In this lesson, we will apply the halving and doubling technique and the distributive property to find the product of two numbers.
The halving and doubling technique involves adjusting two numbers being multiplied, that is:
\(*\) Halve one of the numbers and
\(*\) Double the other number.
This adjustment simplifies calculations without changing the final product because multiplication is commutative and associative.
Example 1
Use the halving and doubling method to find the product of 4 and 7
Solution
Product of 4 and 7
\(\Rightarrow\) 4 \(\times\) 7
Halving 4 and doubling 7
\(\Rightarrow\) \((\frac{1}{2} \times 4)\) \(\times\) \((2 \times7) \)
\(\Rightarrow\) \(2 \times 14\)
\(\Rightarrow\) \(28\)
Multiplication is based on the principle that the product of two numbers remains constant if one number is halved and the other is doubled.
Steps for Halving and Doubling
1. Identify the two numbers to be multiplied.
2. Choose one number to halve (preferably the even number).
3. Double the other number.
4. Multiply the adjusted numbers to get the product.
Example 2
Apply the halving and doubling technique to find the product of 28 and 5.
Solution
Example 3
Apply the halving and doubling technique to find the product of 125 and 4.
Solution
Example 4
Use the doubling and halving technique to find the product of 25 and 7.
Solution
Example 5
Use the doubling and halving technique to find the product of 9 and 21.
Solution
Use the doubling and halving technique to find the product of the following pairs of numbers.
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17 and 18
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29 and 12
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16 and 7
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15 and 11
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9 and 16
We can also use the distributive property of multiplication over addition and subtraction to find the product of two given numbers.
In Mathematics, we say that multiplication is distributive over addition and subtraction because given \(a, b\) and \(c\) belonging to the set of Real numbers \((\mathbb{R})\), the relations below are true.
Addition
\(a (b + c) = ab + ac\)
Subtraction
\(a (b - c) = ab - ac\)
Using the two properties above, we can say that when finding the product of two numbers, we can express one of the numbers as a sum or difference of two other numbers, and apply the distributive property to find the product of our initial numbers.
Consider the examples below:
Example 6
Using the distributive property, find the product of 10 and 14.
Solution
Product of 10 and 14
\(\Rightarrow 10 \times 14\)
\(\Rightarrow 10 \times (4 + 10)\)
\(\Rightarrow 10(4) + 10(10)\)
\(\Rightarrow 40 + 100\)
\(\Rightarrow 140\)
We expressed 14 as the sum of 4 and 10, that is, \(14 = 4 + 10\)
Steps for using the distributive property
1. Identify the two numbers to be multiplied.
2. Choose one number to express as the sum or difference of two other numbers.
3. Apply the distributive property of multiplication over addition and/or subtraction as shown above.
4. Simplify further to get the product.
Example 7
Using the distributive property, find the product of 7 and 15.
Solution
Example 8
Using the distributive property, find the product of 18 and 6.
Solution
Example 9
Using the distributive property, find the product of 25 and 9.
Solution
Example 10
Using the distributive property, find the product of 17 and 7.
Solution
Use the distributive property to find the product of the following pairs of numbers.
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17 and 18
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29 and 12
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16 and 7
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15 and 11
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9 and 16
Test yourself on what you have learnt so far. Click on the link below when you are ready.
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