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I'm solving a math puzzle: "how quickly can you multiply $26$ by $24?$"

I don't know the answer so I use tutorials.

One tutorial say to do it quickly you can round numbers up and down to closest power of $5$ or $10$.

In my example $26\times 24$ I round $26$ down by $1\to 25$, and I round $24$ up by $1\to 25$. I added and removed $1$ from both number so the sum of my change is 0.

Now I have two easy number $25\times 25$ which is $625$.

But now if I use calculator program and multiply $24\times 26$ I get $624$.

Where did 1 go? What did I do wrong?

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    $\begingroup$ It's just not true that $(x-1)(x+1)=x^2$. $\endgroup$
    – M. Van
    Sep 8, 2017 at 0:04
  • $\begingroup$ If you substitute $x=25$ you can see what I mean. In simpler terms I mean that the method you use is not valid. $\endgroup$
    – M. Van
    Sep 8, 2017 at 0:05
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    $\begingroup$ Rounding numbers means that you are changing their value. If I round $4$ to $5$, then I am changing it's value. The tutorial is only telling you how to get "close" to the multiplied value, not the exact number. For the exact number, you will need other tricks. $\endgroup$ Sep 8, 2017 at 0:06
  • $\begingroup$ Why would $26*24=25^2$? We have $26*24=26+...+26$, $26$ added to itself $24$ times, so $26*24=25+...+25+1+..+1=25*24+24$, this is almost $25*25$, but it is off by $1$. $\endgroup$
    – M. Van
    Sep 8, 2017 at 0:08
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    $\begingroup$ $26*24=25^2 - 1 $ since $x^2-1=(x-1)(x+1) $ $\endgroup$ Sep 8, 2017 at 0:11

6 Answers 6

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You missed the blue square.

(not drawn to scale)

enter image description here

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    $\begingroup$ Simply beautiful! $\endgroup$ Sep 8, 2017 at 7:56
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The "quick way" is to use the formula $(x+a)(x-a)=x^2-a^2$. "Quick" does not mean to make approximations.

Here, using $x=25$ and $a=1$ will do the job. Coming to your original question. Why do you expect that adding and subtracting the same number will leave the product unchanged. Think about $5 \times 5=25$. This is the area of a square whose side is 5. Now, if you add 5 to the first term and subtract 5 from the second, then you have a new product $10 \times 0$. This has to be zero, no?

Multiplication just does not work like that. Addition DOES work like that. If you have $5 + 5 = 10$, then if you add and subtract some number from the first and second terms, you get $(5+a) + (5-a)$, the result of your addition is still 10 for any $a$.

For multiplication, the corresponding rule is that if you multiply and divide by the same non-zero number, your result is unchanged. So $(5 \times b)(\frac{5}{b})=25$ for any non-zero $b$.

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Rounding can get us a close answer, but it might not be the same. For example, what is $4\times 6?$ What is $5\times 5?$

In your case, $26\times 24$ is the total of $26$ separate groups of $24$ members each, while $25\times 25$ is the total of $25$ separate groups of $25$ members each. Note that $26\times 25$ has $26$ more than $26\times 24,$ since each of the $26$ groups has exactly one more member. That is: $$26\times 25=26\times 24+26.$$ On the other hand, $25\times 25$ has $25$ fewer members than $26\times 25,$ since it has one fewer group (of $25$). That is: $$25\times 25=26\times 25-25.$$

Consequently, we see that $$25\times 25=26\times 25-25=(26\times 24+26)-25=26\times 24+26-25=26\times 24+1,$$ as you've already discovered.


Put geometrically, you've stumbled upon the fact that a square has a greater area than a non-square rectangle having the same perimeter. Let's see why that is! I'll describe how to picture it, and I suggest that you actually draw pictures.

Start with a rectangle that is $a$ by $b,$ with $b$ bigger than $a.$ Now, draw a line cutting the rectangle into a square (that is $a$ by $a$) and a small rectangle (that is $b-a$ by $a$). Now, cut the small rectangle into two smaller rectangle that are both $\frac12(b-a)$ by $a.$ Move one of the smaller so that the smaller rectangles have an edge coinciding with neighboring edges of the square. Now, if we include a $\frac12(b-a)$ by $\frac12(b-a)$ square touching both of the smaller rectangles, then we have increased the area, and we have created a square whose perimeter is the same as the rectangle we started with.

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The sum of the changes you made to the individual factors was zero, all right, but there is absolutely no reason to think that the product will come out the same.

Let's try another example: $$ 2 \times 99 . $$

Now I find multiplication by $2$ very difficult (not really, but let's pretend I do), so I decide I want to simplify the product by subtracting $1$ from the first factor and adding $1$ back to the second factor: $$ 1 \times 100. $$

The sum of the changes I made ($-1$ and $+1$) is $0,$ so the product is the same, right?

What really happens when you add or subtract $1$ from one of two factors that are multiplied together is that you add or subtract the other factor from the product.

Let's start with $26 \times 24.$ We subtract $1$ from $26,$ and now we have $$ 25\times24 = (26 - 1) \times 24 = \underbrace{26 \times 24}_{\text{our original product}} - \underbrace{1 \times 24.}_{\text{the change we've caused}} $$

So we subtracted $1$ from $26,$ but the result was we subtracted $24$ from the product.

But then we add $1$ to $24,$ so now we have $$ 25\times25 = 25 \times (24 + 1) = \underbrace{25 \times 24}_{\text{what we had after the first change}} + \underbrace{25 \times 1.}_{\text{the additional change}} $$

The first change we made subtracted $24$ from the product, but the second change added $25.$ The two changes do not cancel each other.

If you prefer to see both changes applied at once, it's \begin{align} 25\times25 &= (26 - 1) \times (24 + 1) \\ &= 26 \times (24 + 1) - 1 \times (24 + 1) \\ &= (26 \times 24 + 26 \times 1) - (1 \times 24 + 1 \times 1) \\ &= 26 \times 24 + 26 - 24 - 1 \\ &= 26 \times 24 + 1. \\ \end{align}

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Generally: $(n+1)(n-1) = n^2 - 1.$

Let $n=25:$ $ n^2 =25×25 = 625 .$

Hence: $26×24 = 625 -1 = 624.$

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We would have

$$\tag{1}25=\sqrt{24 \times 26}$$

the latter being called the geometrical mean of 24 and 26. But

$$\tag{2}25=\tfrac{24+26}{2}$$

i.e., the arithmetical mean of 24 and 26.

(1) and (2) are incompatible, because it is known that the geometrical mean of two different numbers is always less than their arithmetical mean. Contradiction.

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    $\begingroup$ To the anonymous downvoter : be courageous, explain what upsets you. Is it a crime to say that the erroneous results can be annalysed by saying that it amounts to assimilate two different means (arithmetical and geometrical mean) ? All right, it introduces a new concept (geometrical mean) that usually is not seen in high school. But nobody can say this concept is unimportant... $\endgroup$
    – Jean Marie
    Sep 11, 2017 at 12:33

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