Question

Ques : ABC+ABC+ABC=BBB

Find the values of A, B and C & give reason for your answer too. ​

Answer 1

AnswEr:

1 4 8

1 4 8

1 4 8

______

4 4 4

_______

ExplaNation:

Here,

A = 1

B = 4

C = 8

$\rule{200}2$

Three cases:

• 3C = B

• 3C = 10 + B

• 3C = 20 + B

$\rule{200}2$

Why these cases?

$\implies$ Let's start putting values from 1-9

• 3C = 1 + B

$\implies$ 3 × 8 = 1 + 4

$\implies$ 24 ≠ 5

$\implies$ Not possible

• 3C = 2 + B

$\implies$ 3 × 8 = 2 + 4

$\implies$ 24 ≠ 8

$\implies$ Not possible

• 3C = 3 + B

$\implies$ 3 × 8 = 3 + 4

$\implies$ 24 ≠ 7

$\implies$ Not possible

• 3C = 4 + B

$\implies$ 3 × 8 = 4 + 4

$\implies$ 24 ≠ 8

$\implies$ Not possible

• 3C = 5 + B

$\implies$ 3 × 8 = 5 + 4

$\implies$ 24 ≠ 9

$\implies$ Not possible

• 3C = 6 + B

$\implies$ 3 × 8 = 6 + 4

$\implies$ 24 ≠ 10

$\implies$ Not possible

Similarly Put the other values, it won't satisfied it. So I've consider three cases.

$\rule{200}2$

★ $\underline{\red{\sf{When\:3C\:=\:B}}}$

Middle figures = B + B + B $\implies$ 3B

So from it, we came to know that B must be a multiple of 3 as 3C = B.

But, Not Possible.

$\rule{200}2$

★ $\underline{\red{\sf{When\:3C\:=\:10\:+\:B}}}$

Middle figures = 3B + 1 $\implies$ 10 + B or 20 + B

From this, we concluded that B is a fraction. so Not Possible.

$\rule{200}2$

★ $\underline{\red{\sf{When\:3C\:=\:20\:+\:B}}}$

Middle figures = 3B + 2 $\implies$ 10 + B or 20 + B

From it, we concluded that, B can be 4 or 10. But we need single digit number. So B = 4.

C = $\dfrac{20\:+\:4}{3}$

: $\implies$ C = $\cancel{\dfrac{24}{3}}$

: $\implies$ C = 8

Also, 3A + 1 = B

: $\implies$ 3A = 4 - 1

: $\implies$ 3A = 3

: $\implies$ A = $\cancel{\dfrac{3}{3}}$

: $\implies$ A = 1

$\rule{200}2$

Let's verify it now,

ABC + ABC + ABC = BBB

Put the values of A, B and C.

: $\implies$ 148 + 148 + 148 = 444

: $\implies$ 444 = 444

Hence, our answer is verified.

Answer 2

AnswEr :

Here we can see that A, B and C are single digit numbers that can be between (1 to 9)

★ So Start From Basics :

↠ ABC + ABC + ABC = BBB

Let's Test what C can be i.e. Last Digit, What is Possible case of C

• C can't be 5, why? Because After Adding 3C i.e. (5 + 5 + 5) = 15 and Last Digit will be 5, but Digit should be B, so C can't be 5 ever.
• C can be 1, 2, 3, 4, 6, 7, 8, 9.

↠ ABC + ABC + ABC = BBB

↠ 3 × ABC = BBB

Let's Test what C can be i.e. Middle Digit and Result too, What is Possible case of B

• Result is Multiple of 3, as we can see, so possible case of B being 1 or 2 isn't possible. why?
• Because when we will Divide 111 or 222 by that will give 2 Digit Quotient Only, But we need 3 Digit.
• B can't be 6 or 9 as they are multiple and give same digits.
• B can be 3, 4, 5, 7, 8.

__________________________

☯ Let's Try value of B one by one :

◗ B = 3

↠ 3 × ABC = BBB

↠ 3 × ABC = 333

• Dividing each term by 3

↠ ABC = 111

⋆ This isn't possible, cuz ABC are different Digits.

◗ B = 4

↠ 3 × ABC = BBB

↠ 3 × ABC = 444

Dividing each term by 3

↠ ABC = 148

⋆ This is Possible Case, as B is 4 in result.

◗ B = 5

↠ 3 × ABC = BBB

↠ 3 × ABC = 555

Dividing each term by 3

↠ ABC = 185

⋆ This isn't possible, cuz C can't be 5

◗ B = 7

↠ 3 × ABC = BBB

↠ 3 × ABC = 777

Dividing each term by 3

↠ ABC = 259

⋆ This isn't possible, cuz B isn't matching.

◗ B = 8

↠ 3 × ABC = BBB

↠ 3 × ABC = 888

Dividing each term by 3

↠ ABC = 296

⋆ This isn't possible, cuz B isn't matching.

━━━━━━━━━━━━━━━━━━━━━━━━

So we seen that value of B = 4 is only accurate and will give Desired Results.

⇾ A = 1

⇾ B = 4

⇾ C = 8

∴ Value of A, B & C is 1, 4 & 8 respectively.

━━━━━━━━━━━━━

★ VERIFICATION :

⇴ ABC + ABC + ABC = BBB

• putting value of each

⇴ 148 + 148 + 148 = 444

⇴ 444 = 444 ⠀ Hence, Verified!