Ivan has 15 yards of green felt 12 yards of blue felt to make 3 quilts, how many yards does he use for each quilt

Problems 2.21, 2.22, 2.23

RoseWind [281]

Statements can be proved by contrapositive, contradiction or by induction.

2.21 and 2.23 are proved by contrapositive
2.22 is proved by induction

2.21: If $n^3$ is even, then n is even (By contrapositive)

The contrapositive of the above statement is that:

If n is odd, then $n^3$ is odd

Represent the value of n as:

$n = 2k + 1$ , where $k \ge 0$

Take the cube of both sides

$n^3 = (2k + 1)^3$

Expand

$n^3 = 8k^3 + 6k^2 + 6k + 1$

Group

$n^3 =[ 8k^3 + 6k^2 + 6k] + 1$

Factor out 2

$n^3 =2[4k^3 + 3k^2 + 3k] + 1$

Assume w is an integer; where:

$w =4k^3 + 3k^2 + 3k$

So, we have:

$n^3 =2w + 1$

The constant term (i.e. 1) means that $n^3$ is odd.

Hence, the statement has been proved by contrapositive.

i.e. If n is odd, then $n^3$ is odd

2.22 $3n + 4$ is even, if and only if n is even

We have: $3n + 4$

Assume that: $n = 2k + 2$ for $k \ge 0$

So, we have:

$3n + 4 = 3(2k + 2) + 4$

Open bracket

$3n + 4 = 6k + 6 + 4$

$3n + 4 = 6k + 10$

Factorize

$3n + 4 = 2(3k + 5)$

The factor of 2 means that $3n + 4$ is even.

Hence, $3n + 4$ is even, if and only if n is even

2.22: $s \ne -1$ and $t \ne -1$ , then $s + t + st \ne -1$

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

We have:

$s + t + st = -1$

Substitute the values of s and t in: $s + t + st = -1$

$-1 -1 -1 \times -1 = -1$

$-1 -1 + 1 = -1$

$-1 = -1$

Hence, by contrapositive:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

Read more about proofs at:

brainly.com/question/19643658

7 0

3 years ago

Please solve for 1,2 and 3

Maslowich

Answer:

1. 750/100

h=7.5 in

4 0

3 years ago

Math hard me need helpp09

balu736 [363]

solution

original price = final price+ decrease percent × final price

original price = $90 +80% ×$90

original price = $90+80/100 ×$90

original price= $90+$72

original price= $162

3 0

3 years ago

could someone help me solve this equations with both comparison and elimination methods: 2(x+y)=120 and 2(2x+y/3)=170

Gnom [1K]

Answer:

y=21 X=39

Step-by-step explanation:

expand brackets on first equation: 2x+2y=120

now divide the second equation by 2 on both sides:

2x+y/3=85

then subtract the second equation from the first equation

so (2x+2y) - (2x+y/3)

the 2x on both sides cancel out and you get 5/3 y

this is equal to the sum of the first equation subtract the second equation which is 35

divide both sides by 5/3 and you get

y=21

substitute y in the first equation and you get X=39

4 0

2 years ago

Can someone please MAKE ME 4 QUESTIONS 2 of the questions will be about Subtracting integers using fractions and the other 2 wil

bearhunter [10]

1/5 - 3/10 = - 1/10
5/8 - 1/2 = 1/8

13/20 + 2/10 = 17/20
3/5 + 3/15 = 12/15

Ask me any questions if u need help

8 0

3 years ago