Answer:
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.
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<em>Check</em>
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.
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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.
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<em>Check</em>
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.
