-
Algebraic Processes | Week 14 Topics1 Quiz
-
Simplification of Algebraic Expressions | Week 24 Topics1 Quiz
-
Simplification of Algebraic Expressions 2 (Use of Brackets) | Week 34 Topics1 Quiz
-
Simple Equations | Week 41 Topic1 Quiz
-
Simple Equations II | Week 53 Topics1 Quiz
-
Plane Shapes I | Week 65 Topics2 Quizzes
-
Plane Shapes II | Week 77 Topics1 Quiz
-
Plane Shapes III | Week 87 Topics1 Quiz
-
Decimals and Percentages I | Week 92 Topics1 Quiz
-
Decimals and Percentages II | Week 103 Topics1 Quiz
Topic Content:
- Multiplication and Division with Decimals
Multiplication and Division by Power of 10:
The table below shows the multiplication of 5.03 by different powers of 10
| 5.03 × 1 | = 5.03 × 100 | 5.03 |
| 5.03 × 10 | = 5.03 × 101 | 50.3 |
| 5.03 × 100 | = 5.03 × 102 | 503 |
| 5.03 × 1000 | = 5.03 × 103 | 5 030 |
| 5.03 × 10000 | = 5.03 × 104 | 50 300 |
If you look closely 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 ÷ 1 | = 5.03 |
| 5.03 ÷ 10 | = 5.03 ÷ 101 = 0.503 |
| 5.03 ÷ 100 | = 5.03 ÷ 102 = 0.0503 |
| 5.03 ÷ 1 000 | = 5.03 ÷ 103 = 0.00503 |
| 5.03 ÷ 10 000 | = 5.03 ÷ 104 = 0.000503 |
If you look at the table above, 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 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 9.2.1:
Write the following as decimal numbers:
(a) 0.063 × 10 000
(b) \( \frac{32}{1000} \)
(c) \( \scriptsize 140 \div 100\:000 \)
(d) \( \scriptsize 0.000271 \: \times \: 100 \)
Solution
(a) 0.063 × 10 000 = 0.063 × 104
= 630
(b) \( \frac{32}{1000} \\ = \scriptsize 32 \; \div \; 100 \\ = \scriptsize 32 \; \div \; 10^3 \\ = \scriptsize 0.032\)
(c) \( \scriptsize 140 \div 100\:000 \\ = \scriptsize 140 \; \div \; 10^5 \\ = \scriptsize 0.00140 = \scriptsize 0.0014\)
Note: It is unnecessary 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 × 100 = 0.000271 × 102
= 0.0271
Multiplication of Decimals:
Example 9.2.2:
Find the product of 28.6 and 1.46
Solution
\( \frac{286}{10} \: \times \: \frac{146}{100} \)Note that 28.6 is same as \( \frac{286}{10} \)
= \( \frac{286 \: \times \: 146}{10 \: \times \: 100} \)
Let’s multiply the numerator (i.e. 286 × 146)
∴ 286 × 146 = 41756
Then let’s multiply the denominator, 10 × 100 = 1000
∴ \( \frac{286 \: \times \: 146}{10 \: \times \: 100} \\ = \frac{41756}{1000} = \scriptsize 41.756\)
Example 9.2.3:
Calculate the cost of 50 shirts at N20.50k each.
Solution
1 shirt costs ⇒ N20 : 50k
50 shirts costs ⇒ N20 : 50k × 50
First, ignore the decimal point and multiply:
Now insert the decimal point in the answer to give 2 decimal places
50 shirts cost ₦1025
Example 9.24:
Calculate 0.25 × 0.008
Solution
Ignoring the decimal:
25 × 8 = 200
There are five digits after the decimal points in the given numbers.
So, 0.25 × 0.008 = 0.00200
∴ 0.25 × 0.008 = 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 9.2.5:
Calculate:
(i) 20 ÷ 0.2
(ii) 6.5 ÷ 0.005
Solution
(i) 20 ÷ 0.2
You need to change 0.2 to a whole number. To change it into a whole number, multiply 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 ÷ 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.
⇒ \( \frac{6.5}{0.005} \: \times \: \frac{1000}{1000} \\ = \frac {6500}{5} \)
= 1300
6.5 ÷ 0.005 = 1300