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4 years ago

7

Wilmer already has 214 cards. he needs to collect 334 cards. he plans to collect 8 cards each day. find the minimum number of da

ys wilmer will take to reach his goal.

1 answer:

Wittaler [7]4 years ago

3 0

The answer is 15 days.

The first step you need to take is to subtract the amount of cards he has from the amount of cards he needs. So 334 (needs) - 214(has) = 120 cards he still needs. So he is collecting 8 cards a day. In order to find out how many days it will take to collect the 120 cards, we need to divide 120 by 8. 120 divided by 8 is 15. So it will take 15 days collecting 8 cards a day in order to reach the 120 cards he still needs to collect.

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The whole numbers a and b are divisible by c. Is a+b divisible by c? Is b−a divisible by c? Explain your reasoning.

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$a+b$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

$b-a$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

Step-by-step explanation:

Let $a$ , $b$ whole number that are divisible by $c$ . Then, we have that $d = \frac{a}{c}$ and $e = \frac{b}{c}$ , which are also whole numbers. We proceed to prove that $a + b$ is divisible by $c$ :

1) $d = \frac{a}{c}$ , $e = \frac{b}{c}$ , $a$ , $b$ , $c$ , $d$ , $e$ $\in \mathbb{Z}$ Given

2) $d + e = \frac{a}{c}+\frac{b}{c}$ Compatibility with addition

3) $d+e = a\cdot c^{-1}+b\cdot c^{-1}$ Definition of division

4) $d + e = c^{-1}\cdot (a+b)$ Commutative and distributive properties

5) $d+e = (a+b) \cdot c^{-1}$ Commutative properties

6) $d+e = \frac{a+b}{c}$ Definition of division/Result

Therefore, $a+b$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

We proceed to prove that $b-a$ is divisible by $c$ :

1) $d = \frac{a}{c}$ , $e = \frac{b}{c}$ , $a$ , $b$ , $c$ , $d$ , $e$ $\in \mathbb{Z}$ Given

2) $d\cdot (-1) = \frac{a}{c}\cdot (-1)$ Compatibility with multiplication

3) $e + d\cdot (-1) = \frac{b}{c}+\frac{a}{c}\cdot (-1)$ Compatibility with addition

4) $e + d\cdot (-1) = b\cdot c^{-1}+(a\cdot c^{-1})\cdot (-1)$ Definition of division

5) $e + d\cdot (-1) = b\cdot c^{-1} + [a\cdot (-1)]\cdot c^{-1}$ Associative and commutative properties

6) $e + d\cdot (-1) = c^{-1}\cdot [b+a\cdot (-1)]$ Commutative and distributive properties.

7) $e+d\cdot (-1) = [b+a\cdot (-1)]\cdot c^{-1}$ Commutative properties.

8) $e + d\cdot (-1) = \frac{b+a\cdot (-1)}{c}$ Definition of division

9) $e-d = \frac{b-a}{c}$ $(-1)\cdot a = -a$ /Definition of subtraction

Therefore, $b-a$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

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