Understanding mathematics, resource 2, chapter nine.
6. if i think of a number, halve it, take away 5 and then add on 3 times the
number i thought of i end up with a number that is 4 more than twice the
number i first thought of. what was the number i first thought of?
7. i think of a number and add it to twice itself. if i then take away one less
than the number first thought of i end up with an answer of 9. find the
number first thought of.
8 i think of a number, add 4 and then multiply the answer by 5. i could
have achieved the same final answer had i instead multiplied my chosen
number by 4 and then added 5. what was the number i thought of?
9. if i take a particular number from 3, double the answer and take the
result of doing this from the answer obtained by multiplying the particular
number by 3 and adding 15, i end up with 25. what is the particular
number i started with?
10. i think of a number, add three, divide the answer by 2 and then add 7
this gives an answer that is the same as i would have obtained had i
instead doubled the number first thought of and then taken away 8. find
the number first thought of
pls answer whichever you can do!

  6.  6 is the number I thought of

  8.  -15 is the number I thought of

Step-by-step explanation:

6. Let x represent the number you thought of.

  x/2 . . . halve it

  x/2 -5 . . . take away 5 (from the previous result)

  x/2 -5 +3x . . . then add on 3 times the number I thought of

  = . . . I end up with ...

  4 + . . . four more than ...

  4 + 2x . . . ... twice the number I first thought of.

So, the equation is ...

  x/2 -5 +3x = 4 +2x

Subtracting 2x we have

  x/2 -5 +3x -2x = 4

Adding 5 and collecting terms, we get

  (3/2)x = 9

Multiplying by the inverse of the coefficient of x, we get ...

  x = (2/3)(9)

  x = 6

The number I first thought of was 6.

__

Check

After I halve it, take away 5 and add back 3 times 6, I have ...

  3 -5 +18 = 16

After I add 4 to twice the number, I have ...

  4 +2·6 = 16

So, the first set of manipulations gives the same result as the second. 6 is the answer.

8. Let x represent the number I thought of.

  x +4 . . . add 4

  5(x +4) . . . multiply the answer by 5

  = . . . . I could get the same final answer by ...

  4x . . . multiplying my number by 4

  4x +5 . . . then adding 5

So, the equation is ...

  5(x +4) = 4x +5

  5x +20 = 4x +5 . . . . eliminate parentheses

  x + 20 = 5 . . . . . . . . subtract 4x

  x = -15 . . . . . . . . . . . subtract 20

The number I thought of was -15.

__

Check

Add 4 and multiply the result by 5: 5(-15+4) = 5(-11) = -55.

Multiply the result by 4 and add 5: 4(-15)+5 = -60+5 = -55, the same final answer.

The answer of -15 checks with the problem statement.