CLASS-7
RATIO - EQUIVALENT RATIO

EQUIVALENT RATIO -

An equivalent ratio refers to two or more ratios that express the same relationship between quantities, even though the numbers may differ. They are essentially ratios that are proportional to each other.

How to Find Equivalent Ratios:-

To find an equivalent ratio, you can multiply or divide both terms of the original ratio by the same non-zero number.

Examples:-

1. Basic Example:- Consider the ratio 2:3. If we multiply both terms by 2, we get 4:6. So, 2:3 and 4:6 are equivalent ratios because they represent the same relationship. Similarly, dividing both terms by the same number (if possible) also gives an equivalent ratio. For instance, dividing the ratio 8:12 by 4 gives us 2:3, which means 8:12 is equivalent to 2:3.
2. Scaling Up:- Starting with the ratio 3:5, multiplying both numbers by 3 gives us 9:15. Therefore, 3:5 is equivalent to 9:15.
3. Scaling Down:- If you have the ratio 10:15, you can simplify it by dividing both terms by 5, resulting in 2:3. Thus, 10:15 is equivalent to 2:3.

Visual Example:-

Imagine you have a recipe that requires 2 cups of flour and 3 cups of sugar (a 2:3 ratio). If you wanted to make twice the amount, you would use 4 cups of flour and 6 cups of sugar, maintaining the equivalent ratio of 4:6.

Checking for Equivalence:-

To check if two ratios are equivalent, you can cross-multiply and compare:-

              a           c

          ------- = -------

              b           d

• For the ratios a and c, check if a × d = b × c .
• If this equality holds, then the ratios are equivalent.

Example:-

• Consider the ratios 2:3 and 4:6. Cross-multiply:- 2 × 6 = 12 and 3 × 4 = 12. Since both products are equal, the ratios are equivalent.

Example.1) Simplify the ratio

             𝟏             𝟏            𝟏

        --------- : -------- : --------

             𝟑             𝟔            𝟖

Ans.)  LCM of 3, 6 and 8 = 24.

Multiplying each fraction by 24 we get,

      1                1               1

   ------ x 24 : ------ x 24 : ------ x 24

      3                6               8

=  8 : 4 : 3       (Ans.)

Example.2)  If A : B = 3 : 4 and B : C = 6 : 7, FIND A : C.  

           𝐴          3              𝐵          6

Ans.)  ------ = ------  and  ------ = ------

           𝐵          4              𝐶          7

Multiplying them we get,

            𝐴           𝐵           𝐴

        ------- × ------- = -------

            𝐵           𝐶           𝐶

          3          6           9

       ------ x ------ = -------

          4          7          14         

So, A : C = 9 : 14.           (Ans.)

Example.3) If A : B = 5 : 6 and B : C = 8 : 9, find A : B : C.

Ans.)

In the 1st ratio, B = 6 and in the 2nd ratio, B = 8.

To make the 2 values of B equal, we take the LCM of 6 and 8 which is 24.

        𝐴           5          5 x 4          20

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

        𝐵           6          6 x 4          24

and

            𝐵            8            8 x 3           24

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

         𝐶            9            9 x 3           27

Therefore, A : B : C = 20 : 24 : 27.      (Ans.)

Example.4) Which ratio is greater: 2:3 or 3:4 ?

Ans.) Convert each ratio into equivalent like fractions with same denominator which is the LCM of 3 and 4 i.e. 12.

       2              2 × 4                8

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

       3              3 × 4               12

         3           3 × 3           9

   ------- = ---------- = -------

       4           4 × 3          12

and, Since  9 > 8,

       9           8

   ------- > -------    

      12          12

So,    3 : 4  > 2 : 3.        (Ans.)

Equivalent ratios are used to solve proportion problems, scale models, recipes, and in many applications where maintaining consistent relationships between quantities is important.