**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**

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.**