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mash [69]
3 years ago
12

How do I do multi digit multiplication?

Mathematics
1 answer:
Alexus [3.1K]3 years ago
3 0

A web search will turn up numerous videos that are available to answer that question. Often, you may find them more satisfactory than the static explanation of a text answer.

The fundamental idea is that the product is the sum of the products of the parts of the number(s). That is, the distributive property applies.

Here is an example.

... 12 × 34

... = (10 +2)×(30 +4)

... = 10(30 +4) +2(30 +4)

... = 10·30 + 10·4 + 2·30 + 2·4

... = 300 + 40 + 60 + 8

... = 408

_____

Here's an interpretation of what we've just seen that is a little unconventional. The product is the following sum:

... (1·3)×100 + (1·4 + 2·3)×10 + (2·4)×1

If you look at the place values of the digits being multiplied, you see that the multiplier here (×100 or ×10 or ×1) is the product of the place values of the digits involved. That is, when a digit in the 10s place is multiplied by another in the 10s place, their product contributes to the 100s place (10×10) of the result.

One method of multidigit multiplication that is taught is to only write down the partial sums obtained by adding products with the same "place" contribution in the result. That is, the product of 1s place digits (2 and 4 in our example) will go in the 1×1=1s place of the result.

The sum of products of the 10s and 1s place digits (1·4 + 2·3) = 10 will go in the 10×1 = 10s place of the result.

The product of the 10s place digits (1·3) = 3 will go in the 10×10 = 100s place of the result.

If you're good at keeping numbers in your head (gets easier with practice), this method can be faster than other more conventional ways to do it.

For numbers of more digits and/or of different lengths (say a 3-digit by 5-digit number), there are more partial sums, but the idea stays the same. It can be helpful to add leading zeros to the short number to make the numbers the same length.

Here's an example with two 5-digit numbers. Digits are chosen to be different so you can see what is being multiplied at each stage. Pay attention to the pattern being used to select digits to play with.

17986\\03524\\\\=(6\cdot 4)\times 1+(8\cdot 4+6\cdot 2)\times 10+(9\cdot 4+8\cdot 2+6\cdot 5)\times 100\\+(7\cdot 4+9\cdot 2+8\cdot 5+6\cdot 3)\times 1000\\+(1\cdot 4+7\cdot 2+9\cdot 5+8\cdot 3+6\cdot 0)\times 10000\\+(1\cdot 2+7\cdot 5+9\cdot 3+8\cdot 0)\times 10^5\\+(1\cdot 5+7\cdot 3+9\cdot 0)\times 10^6+(1\cdot 3+7\cdot 0)\times 10^7+(1\cdot 0)\times 10^8\\=63,382,664

It can be convenient to write down partial sums vertically aligned with the numbers being multiplied. (Put the sum where its place value indicates it should go.) Here, we have proceeded from right to left, but you can also do it proceeding from left to right. (Of course, the product of anything with zero is zero, so can be skipped or ignored.)

Some find it convenient to write the higher-order digits of a sum on separate lines, vertically aligned according to place value. For example, the partial sums we got in the exercise above were 24, 44, 82, 104, 87, 64, 26, and 3. Those might be written like this ...

\begin{array}{cccccccc}3&6&4&7&4&2&4&4\\2&6&8&0&8&4&2\\&&1\\6&3&3&8&2&6&6&4\end{array}

where the last line in this array is the sum of the rows above, hence the result of the multiplication.

_____

When the numbers have decimal digits, the number of decimal places in the result will be the total of the numbers of decimal places in the numbers being multiplied. For example, 8.31×5.6 has 2+1=3 total decimal digits, so the result will have 3 decimal digits. (It is 46.536.) Sometimes such a multiplication will have a product that ends in zeros. Those zeros are counted when placing the decimal point. (1.2×1.5 = 1.80 with 2 decimal digits.)

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Setler79 [48]
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4 0
3 years ago
Simplify algebraic expression 3/4 (1/2X - 12) + 4/5
Nata [24]
\frac{3}{4} ( \frac{1}{2} x-12)+ \frac{4}{5} = 

=\frac{3}{4} ( \frac{x}{2} -12)+ \frac{4}{5}

= \frac{3x}{8} -9+ \frac{4}{5}

= -\frac{41}{5} + \frac{3x}{8}


The answer is: -\frac{41}{5} + \frac{3x}{8}

8 0
3 years ago
What’s the missing factor?
Vesna [10]

The missing factor is 1 5/8

<u>Step-by-step explanation:</u>

Let us consider the missing factor has 'x'

The given expression is 4 1/2 - x = 2 7/8

<u>To find x :</u>

The first step is to bring all the constant terms on on side and x on the other side,

⇒ 4 1/2 - 2 7/8 = x

Now, the numbers are in improper fraction (mixed fraction). So you have to convert them into proper fraction.

To convert the improper fraction 4 1/2 into proper fraction,

Multiply the denominator 2 with 4 and the result 8 is added with the numerator 1.

⇒ 4 1/2 = 9/2.

To convert the improper fraction 2 7/8 into proper fraction,

Multiply the denominator 8 with 2 and the result 16 is added with the numerator 7.

⇒ 2 7/8 = 23/8.

Now replace them in the given expression. The expression becomes as :

⇒ (9/2) - (23/8) = x

Cross multiply 2 and 8,

⇒ (72 - 46) / 16 = x

⇒ 26 / 16 = x

⇒ 13/8 = x

The x value is 13/8. You have to convert it into mixed fraction (since the question is given in the form of mixed fraction).

<u>To convert into mixed fraction,</u>

Divide 13 by 8 and write them in the form of Quotient (Reminder / Divisor).

⇒ 13 ÷ 8

⇒ 1 (quotient)

⇒ 5 (remainder)

The divisor is 8.

Therefore, x = 13/8 can be written as 1 5/8.

The missing factor is 1 5/8.

8 0
3 years ago
Raj tried to solve the system below.
jekas [21]

Answer: Option C) Raj forgot the negative when substituting -15+9x for y.


Solution:

(1) 9x-y=15

(2) 2x+8y=28

Isolating y in the first equation. Subtracting 9x both sides of the equation:

(1) 9x-y-9x=15-9x

Subtracting:

(1) -y=15-9x

Multiplying both sides of the equation by -1:

(1) (-1)(-y)=(-1)(15-9x)

(1) y=-15+9x

Then Raj found the value of y. It's not option D.

Substitutng y by -15+9x in the second equation:

(2) 2x+8(-15+9x)=28

Then option C) is the answer: Raj forgot the negative when substituting -15+9x for y.

Eliminating the parentheses applying the distributive property in the multiplication:

(2) 2x-120+72x=28

Adding similar terms:

(2) 74x-120=28

Solving for x. Adding 120 both sides of the equation:

(2) 74x-120+120=28+120

Adding:

(2) 74x=148

Dividing both sides of the equation by 74:

(2) 74x/74=148/74

Dividing:

(2) x=2

Solving for y: Replacing x by 2 in the first equation:

(1) y=-15+9x

(1) y=-15+9(2)

Multiplying:

(1) y=-15+18

Subtracting:

(1) y=3  

3 0
3 years ago
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Four balls of wool will make 8 knitted hats. How many balls of wool will Janet need to make 6 hats?​
elena55 [62]
3 balls of wool would make 6 hats.
8 0
2 years ago
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