# Multiplying Decimals

**Multiplying decimals** is very similar to multiplying integers (more detailed examples in multiplying whole numbers note) except for only small difference: find number of digits to the right of the first number (number of decimal places), do the same for the second number. Multiply numbers, ignoring decimal points. Then put the decimal point in the answer; it should have as many decimal points as the first two found numbers combined.

**Example 1.** Multiply 4.5 by 2.31

Number 4.5 has 1 decimal place.

Number 2.31 has 2 decimal places.

Sum up: $$${1}+{2}={\color{red}{{{3}}}}$$$.

Multiply numbers without decimal points: $$${45}\times{231}={10395}$$$.

Place dot in the number, so there are $$${\color{red}{{{3}}}}$$$ decimal places: $$${10}.{\color{red}{{{395}}}}$$$.

So, $$${\color{purple}{{{4.5}\times{2.31}={10.395}}}}$$$.

Why does this method works?

Because to obtain integer, we actually move decimal point to the right:

- $$${4.5}{\color{blue}{{\to}}}{45}$$$ (1 move)
- $$${2.31}{\color{blue}{{\to}}}{23.1}{\color{blue}{{\to}}}{231}$$$ (2 moves).

So, total of 3 moves.

After finding result of multiplication, we need to undo that shifting: $$${10395}{\color{blue}{{\to}}}{1039.5}{\color{blue}{{\to}}}{103.95}{\color{blue}{{\to}}}{10.395}$$$ (3 moves).

**Example 2.** Calculate $$${2.57}\times{0.00035}$$$.

Number 2.57 has 2 decimal places.

Number 0.00035 has 5 decimal places.

Sum up: $$${2}+{5}={\color{red}{{{7}}}}$$$.

Multiply numbers without decimal points: $$${257}\times{35}={8995}$$$.

Place dot in the number, so there are $$${\color{red}{{{7}}}}$$$ decimal places (we need to add leading zeros here): $$${8995}={00008995}$$$ becomes $$${0}.{\color{red}{{{0008995}}}}$$$.

So, $$${\color{purple}{{{2.57}\times{0.00035}={0.0008995}}}}$$$.

We can multiply decimal by whole number as well!

**Example 3.** Calculate $$${3}\times{0.04}$$$.

Number 3 has 0 decimal places.

Number 0.04 has 2 decimal places.

Sum up: $$${0}+{2}={\color{red}{{{2}}}}$$$.

Multiply numbers without decimal points: $$${3}\times{4}={12}$$$.

Place dot in the number, so there are $$${\color{red}{{{2}}}}$$$ decimal places (we need to add leading zeros here): $$${12}={0.12}$$$ becomes $$${0}.{\color{red}{{{12}}}}$$$.

So, $$${\color{purple}{{{3}\times{0.04}={0.12}}}}$$$.

**So, how do we multiply decimal decimals?**

- Count number of decimal places in the first number
- Count number of decimal places in the second number
- Combine numbers, found above. You will get some number, let's call it $$${n}$$$
- Multiply numbers without decimal points
- Place the decimal point in the answer, so there are $$${n}$$$ decimal places.

**Exercise 1**. Find $$${3.5}\times{1.7}$$$.

**Answer**: 5.95.

**Exercise 2**. Find $$${0.03}\times{\left(-{50}\right)}$$$.

**Answer**: -1.5.

**Exercise 3**. Multiply -0.36 by -0.7.

**Answer**: 0.252.

**Exercise 4**. Find $$$-{5}\times{0.02}\times{\left(-{3.87}\right)}$$$.

**Answer**: 0.387.

**Exercise 5**. Find $$$-{5.78}\cdot{0.212}\cdot{\left(-{45.5}\right)}\cdot{\left(-{4.018}\right)}\cdot{0.5}$$$.

**Answer**: -112.00954492.