#### Learn the approach to solve the typical questions on Digital Root. A wonderful approach for mathematics as well as DI questions through digital root method.

In this article, we are going to discuss the concept of Digital Root (DR) also known as seed number. This concept is a very powerful tool for saving time and effort in certain long calculations questions both in quant section as well as in data interpretation.

**What is Digital Root?**

Digital Root is the single number obtained by adding the number successively. E.g. Digital Root of 347 = 3 + 4 + 7 = 14, 14 = 1 + 4 = 5. Thus, 5 is the a single digit number, which is the digital root/ seed number of 347.

** How to find the DR**

Simply, add the individual digits of the number in one or more steps until you obtain a single digit

(1, 2, 3… or 9) and that is the digital root of that number. For example:

- 42 = 4+2 = 6
- 84 = 8+4 = 12 = 1+2=3

iii. 737 = 7+3+7 = 17=1+7=8

- 6789 = 6+7+8+9 = 30= 3+0 = 3

**Properties of Digital Root**

Some of the important and useful properties of digital root are –

**When you multiply any number by 9, the digital root will always be 9**

e.g. 7 X 9 = 63. 6 + 3 = 9

13 X 9 = 117. 1 + 1 + 7 = 9

24 X 9 = 216, 2 + 1 + 6 = 9 and so on

**When we add 9 to a number, it does not change the digital root of that number**

e.g. 417 = 4+1+7= 12, 1+2 = 3

Now, if we annex 9 to this number i.e. 4179 = 4+1+7+9 = 21, 2+1 = 3

Hence we observe that adding 9 does not change the DR of that number. So, we can omit 9 while calculating the digital root of a number and we can also omit any 2 or more numbers which add to 9 and still get the same digital root. This property eases out to calculate digital roots of certain numbers.

e.g. = 97 = 9 + 7 = 16, 1 + 6 = 7 so even if we omit 9 still DR is 7

**When we divide any number by 9, the digital root of that number will be the remainder**

e.g. 13/9 remainder is 4 which is same as digital root of 13 (1 + 3 = 4) 47/9 remainder is 2 which is again same as 4 + 7 = 11, 1 + 1 = 2.

**Digital root of any number will be from 1 to 9**

As by definition only digital root of a number is obtained after adding the digits of a number successively until one gets the single digit.

Digital root of any perfect square will fall among 1, 4, 7, 9 only

e.g. 16 = 1 + 6 = 7

36 = 3 + 6 = 9

49 = 4 + 9 = 13, 1 + 3 = 4

64 = 6 + 4 = 10, 1 + 0 = 1 and so on.

**Solved Digital Root Problems**

**Example 1:**3286 x 4783 =

1. 15766938

2. 16716938

3. 15716938

4. 17716948

**Solution:** Now let us apply digital root technique instead of solving it conventionally.

**15716938 has DR = 4.**

**Example 2:** S (x) = Sum of all the digits of natural number x. e.g. S(245) = 2 + 4 + 5 = 11

1. 484

2. 585

3. 47

4. 485

**Solution: Conventional Approach:** At the ten’s place 4 will come 8 times only but 5, 6, 7 & 8 each will come 10 times and finally 9 will come 6 times. So the total at ten’s place [(4 × 8) + (5 + 6 + 7 + 8) 10 + (9 × 6)] = 346 At the units place all the ten digits (from 42 to 51) including zero will come 5 times and then we also include units places of (92 to 95), the total at units place = 5 × (∑ 9) + 14 = 225 + 14 = 239. Now, value of V = 346 + 239 = **585**

**Digital Root Method**

- Using this approach we can simply take the sum of the series as an arithmetic progression with 42 as first term ‘A’, 95 as last term ‘L’ and common difference being 1. The number of terms ‘n’ will be (95 – 42 + 1) = 54.

Now using the formula for summation of A.P. – S_{n}= - Now calculating DR of 27 = 2 + 7 = 9, we need not calculate anything further as we already know that any number when multiplied by 9 gives a DR of 9 only. So, the answer to this calculation will have the DR = 9.
- Now we can simply check through the choice that which choice has DR = 9. This is true only for 2
^{nd }choice i.e. 585. So we observed how simple the calculation become using the digital root approach.

**Directions: what should come in place of question mark (?) in the following questions: –**

**1. 1777 – 2349 – 1345 + 6523 = ?**

** A) 4706 B) 4606 C) 4976 D) 4176 E) 5276 (Digital Root Method)**

Lets solve by DR Method (Digital Root)

Let (1+7+7+7) – (2+3+4+9) – (1+3+4+5) + (6+5+2+3) = x

now (22) – (18) – (13) + (16) = x

now again Digital Root to Single Root

4 – 9 – 4 + 7 = x

so, 9+7 = x ( U can see, There is no value for 9 in digital root)

__ 7 = x (Digital Root of above Equation is 7)__

Now Check the Options

A) 4+7+0+6 = 17 = 8 (not match)

B) 4+6+0+6 = 16 = 7 (Match with digital root of x)

C) 3+9+7+6 = 17 = 8 (Not match)

D) 5+2+7+6 = 20 = 2 (Not Match)

So, B) is the right Answer

Similarly You can solve Below Problems

** 2. 2387 – 123 + 980 = ? – 145 + 945**

**A) 2244 B) 2434 C) 2444 D) 2354 E) 2544 (Digital Root Method)**

2+3+8+7 – 1+2+3 + 9+8+0 = ? – 1+4+5 + 9+4+5

2 – 6 +8 = ? – 1 + 9

4 = ? +8

4 -8 = ?

-4 = ?

(If your final Digital is in Negative Add “9” to it to make it positive)

So, 9-4 = ?

5 = ?

Now Find out the Digital Root of All the Options, if any option whose digital root is exactly “7” that is the Answer

A) 2+2+4+4 = 12 = 3

B) 2+4+3+4 = 13 = 4

C) 2+4+4+4 = 14 = 5 (Match) Exactly

D) 2+5+4+4 = 15 = 6

**3. 1.96*1.96 + 1.04*1.04 – 2.08*1.96 = ?**

** A) 0.7464 B) 0.8464 C) 0.9464 D) 0.8262 E) none of these (Digital Root Method)**

Now Lets Do it Without Pen / Paper

the Digital Root of Above Equation is as follows

(Note : Wherever the “9” comes in Digital Root just neglect it)

Now, 7*7 + 5*5 – 1*7 = 49 + 25 – 7 = 4 + 7 – 7 = X, now final Digital Root x = 4

Match with Options

(neglect the Decimal Points)

A) 7+4+6+4 = 21 = 3 (Not Match)

B) 8+4+6+4 = 22 = 4 (Match)

C) 9+4+6+4 = 23 = 5 (Not Match)

D) 8+2+6+2 = 18 = 9 (not Match)

So, B) is the Right Choice

**4. 41% of 801 – 150.17 = ? – 57% of 910 (Approximation method)**

** A) 693 B) 694 C) 697 D) 707 E) none of these**

You Can’t Apply Digital Root For Approximations, you can do it by Normal Method

**5. 5016 x 1001 – 333×77+22 = ? x 11 (Digital Root Method)**

** a. 435570 b. 454127 c, 527240 d. 366531 e. 511990**

Now according to Digital Root Method

The Digital Root of above Equation is

6 – 9 + 4 = x * 2

1 = x * 2 (Just Neglect 9)

Now we have to Make LHS = RHS, by suitable “x” value putting from 1 to 9)

So, If we Substitute X = 5

We get 1 = 5 * 2

1 = 10

1 = 1

SO, Clearly the Digital Root of above Equation is 5 (as X = 5)

Now Find out the digital root of given options

A) 4+3+5+5+7+0 = 24 = 6

B) 4+5+4+1+2+7 = 5 (Match)

C) 5+2+7+2+4+0 = 20 = 2

D) 3+6+6+5+3+1 = 6

E) 5+1+1+9+9+0 = 7

### Digital Root Method for Percentage Questions

**1.) 50% of 450 + x = 300**

**a) 52 b) 75 c) 42 d) 56 e) None**

Now According to Digital Root Method

(5+0) of (4+5+0) + x = 3+0+0

5 of 9 + x = 3

45 + x = 3

9 + x = 3

Now Choose the value of x B/w 1 to 9 such that LHS = RHS

If we Put X value as 3 then LHS = RHS

**So DIgital Root Of equation is = 3**

now Apply Digital Root to Options

A) 52 = 5+2 = 7

**B) 75 = 7+5 = 12 = 3 (Only This will match with above digital root)**

c) 42 = 4+2 = 6

d) 56 = 5+6 = 11 = 2

**2) 125% of 320 + x% of 125 = 440**

**a) 32 b) 51 c) 60 d) 30 e) None **

By Digital Root Method

(1+2+5) of (3+2+0) + x of 1+2+5 = 4+4+0

8 of 5 + x of 8 = 8

40 + 8x = 8

4 + 8x = 8

8x = 8-4

8x = 4

now Choose the X value b/w 1 to 9 such that LHS = RHS

**so, x= 5 **

8*5 = 4

40 = 4

4 = 4

**SO, Value is 5.**

**YOU CAN SOLVE THESE PROBLEMS BY DIGITAL ROOT METHOD**

**1. What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12 p.c.p.a?**

**a) 10101 b) 10110 c) 21212 ****d) 10123.2**** e) None of these**

Let P = 25000 n = 3yr R = 12%

**Now Amount (A) = P (1+R/100)^n**

Now A = 25000 (1+12/100)^3

Now according to Digital Root Method

A = 2+5+0+0+0 (1+(1+2))^3

A = 7 (1+3)^3

A = 7 (4)^3

A = 7 (64) = 7(6+4) = 7(10) = 7(1+0) = 7

Now Amount Digital Root is = 7

And we know that **Amount = P + CI ————– (1)**

Substitute the Digital Root of Amount and Principal in **Eq —–(1)**

**Now 7 = 25000 + Ci**

** 7 = 7+ Ci**

**So, Clearly If CI Digital Root is 9 then only LHS = RHS (According to Digital Root Method).**