Question

A house wife forgot her bank 'ATM PIN' which is a four digit number, but luckily she remembered some hints on how to recall this 'PIN'
Here are some of the clues
(1) The 1st digit is half of the 2nd
(2) The sum of the 2nd and 3rd is 10
(3) The 4th is equal to the 2nd plus 1
(4) The sum of all the digits is 23

What is the ATM PIN ?

Answer 1

Given

• The 1st digit is half of the 2nd.
• The sum of the 2nd and 3rd is 10.
• The 4th is equal to the 2nd plus 1.
• The sum of all digits is 23.

_____________________________

To Find

• The ATM pin with the given clues.

_____________________________

Solution

Let's consider the 1st digit to be → x

So, the second digit would be → 2x [1st digit is half of the second so we can say that the second digit is twice the 1st digit]

Sum of the 2nd and 3rd digit is 10

Let 3rd digit be → y  

⇒ 2x + y = 10

⇒ y = 10 - 2x  

∴ 3rd digit is → 10 - 2x

The 4th digit is equal to the 2nd digit plus 1.  

So, 4th digit would be → 2x + 1

Now we have the value of all 4 digits as common variables.

So, now let's solve the below equation to find the 3 digits.

x + 2x + 10 - 2x + 2x + 1 = 23

Step 1: Combine Like Terms

⇒ x + 2x + 10 - 2x + 2x + 1 = 23

⇒ (x + 2x - 2x + 2x) + (10 + 1) = 23

Step 2: Simplify the equation.

⇒ (x + 2x - 2x + 2x) + (10 + 1) = 23

⇒ 3x + 11 = 23

Step 3: Subtract 11 from both sides of the equation.

⇒ 3x + 11 - 11 = 23 - 11

⇒ 3x = 12

Step 4: Divide 3 from both sides of the equation.

⇒ 3x ÷ x = 12 ÷ 3

⇒ x = 4

∴ x = 4 in the above equation.

∴ 1st digit ⇒ x = 4

∴ 2nd digit ⇒ 2x = 2(4) = 8

∴ 3rd digit ⇒ 10 - 2x = 10 - 2(4) = 10 - 8 = 2

∴ 4th digit ⇒ 2x + 1 = 2(4) + 1 = 8 + 1 = 9

∴ The house wife's ATM PIN is 4829.

_____________________________

Answer 2

Answer:

${ \large{ \underline{ \sf{ \pmb{Given...}}}}}$

★ The first digit is half of the second

★ The sum of second digit and third digit is 10

★ The 4th is equal to the second plus 1

★ The sum of all digits is 23

${ \large{ \underline{ \sf{ \pmb{To \: Find ...}}}}}$

★ The ATM pin [ The number]

${ \large{ \underline{ \sf{ \pmb{Solution ...}}}}}$

★ The ATM pin is 4829

${ \large{ \underline{ \sf{ \pmb{Full \: Solution ...}}}}}$

~ As we have given the clues about the digits of the ATM pin let's assign a suitable variable with some constants which make the given statement true

➼ Let's assign the following values to the digits

• 1st digit = x
• 2nd digit = 2x
• 3rd digit = 10 - 2x
• 4th digit = 2x + 1

★According To The Question :

✪ The sum of all the digits is 23

~Now let's frame an equation adding up the assigned values letting the sum to be 23

${ \underline{ \bf{ \bigstar{Framing \: an \: equation \: we \: get : }}}}$

${ : \implies} \sf \: x + 2x + 10 - 2x + 2x + 1 = 23 \\ \\ \\ { : \implies} \sf 3x + 10 - 2x + 2x + 1 = 23 \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \sf 10 + 1 + 3x - 2x + 2x = 23 \: \: \: \: \: \\ \\ \\ { : \implies} \sf 11 + x + 2x = 23 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \sf 11 + 3x = 23 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \sf 3x = 12 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \sf x = \cancel\frac{12}{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \sf { \purple{ \underline{ \boxed{ \pmb{ \frak{x = 4}}} }\bigstar}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

• Henceforth the Value of x is 4

~Now let's find the digits of the pin

• First Digit :

${ : \implies} \sf 1st \: digit = x = 4$

• Second Digit :

${ : \implies} \sf 2nd \: digit = 2x = 2(4) = 8$

• Third Digit :

${ : \implies} \sf 3rd \: digit = 10 - 2x = 10 - 2(4) = 2$

• Fourth Digit :

${ : \implies} \sf 2x + 1 = 2(4) + 1 = 9$

• Henceforth the ATM pin number is 4829