students = 720

Girls : boys = 3 : 5

Sum of the ratios = 3 + 5 = 8

                        3

Number of girls = ------- × 720 = 270.

                        8

So, number of boys = 720 - 270 = 450

If 18 new boys are admitted, new number of boys = 450 + 18 = 468

Let number of new girls joining be x.

So, new number of girls = (270 + x)

According to the sum, girls : boys = (270 + x) : 468 = 2 : 3

Or,

              (270 + 𝑥)          2

           ------------ = -------

                468              3

Or,   3 (270 + x) = 2 × 468

Or,     810 + 3x = 936

Or,       3x = (936 - 810) = 126

Or,        x = 126/3 = 42

So,the number of new girls to be admitted is 42.   (Ans.)