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Please help.

What is $£3.99*114$ in long multiplication?

I have done the first part $3.99*4$, which is $15.96$. I am not sure about the rest.

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    You need $15.96$, when multiplying by 4 (units digit). Then your next row will end in a zero: 39.90, and last, your third row will end in two zeros...399.00. Now add $15.96 + 39.9+ 399$. – 2017-01-14
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    https://en.wikipedia.org/wiki/Multiplication_algorithm#Long_multiplication – 2017-01-14
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    Also, you may notice that $3.99 = 4 - 0.01$, and this means that $3.99 \times 114 = 4 \times 114 - 114 \times 0.01 = 456 - 1.14 = 454.86$. – 2017-01-14

1 Answers 1

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Because $3.99 = 4 - 0.01$, you can simply do:

$$\begin{align} (4 \cdot 114) - (0.01 \cdot 114) = \\ 456 - 1.14 = \\ 454.86 \end{align}$$ Therefore, $£3.99 \cdot 114 = £454.86$

This is an easier way of calculating Multiplications where $a_1$ ($3.99$ in this case) is a close to another number which is easier to multiply with $a_2$ ($114$ in this case):

$$\begin{align} \underbrace {a_1}_{a_1\lt a_2} \cdot \underbrace{a_2}_{a_2 \gg a_1}=\\ (\lceil a_1 \rceil \cdot a_2)-((1-\{a_1\}) \cdot a_2) \end{align}$$

Now what are those have square brackets? The bracket $\lceil x\rceil$ is the Ceiling Function. The squiggly bracket ($\{x\}$) is the Fractional Part Function.

Example No.1:

$$\begin{align} 15.87 \cdot 80= \\ (16 \cdot 80)-(0.13 \cdot 80)= \\ 1280-10.4= \\ 1269.6 \end{align}$$

Checking this on a calculator, does give us $1269.6$ indeed.

Example No.2:

$$\begin{align} 6.177 \cdot 97= \\ (7 \cdot 97)-(0.823 \cdot 97)= \\ 679 - 79.831= \\ 599.169 \end{align}$$

Even this one does equal to $599.169$ even when checked on a calculator.