# Four Digits

There are several 4-digit numbers that end with 5. But with one such number by taking the 5 from the end and putting it at the start of the number gives a number 1.2 times larger. What is the original number?

Ans: 4545

Taking the 5 around to the start gives 5454 and 5454/4545 = 1.2

We can express everything algebraically in other words with symbols and work on these statements:

We can express the digits in a 4-digit number as wxyz

This means that the number N_{1} = 1000w + 100x + 10y + z

If we take the end digit and put it at the front we get

N_{2} = 1000z + 100w + 10x + y

We are told that N_{2} = 1.2N_{1}

So assembling all our information:

1000z + 100w + 10x + y = `1.2(1000w + 100x + 10y + z)

And we know that z the end digit is 5

1000(5) + 100w + 10x + y = `1.2(1000w + 100x + 10y + 5)

5000 + 100w + 10x + y = 1200w + 120x + 12y + 6

Subtracting 100w + 10x + y from both sides preserves the equality.

5000 – 6 = 1100w + 110x + 11y

4994 = 1100w + 110x + 11y

Dividing each side by 11 preserves the equality-

454 = 100w + 10x + y

This means that y is the units place, x is the tens place and w is the hundreds place in 454.

w = 4

x =5

y =4

and we know that z = 5

therefore original number is 4545

And verify that new number 5454 divided by 4545 is 1.2