# Problems from Sum of natural numbers and its properties

a) Is the following equality true? What property is used?

$$15+(96+4)=(15+96)+4$$

b) Is the following equality true? What property or properties is/are used?

$$(1+6)+9=1+(9+6)$$

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### Development:

a) On both sides of the equality, the addends are the same, but the additions are done in a different order. The associative property tells us that the result is the same in both cases.

b) On the left side of the equality, the first sum that we do is $$1+6$$ and then we add $$9$$. On the right side, we first add $$9+6$$, which is the same as $$6+9$$ (using the commutative property), and then $$1$$ is added. Using the associative property, we know that the result is the same.

### Solution:

a) The equality is true, and the property used is the associative property.

b) The equality is also true. Here the commutative and the associative properties are used.

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Write the following sums, without calculation, in two different ways without changing the addends:

a) $$4+18+75$$

b) $$12+36+4+100$$

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### Development:

In both cases, using the commutative property, ordering the addends in two different ways we know that we will get the same number:

a) For example, you may decide to firstly write the $$18$$, then $$4$$ and finally $$75$$. This is also correct: first the $$4$$, then $$75$$ and finally $$18$$.

b) In this case, you can choose: first $$100$$, then $$12$$, $$4$$ and at last $$36$$. Or: $$4, 36,12$$ and $$100$$.

### Solution:

a) One possible solution, then, is: $$4+18+75=18+4+75=4+75+18$$ but you can choose any order, while the addends are $$4,18$$ and $$75$$.

b) A possible solution is: $$12+36+4+4+100=100+12+4+36=4+36+12+100$$ but you can choose any order, while the summands are $$12,36,4$$ and $$100$$.

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Say which sum has to be done first in each case:

a) $$(114+168)+223$$

b) $$4+(10+2)$$

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### Development:

In both cases the sum that is in brackets must be done first of all.

### Solution:

a) First, we do the sum $$114+168=282$$, and later $$282+223=505$$.

b) First, $$10+2=12$$, and later $$4+12=16$$.

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