How do I explain why multiplying $0.8 \times 0.8$ is less than $0.8$?

Question

I'm math is very rusty so forgive me for a trivial question.

In my daughters home work she had the sum $4.8 \times 4.8$. Her thought process was to multiply $4 \times 4 =16$ and then $0.8 \times 0.8$ (which she arrived at $6.4$) and she added the two values together giving $22.4$.

Now I know you multiply by ignoring the decimal place $48 \times 48 = 2304$ and add in the decimal place afterwards 23.04. I struggled to explain why.

She then asked why $0.8 \times 0.8 = 0.64$. I'm at a loss why.

Can anyone explain this to me so I can explain it to my daughter.?

Thanks Ste

Answer 1

You have a few good answers. I will try to explain this in an intuitive way.

Think of these numbers relative to 1. If .8 is your starting number and you want to multiply it by .8 then what you are really doing is answering the question:

what is $80$% of $.8$?

Fractionally, you're trying to answer: What is $\frac{4}{5}$ of .8?

You know that any number multiplied by one will yield that number. For example $a*1=a$.

You know that multiplying a number by anything larger than one will increase that number. For example $A*3=3a$.

And finally you can see that multiplying $a$ by any number less than one (maybe most easily thought about as a fraction less than one) would give an answer less than the original. For example $a*\frac{4}{5}=\frac{4a}{5}$

Now let $a=.8$ and think about $.8=\frac{4}{5}$

and now you can see the answer to your question of why does $.8*.8=x$ where $x<.8$

Answer 2

$0.8 \times 1 = 0.8$
$0.8 \times 1.5 > 0.8$
$0.8 \times 0.8 < 0.8$

$0.8 < 1$, so naturally, any number you multiply it by will be reduced.

Answer 3

$0.8 \times $ something is the same as taking $80\%$ and this is less than $100\%$.

Answer 4

$0.8=\frac{4}{5}$. four-fifths of four-fifths is sixteen-twenty-fifths.

Also, $$4.8\times 4.8 = (4+0.8)\times(4+0.8)=(4\times 4)+(4\times 0.8)+(0.8\times 4)+(0.8\times 0.8)$$ Hence your daughter's calculation missed the middle terms, which can be simplified as $$2\times (4\times 0.8)$$

Answer 5

Say you want to multiply $3$ by $4$. $$ \begin{array}{cccccccccccccccc} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ | & | & | & | & | & | & | &| &| & | & | & | & | \\ 0 & & & 1 & & & 2 & & & 3 & & & 4 \end{array} $$ The "$1$" on the second line goes where the "$3$" is on the first line. Then you go to where $4$ is on the second line and see the answer: $12$.

Now say you want to multiply $50$ by $0.8$: \begin{array}{cccccccccccccccc} 0 & 10 & 20 & 30 & 40 & 50 \\ | & | & | & | & | & | \\ 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \end{array} The "$1$" on the second line goes where the "$50$" is on the first line. Then you go to where $0.8$ is on the second line and see the answer: $40$.

And you get $40$, which is smaller than $50$.

Answer 6

You could use an area model. Draw a square and cut it into tenths in each direction. Then shade $\frac{8}{10}$ by $\frac{8}{10}$. This will show $64$ of the $100$ subsquares being shaded.

Answer 7

No problem- thanks for taking the initiative to follow up with your questions.

As for your first question, your daughter forgot to add a term. Allow me to illuminate: Consider two numbers, $a,b$. They can be anything you want.

$$(a+b)(a+b) = a^2+2ab+b^2.$$

If you forgot why, here's a refresher- if it's obvious then just skip over it:

\begin{align*} (a+b)(a+b) & =(a\cdot a) + (a\cdot b) +(b \cdot a) + (b \cdot b)\\ & =a^2 + (a\cdot b) +(a \cdot b) + b^2\\ & =a^2 + 2(a\cdot b) + b^2\\ & =a^2 + 2ab + b^2 \end{align*}

In your daughter's example, $a+b=4.8$, where $a=4$ and $b=0.8$. She computed $a^2=16$ properly, and $b^2=0.64$ correctly. However, you forgot the term $2ab = 2\cdot 4 \cdot 0.8= 6.4.$

Once you add that, you'll get: $$6.4+16+0.64 = 23.04,$$ which is the square of $4.8$.

Answer 8

The others have answered the question in your title, but I'll address why the approach that you memorized from elementary school actually works in the first place.

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Recall that:

• To divide by $10$, you simply move the decimal point once to the left.
• To divide by $100$, you simply move the decimal point twice to the left.
• To multiply by $10$, just move the decimal point once to the right.
• To multiply by $100$, just move the decimal point twice to the right (once for each zero).

Thus, notice that: \begin{align*} 4.8 \times 4.8 &= \frac{48}{10} \times \frac{48}{10} \\ &= \frac{48 \times 48}{100}\\ &= \frac{2304}{100} \\ &= 23.04 \end{align*}

Answer 9

$0.8$ is analogous to $80\%$. If you have $80\%$ of a whole pie--that is to say, the pie was cut into five equally sized pieces, and one piece was already eaten, leaving four left--then say you want to divide the remaining pieces among yourself and four friends. You would cut each piece into five more equal pieces, giving you $20$ such pieces. You give each of your four friends four pieces, distributing $16$ of the $20$ pieces, and you get the remaining four.

So, as a fraction of the whole pie, how much did your four friends get? They got $80\%$ of $80\%$ of the whole pie; i.e., $$0.8 \times 0.8 = (4/5) \times (4/5) = 16/25 = 64\%$$ of the whole pie, because the pie was divided into five large pieces, then each piece was subdivided into another five pieces, meaning the smallest pieces were each $1/25$th of the whole pie, and $16$ of them were given to your friends.

Now, let's consider the original question of multiplying $4.8 \times 4.8$. In fact, you must write $$\begin{align*} 4.8 \times 4.8 &= (4 + 0.8) \times (4 + 0.8) \\ &= 4(4 + 0.8) + 0.8(4 + 0.8) = 4 \times 4 + 4 \times 0.8 + 0.8 \times 4 + 0.8 \times 0.8 \\ &= 16 + 3.2 + 3.2 + 0.64 \\ &= 23.04, \end{align*}$$ because you must distribute.