Follow the instructions for each “trick” starting with several different numbers of your choice. Make a conjecture as to what result you get in each case. Use algebra and deductive reasoning, as we did in Example
Explaining a Number Trick by Using Deductive Reasoning
Consider the following number trick:
1. How many days a week do you eat out?
2. Multiply this number by 2.
3. Add 5 to the number you got in step 2.
4. Multiply the number you obtained in step 3 by 50.
5. If you have already had your birthday this year, add 1,764; if you haven’t, add 1,763.
6. Subtract the four-digit year that you were born.
7. You should now have a three-digit number. The first digit is the number of times that you eat out each week.
Use deductive reasoning to explain why this trick works.
SOLUTION:
We will use general algebra principles and deductive reasoning to explain this trick.
Let’s begin by calling the number of days that you eat out n and go through the trick
step-by-step.
Step 2: Multiplying the number by 2 will give us 2n.
Step 3: Adding 5, we get 2n + 5.
Step 4: Multiplying by 50 gives us the expression (2n + 5)50.
Step 5: Let’s assume that you haven’t had your birthday yet, so we will add 1,763 to get (2n + 5)50 + 1,763.
Step 6: Let’s assume that you were born in 1995, so we will subtract 1,995 to get
(2n + 5)50 + 1,763 - 1,995.
When we simplify the expression in step 6, we get
(2n + 5)50 + 1,763 - 1,995 = 100n + 250 + 1,763 - 1,995
= 100n + 2,013 - 1,995
= 100n + 18.
Notice that when we look at this expression in this form, the hundred’s digit is n, which is the number of days that you eat out each week. Notice that the difference 2,013 - 1,995 has no effect on the hundred’s digit. We will have more to say about this difference in the Challenge Yourself Exercises, to explain why your conjecture is correct.
a. Choose any natural number.
b. Multiply the number by 8.
c. Add 12 to the product you just found.
d. Divide the results of part (c) by 4.
e. Subtract 3.
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