2What 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
Also
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
=450
Now putting the value of b in equation 1
a+b=810
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
Therefore
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.
