MATHEMATICS
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Answer: (2×10^15)/(4×10^9) = (2/4)×10^(15-9) = 0.5×10^6 = 5×10^5

The first number is 5×10^5 = 500,000 times larger than the second.

The first number is 5×10^5 = 500,000 times larger than the second.

HIGH SCHOOL

Prove that if n is an integer and 3n + 2 is even, then n is even using a.a proof by contraposition.

b.a proof by contradiction.

Part (a):

A proof by contrapositive means that we will prove the opposite of the given statement. In this case, we will prove that when n is odd, the given statement 3n+1 is odd.

Steps:

1- Assume n is odd, this means that n = 2m + 1 where m is an integer

2- Substitute with the assumed n in the given expression:

3n + 2 = 3(2m+1) + 2

= 6m + 3 + 2

= 6m + 5

= (6m+4) + 1

= 2(3m+2) + 1

3- Now, 2(3m+2) means that we will multiply a number (3m+2) by 2 which will always give an even number. Adding one to the result means that the final result is odd.

Therefore, when n is odd, the expression 3n+2 is odd.

Part (b):

A proof by contradiction means that we will prove that if the given statement is F, then this will lead to a contradiction.

The given statement is:

If 3n+2 is even, then n is even

For this statement to be F, 3n+2 has to be even while n is odd.

So, the steps of the proof will be as follows:

1- Assume 3n+2 is even

2- Assume n is odd. This means that n = 2m+1

3- Substitute with the assumed n in the given expression.

3n + 2 = 3(2m+1) + 2

= 6m + 3 + 2

= 6m + 5

= 2(3m+2) + 1

Now, again, 2(3m+2) means that we will multiply a number (3m+2) by 2 which will always give an even number. Adding one to the result means that the final result is odd.

This means that 3m+2 is odd which contradicts our assumption in step 1 that it is even.

Hope this helps :)

A proof by contrapositive means that we will prove the opposite of the given statement. In this case, we will prove that when n is odd, the given statement 3n+1 is odd.

Steps:

1- Assume n is odd, this means that n = 2m + 1 where m is an integer

2- Substitute with the assumed n in the given expression:

3n + 2 = 3(2m+1) + 2

= 6m + 3 + 2

= 6m + 5

= (6m+4) + 1

= 2(3m+2) + 1

3- Now, 2(3m+2) means that we will multiply a number (3m+2) by 2 which will always give an even number. Adding one to the result means that the final result is odd.

Therefore, when n is odd, the expression 3n+2 is odd.

Part (b):

A proof by contradiction means that we will prove that if the given statement is F, then this will lead to a contradiction.

The given statement is:

If 3n+2 is even, then n is even

For this statement to be F, 3n+2 has to be even while n is odd.

So, the steps of the proof will be as follows:

1- Assume 3n+2 is even

2- Assume n is odd. This means that n = 2m+1

3- Substitute with the assumed n in the given expression.

3n + 2 = 3(2m+1) + 2

= 6m + 3 + 2

= 6m + 5

= 2(3m+2) + 1

Now, again, 2(3m+2) means that we will multiply a number (3m+2) by 2 which will always give an even number. Adding one to the result means that the final result is odd.

This means that 3m+2 is odd which contradicts our assumption in step 1 that it is even.

Hope this helps :)

HIGH SCHOOL

a family spends 1/3 of its weekly budget on rent 1/4 on food 1/8 on clothes 1/12 on entertainment and the remainder is banked how much is banked if the weekly income is €846.72

The amount banked is €176.4.

Given,

A family spends

- 1/3 of its weekly budget on rent

- 1/4 on food

- 1/8 on clothes

- 1/12 on entertainment

- remainder is banked

We need to find how much is banked if the weekly income is €846.72.

How do we find fractional value from an amount?We multiply the fraction with the total amount.

Example:

50% of 100

50% = 1/2

= 1/2 x 100

= 50

We have,

Total budget = €846.72

Amount spend:

On rent - 1/3 x 846.72 = 282.24

On food - 1/4 x 846.72 = 211.68

On clothes - 1/8 x 846.72 = 105.84

On entertainment - 1/12 x 846.72 = 70.56

Find the amount that remained:

= 846.72 - ( 282.24 + 211.68 + 105.84 + 70.56 )

= 846.72 - 670.32

= 176.4

Thus the amount banked is €176.4.

Learn more about finding monthly income with fraction of different amount spend is given here:

brainly.com/question/15108046

#SPJ2

1/3+1/8+1/12+1/4=19/24

19/24=791.66

846.72-791.66=55.06

19/24=791.66

846.72-791.66=55.06

MIDDLE SCHOOL

If marching bands vary from 21 to 49 player, which numbers of players can be arranged in the greatest number of rectangles?

14 numbers of players can be arranged in the greatest number of rectangles.

Given that,

Marching bands vary from 21 to 49 players,

To determine the number of players can be arranged in the greatest number of rectangles.

The rectangle is 4 sided geometric shape whose opposites are equal in length and all angles are about 90°.

here,

In a marching band, there would be at least 2 rows of the player can be start with, so if the number of rows of players is 2 and the length of the row could be 14 the players can be arranged in the greatest number of rectangles,

Players arranged in this phenomenon = 2 * 14 = 48.

Thus, 14 numbers of players can be arranged in the greatest number of rectangles.

Learn more about rectangles here:

brainly.com/question/15019502

#SPJ2

Hmm. a rectangle?ok then, least number of rectangles needed-6.

. . .

. . . ( rectangle)

so, 6 x 8=48(greatest multiple of 6 from between 21 and 49)

48 players are most, so 8 rectangles

. . .

. . . ( rectangle)

so, 6 x 8=48(greatest multiple of 6 from between 21 and 49)

48 players are most, so 8 rectangles

MIDDLE SCHOOL

In a class 4/5 of the girls had brown hair. Of the girls with brown hair, 3/4 of them had long hair. of the girls with long brown hair 1/3 of them had green eyes. What fraction of the girls in the class had long brown hair and green eyes?

Lets say there are 100 girls.

4/5 of 100 is 80, so 80 girls have brown hair.

3/4 of the 80 who have brown hair have long hair.

3/4 of 80 is 60.

so we know that out of 100, 60 have long brown hair.

in simplest fraction form, 60/100= 3/5

4/5 of 100 is 80, so 80 girls have brown hair.

3/4 of the 80 who have brown hair have long hair.

3/4 of 80 is 60.

so we know that out of 100, 60 have long brown hair.

in simplest fraction form, 60/100= 3/5