Question

Hilda was simplifying some numerical expressions and made each of the following sequences of calculations. Name the mathematical property, operation, or idea that justifies how Hilda went from each step to the next step, some of the steps are done for you. Look at the Math Notes found on page 190 for definitions. Problem

Problem 1: 5 · (−4/3) · (2/5)
= (−4/3) · 5 · (2/5)
= (−4/3) · (5 · (2/5))
= (−4/3) · (2/1) = −8/3 = −2*2/3
Problem 2: 17 + 29 + 3+ 1
= 17 + 3 + 29 + 1
= (17 + 3) + (29 + 1)
= 20 + 30 Added groups
= 50 Added terms

Answer 1

The reason for each statement is shown below.

Explanation:

Commutative Property of Multiplication:

`a* b=b* a`

Associative Property of Multiplication:

`a* (b* c)=(a* b)* c`

Commutative Property of Addition:

`a+b=b+a`

Associative Property of Addition:

`a+(b+c)=(a+b)+c`

Problem 1:

The given expression is

`5\cdot (-(4)/(3))\cdot((2)/(5))`

It can be written as

`[5\cdot (-(4)/(3))]\cdot((2)/(5))`

`[(-(4)/(3))\cdot 5]\cdot ((2)/(5))` (Commutative Property of Multiplication)

`(-(4)/(3))\cdot [5\cdot ((2)/(5))]` (Associative Property of Multiplication)

`(-(4)/(3))\cdot (2)/(1)=-(8)/(3)=-2(2)/(3)` (Simplification)

Problem 2:

The given expression is

`17 + 29 + 3+ 1`

It can written as

`17 + (29 + 3)+ 1`

`17 + (3+29)+ 1` (Commutative Property of Addition)

`(17 +3)+(29+ 1)` (Associative Property of Addition)

`20+30` (Added groups)

`50` (Added terms)

Answer 2

Problem 1:

5 · (−4/3) · (2/5)

= (−4/3) · 5 · (2/5) Conmutative Property of Multiplication

= (−4/3) · (5 · (2/5)) Associative Property of Multiplication

= (−4/3) · (2/1) = −8/3 = −2*2/3 Multiplied fractions and extracted common factor

Problem 2:

17 + 29 + 3+ 1

= 17 + 3 + 29 + 1 Conmutative Property of Addition

= (17 + 3) + (29 + 1) Associative Property of Addition

= 20 + 30 Added groups

= 50 Added terms

Explanation:

For the Problem 1:

In the first step, Hilda applied the Conmutative Property of Multiplication, because she changed the order of the numbers in the product

In the second step, she applied the Associative Property of Multiplication, because she agrouped the product of 5 and 2/5 to perform it sepparately

In the third step, she calculated the product of the fractions -4/3 and 2/1, then she extracted 2 as a common factor to express the fraction as -2*2/3

For the Problem 2:

In the first step, Hilda applied the Conmutative Property of Addition, because she changed the order of the numbers in the sum

In the second step, she applied the Associative Property of Addition, because she associated the addition of 17 and 3 and the addition of 29 and 1, to calculate them in groups.