Question

Use distributive property to find:
(a) $ 1001 \times 87 $
(b) $ 597 \times 1003 $
(c) $ 996 \times 265 $

Answer 1

Hint: According to the distributive property, multiplying the sum or difference of two or more addends by another number will give the same answer as multiplying each addend individually by the number and then adding or subtracting the products together. So here divide the given number which is close to 1000 or 100 into two by adding or subtracting the appropriate number from it. And then apply distributive law.

Complete step by step solution:
We are given three products and we have to solve their values using distributive property.
(a) $ 1001 \times 87 $
Here 1001 is closer to 1000 than 87and it can also be written as the sum of 1000 and 1.
 $
  1001 = 1000 + 1 \\
  1001 \times 87 = \left( {1000 + 1} \right) \times 87 \;
  $
According to distributive property of addition, $ \left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right) $
Here a is 1000, b is 1 and c is 87.
 $
  \left( {1000 + 1} \right) \times 87 = \left( {1000 \times 87} \right) + \left( {1 \times 87} \right) \\
   = 87000 + 87 = 87087 \\
  \therefore 1001 \times 87 = 87087 \\
  $
(b) $ 597 \times 1003 $
Here 1003 is closer to 1000 than 597 and it can also be written as the sum of 1000 and 3.
 $
  1003 = 1000 + 3 \\
  597 \times 1003 = 597 \times \left( {1000 + 3} \right) = \left( {1000 + 3} \right) \times 597\left( {\because a + b = b + a,{\text{ Commutative property}}} \right) \;
  $
According to distributive property of addition, $ \left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right) $
Here a is 1000, b is 3 and c is 597.
 $
  \left( {1000 + 3} \right) \times 597 = \left( {1000 \times 597} \right) + \left( {3 \times 597} \right) \\
   \Rightarrow 597000 + 1791 = 598791 \\
  \therefore 597 \times 1003 = 598791 \\
  $
(c) $ 996 \times 265 $
Here 996 is closer to 1000 than 265 and it can also be written as the difference of 1000 and 4.
 $
  996 = 1000 - 4 \\
  996 \times 265 = \left( {1000 - 4} \right) \times 265 \;
  $
According to distributive property of subtraction, $ \left( {a - b} \right) \times c = \left( {a \times c} \right) - \left( {b \times c} \right) $
Here a is 1000, b is 4 and c is 265.
 $
  \left( {1000 - 4} \right) \times 265 = \left( {1000 \times 265} \right) - \left( {4 \times 265} \right) \\
   = 265000 - 1060 = 263940 \\
  \therefore 996 \times 265 = 263940 \;
  $

Note: Many people confuse distributive property with associative property. So we must be careful. Associate property does not work for subtraction and division whereas distributive property works for all arithmetic operations. Distributive law is multiplying everything inside the parenthesis with the outside number whereas associative law is just changing the place of parenthesis wherever you like but it does not change the result.