need help with math guessing number problem?

Question

Ask your friend to guess a number, and then ask her to:

Multiply it by 5; add 6 to the result; multiply the result by 4; add 10 to the result; and then multiply the result by 5.

Now ask your friend to tell you the result. The result should always end with 70 in the least significant positions (e.g.: 370, 570, 1270, 45670, etc...).

Scratch the ‘70’ from the last two digits and subtract one from whatever is left. That would be the number your friend had guessed! For example, if the result is 2370, the number she had guessed is 22 (23 -1), if the result is 470, the number is 3 (4-1), and so on.

Note: if the result does not end with 70, you know that she has made a mistake in her calculations!

1. Why are the two least significant digits are always 7 and 0, respectively.
2. Why can you come up with the guessed number if you subtract 1 from the left over number after scratching 70?

Answer 1

Let x be the number.
After each step, the number becomes:

Multiply it by 5: 5x
add 6 to the result: 5x+6
multiply the result by 4: 20x + 24
add 10 to the result: 20x + 34
multiply the result by 5: 100x + 170

[ Notice that: 100x + 170 = 100(x+1) + 70
this answers question 1. ]

"Scratching 70" from this number is the same as subtracting 70 and dividing the number by 100. So what is left is:

subtract 70: 100(x+1)
divide by 100: x+1
Subtract 1 from this and you have the original number x.

Answer 2

Suppose the number your friend selected is x.

x
5x = Multiply by 5
5x + 6 = Add 6
4(5x + 6) = 20x + 24 = Multiply by 4
20x + 34 = Add 10
5(20x + 34) = 100x + 170 = Multiply by 5

Notice that the original number has shifted to the left two places since it was multiplied by 100. So the only thing left in the last two places is 70.

x + 1 = Scratch the 70 (really subtract 70 then divide by 100)

x = Subtract 1

And you have your original number back.

Answer 3

Because

((x*5 + 6) * 4 + 10) * 5 = 100 * (x + 1) +70

Answer 4

x=6. the final thank you to remedy this is to translate the word difficulty into an algebra equation. "two times the variety greater desirable by skill of one" is translated to => "2x + a million" "is" interprets mathematically to => "=" "5 below thrice the variety" interprets to => "3x - 5" So placed all of it at the same time and we've the equation "2x + a million = 3x - 5" Then purely remedy this equation 2x + a million + 5 = 3x a million + 5 = 3x - 2x 6 = 1x and then we've the respond: x = 6