Depth · one concept from first principles · step 2 of 4

A birthday trick is just 100x+y in disguise

Read the conversation below and answer the question.

Grace: Hannah! I'll guess your birthday. If I get it right, you buy me ramen; if I'm wrong, I'll buy you spaghetti. Deal?
Hannah: Sure. But how are you going to guess it?
Grace: First, multiply your birth month by $5$ and add $12$ . Then multiply that by $20$ !
Hannah: Okay.
Grace: Now add the number of your birth day!
Hannah: Done.
Grace: Subtract $300$ from that. What do you get?
Hannah: (A)
Grace: (thinking to herself: adding $60$ to (A) gives (B)) Your birthday is November $6$ , right?
Hannah: What? How did you know?

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Expressions

Your answer

How to solve

Strategy Convert to Algebra — Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

1STEP 1

Times 5, plus 12, times 20

Write the month steps and distribute the 20.

(5x+12)×20=100x+240

2STEP 2

Add the day, subtract 300

This is Hannah's answer (A).

100x+240+y-300=100x+y-60

3STEP 3

Grace adds 60

Adding 60 cancels the -60, giving (B).

100x+y-60+60=100x+y

4STEP 4

Why it reveals the birthday

100x+y puts the month in the hundreds (and thousands) place and the day in the last two digits. For November 6 it is 1106: month 11, day 06.

100×11+6=1106

Answer

(A) 100x+y-60, (B) 100x+y

Check with November 6: (5×11+12)×20=67×20=1340, then 1340+6-300=1046, and 100×11+6-60=1046. Adding 60 gives 1106, which reads as 11/06.

Takeaway

A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Times 5, plus 12, times 20
Add the day, subtract 300
Grace adds 60
Why it reveals the birthday

Where next?

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Measure a border path strip by strip Split off a multiple of 5 to read the quotient

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▶ Practice — 12 problems