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3 years ago

Tamara has 20000. If 4/5 of what Tamara has equals 8/9 of what Martina has, how much does Martina have

Mathematics

1 answer:

lilavasa [31]3 years ago

7 0

Answer:

Martina has 18,000

Step-by-step explanation:

Set up the 4/5 the amount Tamara has as a proportion. 4/5 = x/20000
Then cross multiply
4 * 20,000 = 80,000
80,000 = 5x
Divide and get 16,000 (This is 4/5 of what Tamara has)
Set up what Martina has as a proportion. 8/9 = 16,000/x
Cross multiply 8x = 16,000 * 9
8x = 144,000
Divide by 8
18,000 :)

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A number n plus 7 is less than or equal to 9

devlian [24]

N ≤ 2, if n plus 7 is less than or equal to 9 (n+7 ≤ 9) then carry the 7 over to create n ≤ 9-7 giving the answer n ≤ 2 (n less than or equal to 2)

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3 years ago

Prove each of the following statements below using one of the proof techniques and state the proof strategy you use.

pochemuha

Answer:

See below

Step-by-step explanation:

a) Direct proof: Let m be an odd integer and n be an even integer. Then, there exist integers k,j such that m=2k+1 and n=2j. Then mn=(2k+1)(2j)=2r, where r=j(2k+1) is an integer. Thus, mn is even.

b) Proof by counterpositive: Suppose that m is not even and n is not even. Then m is odd and n is odd, that is, m=2k+1 and n=2j+1 for some integers k,j. Thus, mn=4kj+2k+2j+1=2(kj+k+j)+1=2r+1, where r=kj+k+j is an integer. Hence mn is odd, i.e, mn is not even. We have proven the counterpositive.

c) Proof by contradiction: suppose that rp is NOT irrational, then rp=m/n for some integers m,n, n≠. Since r is a non zero rational number, r=a/b for some non-zero integers a,b. Then p=rp/r=rp(b/a)=(m/n)(b/a)=mb/na. Now n,a are non zero integers, thus na is a non zero integer. Additionally, mb is an integer. Therefore p is rational which is contradicts that p is irrational. Hence np is irrational.

d) Proof by cases: We can verify this directly with all the possible orderings for a,b,c. There are six cases:

a≥b≥c, a≥c≥b, b≥a≥c, b≥c≥a, c≥b≥a, c≥a≥b

Writing the details for each one is a bit long. I will give you an example for one case: suppose that c≥b≥a then max(a, max(b,c))=max(a,c)=c. On the other hand, max(max(a, b),c)=max(b,c)=c, hence the statement is true in this case.

e) Direct proof: write a=m/n and b=p/q, with m,q integers and n,q nonnegative integers. Then ab=mp/nq. mp is an integer, and nq is a non negative integer. Hence ab is rational.

f) Direct proof. By part c), √2/n is irrational for all natural numbers n. Furthermore, a is rational, then a+√2/n is irrational. Take n large enough in such a way that b-a>√2/n (b-a>0 so it is possible). Then a+√2/n is between a and b.

g) Direct proof: write m+n=2k and n+p=2j for some integers k,j. Add these equations to get m+2n+p=2k+2j. Then m+p=2k+2j-2n=2(k+j-n)=2s for some integer s=k+j-n. Thus m+p is even.

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3 years ago

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3 years ago

#1 #2 #3 & #4 thank you

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