Math Revolution Approach (PS)

It’s well known that a company's profit=10,000x-50x^2, where x is the number of workers. If the profit of the company is the greatest, what is the value of x?

A. 5
B. 10
C. 50
D. 80
E. 100

==> From \(y=f(x)=ax^2+bx+c\), if \(x=\frac{-b}{2a}\), f(x) becomes either maximum or minimum. Thus, profit=\(-50x^2+10,000x\), and in order for the profit to become the greatest, you get \(x=\frac{-10,000}{2(-50)}=100\).

The answer is E.
Answer: E

What is the range of 25 consecutive even numbers?

A. 44
B. 46
C. 48
D. 50
E. 52


==> The number of consecutive even numbers=(The last number-the first number)/2 +1=range/2+1, so from range/2+1=25, range/2=25-1=24, you get range=2(24)=48.

Therefore, the answer is C.
Answer: C

If 1 male, 2 females, and 1 child are to be selected at random from 8 males, 10 females, and 8 children, respectively, how many such cases are possible?

A. 980
B. 1,440
C. 1,880
D. 2,480
E. 2,880

==> From (8C1)(10C2)(8C1)=(8)(10*9/2!)(8)=2,880, the answer is E.
Answer: E

When \(5^{11}\) and \(2^n7^2\) have the same number of factors, what is the value of n?

A. 2
B. 3
C. 4
D. 5
E. 6

==> From 11+1=(n+1)(2+1), you get 12=3(n+1), and n-3.

Therefore, the answer is B.
Answer: B

If \(x^{-2}+x^{-1}=0\), \(x=\)?

A. -1
B. 0
C. 1
D. 0, 1
E. -1, 0

==> If you multiply x^2 on both sides, you get 1+x=0, x=-1.

The answer is A.
Answer: A

If the distances between two consecutive points are the same, what is the value of A in terms of x and y?

A. 3y-2x
B. 3x-2y
C. 2x-3y
D. 2y-3x
E. 3y+2x

==> The distance between 2 consecutive points=d, hence d=y-x and A=y+2d=y+2(y-x)=3y-2x.

The answer is A.
Answer: A

If 3 juniors from 8 juniors are selected at random to make a committee, how many cases are possible?

A. 8
B. 28
C. 35
D. 42
E. 56

==> In general, you solve probability questions using nCr, a combination. In other words, from 8C3=56, the answer is E.
Answer: E

If xy>0, which of the following must be positive?

A. \(x\)
B. \(y\)
C. \(x^2y\)
D. \(y^2x\)
E. \(x^3y\)

==> Squares are always positive, so the inequality sign doesn’t change even if you multiply or divide. Thus, from xy>0 and \((x^2)xy>(x^2)0\), you get \(x^3y>0\).

The answer is E.
Answer: E

If \(a>b>0\), \(a^2+b^2=6ab\), \(\frac{(a+b)}{(a-b)}=\)?

A. 1

B. \(\sqrt{2}\)

C. \(\sqrt{3}\)

D. \(\sqrt{5}\)

E. 2

==> From \((a+b)^2=a^2+b^2+2ab=6ab+2ab=8ab\), you get \(a+b=\sqrt{8ab}\)
And from \((a-b)^2=a^2+b^2-2ab=6ab-2ab=4ab\), you get \(a-b=\sqrt{4ab}\) and
\(\frac{(a+b)}{(a-b)}=\sqrt{8ab}/\sqrt{4ab}=2\).

Therefore, the answer is B.
Answer: B

From the above, you get \(h:(s-r)/2=√3:1\) and \(h=√3(s-r)/2\). The answer is A.
Answer: A

1-0.000001=?

A. 1.001*0.999
B. 1.0001*0.9999
C. 1.001*0.9999
D. 1.0001*0.999
E. 1.01*0.99

==> You get \(a^2-b^2=(a+b)(a-b)\), and thus \(1-0.000001=1^2-0.001^2=(1+0.001)(1-0.001)=1.001*0.999\).

The answer is A.
Answer: A

When a certain coin is flipped, the probability that the coin will land on head or tail is 1/2 each. If the coin is flipped 4 times, what is the probability that it will land on tail at least twice on 4 flips?

A. \(\frac{3}{8}\)
B. \(\frac{1}{16}\)
C. \(\frac{1}{2}\)
D. \(\frac{5}{8}\)
E. \(\frac{11}{16}\)

==> In general, you solve probability questions using nCr combination. In other words, from TTHH, there are (4!/2!/2!)=6 possibilities, so you get (1/2)(1/2)(1/2)(1/2)*6=3/8. From TTTH, there are (4!/3!)=4 possibilities, so you get (1/2)(1/2)(1/2)(1/2)*4=1/4. From TTTT, there is only 1 possibility, so you get (1/2)(1/2)(1/2)(1/2)=1/16 and (3/8)+(1/4)+(1/16)=(6+4+1)/16=11/16.

The answer is E.
Answer: E

Which of the following is equal to \(\frac{(n+2)!}{n!}\)?

A. \(n(n+1)\)
B. \(1+(\frac{2}{n})\)
C. \(n^2+3n+2\)
D. \(n^2+2n\)
E. \(n+2\)

==> Since (n+2)!=(n+2)(n+1)n!, from (n+2)!/n!=(n+2)(n+1)n!/n!=(n+2)(n+1)
=\(n^2+3n+2\), the answer is C.
Answer: C

When a/b=1.2, (a-b)/(a+b)=?

A. \(\frac{1}{15}\)
B. \(\frac{1}{11}\)
C. \(\frac{1}{9}\)
D. \(\frac{3}{8}\)
E. \(\frac{1}{3}\)

==> From a/b=1.2=6/5, if you substitute a=6 and b=5, you get (a-b)/(a+b)=(6-5)/(6+5)=1/11.

Therefore, the answer is B.
Answer: B

If two times x is 5 greater than three times y, what is the value of y, in terms of x?

A. y=2x-5
B. y=6x-5
C. y=(x/2)-5
D. y=(x/3)-5
E. y=(2x-5)/3

==> According to the ivy approach, is:”=”, greater than:”+”, hence you get 2x=5+3y. Thus, from 3y=5-2x, y=(2x-5)/3, the answer is E.

Answer: E

If the sum of the annual salary of n persons is $x and the monthly salary per person is $y, what is the value of n in terms of x and y?

A. \($\frac{x}{12y}\)
B. \($\frac{12x}{y}\)
C. \($12xy\)
D. \($\frac{12y}{x}\)
E. \($\frac{xy}{12}\)

==> If the annual salary per person: s, you get ns=x, s=12y. If you substitue this, you get n(12y)=x, \(n=\frac{x}{12y}\).

Therefore, the answer is A.
Answer: A

\((5+\frac{1}{5})^2-(5-\frac{1}{5})^2\)=?

A. 2
B. 3
C. 4
D. 5
E. 9

==> Since \(a^2-b^2=(a-b)(a+b)\), you get \((5+\frac{1}{5})^2-(5-\frac{1}{5})^2=[(5+\frac{1}{5})-(5-\frac{1}{5})][(5+\frac{1}{5})+(5-\frac{1}{5})]=(\frac{2}{5})(10)=4\).

The answer is C.
Answer: C