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Mathematics Ratio (School Exam and Competitive Exam)

Sonal Umesh Bamne
28 Jul 0 2

What is Ratio?

The ratio or comparison of any two quantities say a and b in the same units, is the fraction a/b, and we write it as a:b.

In the ratio a:b, we call the first term or antecedent and b the second term or consequent.

Ex: The ratio 4:7 represents 4/7 with antecedent =4 and consequent =7.

Rules of ratio:

1. The ratio or comparison of two quantities is meaningless if they are not of the same kind or in the unit ( of length, volume, currency, etc.) We do not compare five girls with three toys or 15 kilometres with 10 kilograms.

2. The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Ex. If the given ratio is 4:5 then

4:5 = 4/5 = 4×2/5×2 = 8/10 also

4:5 = 4/5 = 4×3/5×3 = 12/15 it means

4:5 = 8:10 = 12:15.

In the same way

If the given ratio is 10: 12 then

10:12 = 10/12 = (10/2)/(12/2) = 5/6 it means

10:12 = 5:6.

Generally we can say that

a/ b= ak/bk=am/bm


a/b= (a/k)/ (b/k)= (a/m) / (b/m)

Solved Problems

Example 1:

Two numbers are in the ratio of 4:5. If the sum of these two numbers is 810. Find the number?

School method

Let the two numbers be a and b. 

Given a+b=810 .........equation 1

And a:b=4:5 it means a/b =4/5

Therefore a = (4/5) × b .....equation 2

Putting the value of an in equation 1

            a + b = 810

becomes   (4/5)×b + b =810

Therefore   (4b+5b)/5=810

Therefore   9b = 810 ×5

Therefore    b= (810×5)/9


Now putting the value of b in equation 1


Therefore   a+450=810

Therefore   a=810 - 450 =360.

2. Shortcut Method ( scholarship/ competitive exams)

          a: b    

          4: 5...........Total=4+5=9

Given total =810

Since 9 is equivalent to 810

      1 is equivalent to 810/9=90

Therefore 4 is equivalent to 4×90=360

      and 5 is equivalent to 5×90=450.

Let us consider another example

Example 2:

The sum of the three numbers is 98. If the ratio between the first and second is 2:3 and that between second and third is 5:8. Then find all three numbers?

Solution : 

1. School method

Let three numbers be a, b and c

Now as given a+b+c=98..... Equation 1

And a:b=2:3 therefore a/b=2/3

Therefore a=(2/3)×b....... Equation 2

Now b:c=5:8 therefore b/c=5/8

Therefore c=(8/5)×b ........ Equation 3

Now putting the values of a and c in equation 1

That is a+b+c=98

 It means  (2/3)×b + b +(8/5)×b=98

Therefore (10b+ 15b+ 24b)/15=98

Therefore 49b=98×15

Therefore b=30

  Now putting the value b=30 in equation 2 we get a= (2/3)×b=(2/3)×30=20

Similarly, by putting value b=30 in equation 3, we get c=(8/5)×b=(8/5)×30=48.

Thus the required three numbers are 20,30 and 48

2. Shortcut method(Scholarship/ competitive exam)

          A : B : C

          2 : 3

                 5 : 8

since B is common, we make B equal. So taking the LCM of 3 and 5, which is 15. We make B =15. Since A: B and B: C are ratios, whatever changes applied to B those also applied to A and C


           A : B : C

     2×5 --------: 3×5

                 5×3------: 8×3

     10: 15 : 24

Here Total =49(10+15+24)

Given total =98 

Since      49 is equivalent to 98

            1 is equivalent to 2

Therefore 10 is equivalent to 10×2=20

            15 is equivalent to 15×2=30

And 24 is equivalent to 24×2=48.

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Upalimitra | 30 Jul

Thank you! You made it very easy to understand.

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Mrunal Sawant | 29 Jul

Great explanation with appropriate examples 👍🏻

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