Simplify the following using the properties of whole numbers:
(a) $113 \times 25 \times 4$
(b) $400 \times 37-400 \times 7$
(c) $300+195+5$
(d) $12 \times 202$

For (a), use the associative property: 113 × 25 × 4 = 113 × ( 25 × 4 ) = 113 × 100 = 11300 .
For (b), use the distributive property: 400 × 37 − 400 × 7 = 400 × ( 37 − 7 ) = 400 × 30 = 12000 .
For (c), use the associative property: 300 + 195 + 5 = 300 + ( 195 + 5 ) = 300 + 200 = 500 .
For (d), use the distributive property: 12 × 202 = 12 × ( 200 + 2 ) = ( 12 × 200 ) + ( 12 × 2 ) = 2400 + 24 = 2424 .
( a ) 11300 , ( b ) 12000 , ( c ) 500 , ( d ) 2424 ​

Explanation

Understanding the Problem We are asked to simplify the given expressions using properties of whole numbers. We will use the associative and distributive properties to make the calculations easier.

Simplifying (a) (a) We have 113 × 25 × 4 . Using the associative property of multiplication, we can rearrange the terms: 113 × 25 × 4 = 113 × ( 25 × 4 ) . We know that 25 × 4 = 100 . Therefore, 113 × ( 25 × 4 ) = 113 × 100 = 11300 .

Simplifying (b) (b) We have 400 × 37 − 400 × 7 . Using the distributive property, we can factor out 400: 400 × 37 − 400 × 7 = 400 × ( 37 − 7 ) . We know that 37 − 7 = 30 . Therefore, 400 × ( 37 − 7 ) = 400 × 30 = 12000 .

Simplifying (c) (c) We have 300 + 195 + 5 . Using the associative property of addition, we can rearrange the terms: 300 + 195 + 5 = 300 + ( 195 + 5 ) . We know that 195 + 5 = 200 . Therefore, 300 + ( 195 + 5 ) = 300 + 200 = 500 .

Simplifying (d) (d) We have 12 × 202 . Using the distributive property, we can rewrite 202 as 200 + 2 : 12 × 202 = 12 × ( 200 + 2 ) . Applying the distributive property, we get 12 × ( 200 + 2 ) = ( 12 × 200 ) + ( 12 × 2 ) . We know that 12 × 200 = 2400 and 12 × 2 = 24 . Therefore, ( 12 × 200 ) + ( 12 × 2 ) = 2400 + 24 = 2424 .

Final Answer Therefore, the simplified expressions are: (a) 11300 (b) 12000 (c) 500 (d) 2424


Examples
Understanding the properties of whole numbers like the associative and distributive properties can help simplify calculations in everyday life. For example, when calculating the total cost of multiple items at a store, you can use the distributive property to quickly estimate the total. If you buy 12 items that cost $202 each, you can calculate $12 \times 200 + 12 \times 2 = 2400 + 24 = $2424 to find the total cost.

Answered by GinnyAnswer | 2025-07-08

We simplified the expressions using associative and distributive properties to yield the results: (a) 11300, (b) 12000, (c) 500, and (d) 2424. Understanding these properties helps make calculations easier. This approach is beneficial when working with whole numbers in mathematics.
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Answered by Anonymous | 2025-07-15