Math trick and its proof: square a number ending in 5

I will be hosting the blog carnival Math Teachers at Play next week. (You can send in submissions here.)

One submission I got about various multiplication tricks or shortcuts got me inspired to write a proof of the particular trick.

You could definitely use this in algebra class. First explain the shortcut or trick itself. Then ask students to prove it, or to explain WHY it works, using algebra.

You could also explain this to younger students as an additional "neat trick" and let them explore and play with it.

THE "TRICK"

If a number ends in 5, then its square can be calculated using this "trick" (I like to call it a shortcut because there's nothing magic about it):

Let's say we have 75 × 75 => Go 7 × 8 = 56. Then tag 25 (or 5 × 5) into that. You get 5625.

Let's say we have 35 × 35 => Go 3 × 4 = 12. Then tag 25 into that. You get 1225.

Let's say we have 115 × 115 => Go 11 × 12 = 132. Then tag 25 into that. You get 13,225.

Let's say we have 245 × 245 => Go 24 × 25 = 600. Then tag 25 into that. You get 60,025.

So, you simply take the digit or digits in front of the 5 and consider those as a number in itself. Multiply that by the next number. Then "tag" 25 to the answer you got in the previous step.

Photo by Lazy butterfly

PROOF

Any whole number that ends in five is of the form A + 5, and A is a multiple of 10. Since A is a multiple of 10, we can write A = 10b, where b is now some whole number. So, our number is of the form 10b + 5. Now, let's square it.

(10b + 5)(10b + 5) and we use the distributive property to multiply this out.

(10b + 5)(10b + 5) = 100b² + 50b + 50b + 25 = 100b² + 100b + 25

Now, notice those 100b's there. We can gather that as a common factor for the first two terms:

= 100 b (b + 1) + 25

This is now essentially in the form that the trick is using.

The trick says to take b, or the number formed by the digits in front of the 5. That corresponds exactly with our b! (For example, in number 645, b is 64. Our number 645 is 10b + 5, or 640 + 5.)

So we take b, multiply it by (b + 1) which is the next number, and also by 100, and lastly add 25.

Now, b × (b+1) is the part of the trick where you multiply the digits in front of the 5 by the next number. To "tag" 25 to those digits means you add 25 only after having multiplied the number by 100 so that it would end in "00". Once it ends in "00" you can add 25 (or any two-digit number) and it is the same as "tagging" 25 to the digits without the "00".

I hope this is clear enough.

By the way, I do not feel all students must learn this shortcut for finding the square of numbers ending in 5. It is a nice addition to one's mathematical knowledge, but not any necessity. However, it is useful as an algebra problem, and of course has been used as such over the course of centuries, I'm sure.

A math review or reference book for adults & students

I just stumbled upon the book Math on Call: A Mathematics Handbook:

Based on the reviews and from the "Look Inside" feature at Amazon it looks like a GREAT math reference book. I've been on the lookout for something like this, because I get emails from adults who want to review or refresh their basic math before going to college or some other course they're taking. So now I have something to refer them to.

The table of contents includes these chapters:

NUMERATION

• The Base Ten System
• Rational Numbers
• Irrational Numbers

NUMBER THEORY

• Factors and Multiples
• Powers and Roots

COMPUTATION

• Addition
• Subtraction
• Multiplication
• Division

ALGEBRA

• Algebraic Notation
• Properties
• Relations and Functions
• Equations
• Inequalities

GRAPHS AND STATISTICS

• Gathering Data
• Organizing and Summarizing Data with Statistics
• Displaying Data in Tables and Graphs
• Interpreting Data

GEOMETRY

• Elements of Geometric Figures
• Angles
• Plane Figures
• Solid Figures

RATIO, PROPORTION, AND PERCENT

• Ratio
• Proportion
• Percent

PROBABILITY AND ODDS

• Permutations and Combinations
• Probability
• Odds

ALMANAC

• Prefixes
• Suffixes
• Problem Solving
• Study Tips
• Using a Computer for Math
• Geometric Constructions
• Test-Taking Skills
• Handy Tables
• The Metric System
• The Customary System
• General Measurement
• Fraction/Decimal Equivalents
• Multiplication Table
• Square and Cube Roots
• Right-Triangle Ratios
• Prime Numbers Less Than 500
• Mathematical Symbols
• Number Patterns
• Number Systems

YELLOW PAGES

• Glossary of Mathematical Formulas
• Glossary of Mathematical Terms

INDEX

A total of 608 pages. I can hardly believe how good reviews it has!

I was unclear if Math on Call contained exercises or not, but I'm thinking not, since I didn't see anyone mentioning such in the reviews or the description.

Math Teachers at Play Carnival

There is lots of interesting stuff today at Math Teachers at Play Carnival #14 at Math Mama Writes. Quite a nice assortment - surely something for everyone!

I particularly recommend Three Applications of High, Abstract Math. We teachers can always use other ideas as to how mathematics is useful in real life.