### Multiplication and Division by Power of 10

The table below shows the multiplication of 5.03 by different powers of 10

5.03Â Ã—Â 1 | = 5.03Â Ã—Â 10^{0} | 5.03 |

5.03Â Ã—Â 10 | = 5.03Â Ã—Â 10^{1} | 50.3 |

5.03Â Ã—Â 100 | = 5.03Â Ã—Â 10^{2} | 503 |

5.03Â Ã—Â 1000 | = 5.03Â Ã—Â 10^{3} | 5 030 |

5.03Â Ã—Â 10000 | = 5.03Â Ã—Â 10^{4} | 50 300 |

If you look very well at the table, you will notice:

**1.** As the power of 10 increases, it appears as if the decimal point stays where it is and the digits in the number move to the left.

**2.** The digits move as many places to the left as the power of 10 (or as the number of zeros in the multiplier.

**3.** As each place to the right of the digits becomes empty, we fill it with a zero to act as a placeholder.

**4.** If the fraction of the right of the decimal point becomes zero, there is no need to write anything after the point.

Similarly, we divide 5.03 by increasing power of 10.

5.03 Ã· 1 | = 5.03 |

5.03 Ã· 10 | = 5.03 Ã· 10^{1} = 0.503 |

5.03 Ã· 100 | = 5.03 Ã· 10^{2 }= 0.0503 |

5.03 Ã· 1 000 | = 5.03 Ã· 10^{3 }= 0.00503 |

5.03 Ã· 10 000 | = 5.03 Ã· 10^{4} = 0.000503 |

If you look at the table above you, you will notice that when dividing by powers of 10:**1. **As the power of 10 increases, it appears as if the decimal point stays where it is and the digits in the number move to the right.

**2.** The digits move as many places to the right as the power of 10 (or as the number of zeros in the division; i.e. the dividing number).

**3.** As each place to the left of the digits becomes empty, we fill it with a zero to act as a placeholder.

**4.** If the number to the left of the decimal point becomes zero, it is usual to write a zero there.

### Example 1:

Write the following as decimal numbers:**(a)** 0.063Â Â Ã—Â 10 000**(b)** \( \frac{32}{1000} \)**(c)** \( \scriptsize 140 \div 100\:000 \)**(d)** \( \scriptsize 0.000271 \: \times \: 100 \)

**Solution**

**(a)** 0.063 Â xÂ 10 000 = 0.063 Â x 10^{4}

= 630

**(b) **\( \frac{32}{1000} \\ = \scriptsize 32 \; \div \; 100 \\ = \scriptsize 32 \; \div \; 10^3 \\ = \scriptsize Â Â 0.032\)

**(c)** \( \scriptsize 140 \div 100\:000 \\ = \scriptsize 140 \; \div \; 10^5 \\ = \scriptsize 0.00140 = \scriptsize 0.0014\)

**Note:** It is not necessary to write zeros to the right of a decimal fraction. For example, 0.800 000 is just the same as 0.8.

**d)** 0.000271 Â Ã—Â 100 = Â 0.000271Â Ã—Â 10^{2}

= 0.0271

### Multiplication of Decimals:

### Example 2:

Find the product of 28.6 and 1.46

**Solution**

Note that 28.6 is same as Â \( \frac{286}{10} \)

= \( \frac{286 \: \times \: 146}{10 \: \times \: 100} \)

Let’s multiply the numerator (i.e. 286 Ã— 146)

âˆ´ 286 Ã— 146 = 41756

Then let’s multiply the denominator, 10 Ã— 100 = 1000

âˆ´ \( \frac{286 \: \times \: 146}{10 \: \times \: 100} \\ = \frac{41756}{1000} = \scriptsize 41.756\)

### Example 3:

Calculate the cost of 50 shirts at N20.50k each.

**Solution**

1 shirt costs â‡’ N20 : 50kÂ

50 shirts costs â‡’ N20 : 50k Â Ã—Â 50

First, ignore the decimal point and multiply:

Now insert the decimal point in the answer to give 2 decimal places

50 shirts cost N1025

### Example 4:

Calculate 0.25 Ã— 0.008

**Solution**

Ignoring the decimal: Â Â Â

25 Â Ã— Â 8 Â = Â 200

There are five digits after the decimal points in the given numbers.

So, 0.25Â Ã—Â 0.008 Â = Â 0.00200

âˆ´ 0.25 Â Ã—Â 0.008 Â =Â 0.002

### Division of Decimals:

You can easily divide a decimal number by a whole number. If the divisor is a whole number, divide it in the usual way. Be careful to include the decimal point in the correct place.

### Example 5:

Calculate: Â **(i) **Â Â 20 Â Ã· Â 0.2 Â **(ii)**Â Â 6.5 Â Ã·Â 0.005

**Solution**

**(i)** 20Â Ã· Â 0.2Â

You need to change 0.2 to a whole number. To change it into a whole number multiply both the numerator and denominator by 10.

i.e.\( \frac{20}{0.2} = \frac{20 \: \times \: 10}{0.2 \: \times \: 10} = \frac{200}{2} = \ \scriptsize 100 \)

**(ii)** 6.5 Â Ã· Â 0.005

You need to change 0.005 to a whole number. To change it into a whole number multiply both the numerator and denominator by 1000.

â‡’ \( \frac{6.5}{0.005} \: \times \: \frac{1000}{1000} \\ = \frac {6500}{5} \)

= 1300

6.5 Ã· 0.005 = 1300

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