Jill asks each of her classmates to choose a number, then multiply the number by -3, add 2 to the product, multiply the result by -2, and then subtract 14. Finally, each student is asked to divide the result by 6 and record the answer. When Jill gets an answer from a classmate, she just adds three to it in her head and announces the number that classmate originally chose. How did Jill know to add 3 to each answer?
[(-3 x + 2 )(-2) -14 ]/6
[(6 x - 4 ) -14 ]/6
[6 x -18 ]/6
x - 3 write it down
add three and you get x back :)
X-3
To understand why Jill knew to add 3 to each answer, let's break down the steps Jill's classmates followed:
1. Each classmate chose a number.
2. The chosen number was multiplied by -3.
3. 2 was added to the product.
4. The result was multiplied by -2.
5. 14 was subtracted from the product.
6. The final result was divided by 6 and recorded.
Now, let's backtrack and analyze what happens when Jill adds 3 to each classmate's answer:
7. Jill adds 3 to each recorded answer.
If Jill's result after adding 3 matches the number initially chosen by the classmate, it means that the steps leading up to the division by 6 can be reversed to find the original number.
Let's reverse the operations to find the original number:
1. Start with Jill's final result (after adding 3).
2. Multiply it by 6 (reverse the division).
3. Add 14 (reverse the subtraction).
4. Multiply the result by -0.5 (reverse the multiplication by -2).
5. Subtract 2 (reverse the addition).
6. Multiply the result by -0.3333 (reverse the multiplication by -3).
Now, if we simplify the expression, we should end up with the original number chosen by the classmate. Let's calculate:
Jill's result after adding 3: X + 3
1. (X + 3) * 6 = 6X + 18
2. 6X + 18 + 14 = 6X + 32
3. (6X + 32) * -0.5 = -3X - 16
4. -3X - 16 - 2 = -3X - 18
5. (-3X - 18) * -0.3333 = X + 6
As we can see, by simplifying the expression, we end up with X + 6, which equates to the original number chosen by the classmate. Therefore, Jill knew to add 3 to each answer because it allowed her to reveal the original number chosen by her classmates.