We have : \(P+Q>P-Q\ge2\), so P + Q is odd and P > Q

Then, one of them is odd and the another is even. Because 2 is the only even prime and is the smallest prime, Q = 2

If P - 2 = 3k + 1, then P + 2 = 3k + 3 is a composite

If P - 2 = 3k + 2, then P + 2 = 3k + 6 is a composite

So, P - 2 = 3k <=> k = 1 <=> P - 2 = 3 <=> P = 5

The answer is : P + Q + P + Q + P - Q = 3P + Q = 17

The distance Rob runs is \(\sqrt{2}\) times as long as the distance Bob runs and they arrive at their respective corners at the same time, so Rob's speed is \(8\sqrt{2}\approx11.3\) ( mi/h)

Let y = ax + b be the equation of the segment AB. We have :

\(\left\{{}\begin{matrix}-8.1a+b=4.9\\-7.6a+b=2.9\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}0,5a=-2\\b=4.9+8.1a\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}a=-4\\b=-27.5\end{matrix}\right.\)

So, the equation of AB is y = -4x - 27.5

The answer is : \(\dfrac{-1}{-4}=\dfrac{1}{4}\)

From the question, we deduce that the radius of the base of the cone is triple the cylinder's. Since the cone and the cylinder have the same height, the answer is :

\(\dfrac{1^2\pi}{\dfrac{1}{3}.3^2\pi}=\dfrac{1}{3}\)

\(\dfrac{x-y}{z-y}=-2\Rightarrow x-y=2\left(y-z\right)\Rightarrow x-y=2y+z-3z\)

\(\Rightarrow x-z=2y-3z+y\Rightarrow x-z=3\left(y-z\right)\Rightarrow\dfrac{x-z}{y-z}=3\)

The diagonal of the square is : 4 x 2 = 8 (in)

The edge of the square is : \(\dfrac{8}{\sqrt{2}}=4\sqrt{2}\) (in)

The area of the square is : \(\left(4\sqrt{2}\right)^2=32\) (in²)

The area of the large circle is : \(8^2\pi=64\pi\) (in²)

The area of the small circle is ; \(4^2\pi=16\pi\) (in²)

The answer is : \(64\pi-16\pi+32=48\pi+32\) (in²)

\(S_{ABC}=\dfrac{AB.AC}{2}=\dfrac{\left(10-2\right)\left(16-1\right)}{2}=\dfrac{8.15}{2}=60\)

Let a,b (ounces) be the quantity of 20% acid solution and 30% acid solution respectively which is needed to mix with each other. We have :

\(\left\{{}\begin{matrix}a+b=45\\20\%a+30\%b=45.24\%\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}20\%a+20\%b=9\\20\%a+30\%b=10.8\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a=45-b\\10\%b=1.8\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=27\\b=18\end{matrix}\right.\)

So, 27 ounces of 20% acid solution is mixed with 18 ounces of 30% acid solution to obtain 45 ounces of 24% acid solution 

(The answer is 27 ounces)

Let a, b be the length and the width of the rectangle. The new area of the rectangle is :

\(\left(a+75\%.a\right)\left(b+25\%.b\right)=\dfrac{7}{4}a.\dfrac{5}{4}b=\dfrac{35}{16}ab=218.75\%.ab\)

The answer is : 218.75% - 100% = 118.75%

Let x (ml) be the quantity of 10% acid solution that she has to add.

We have :

\(\dfrac{\left(50.50\%+10\%x\right).100}{50+x}\%=20\%\)

\(\Leftrightarrow\dfrac{2500+10x}{50+x}=20\Leftrightarrow2500+10x=1000+20x\)

\(\Leftrightarrow2500-1000=20x-10x\Leftrightarrow1500=10x\Leftrightarrow x=150\)

So, the answer is : \(50+150=200\) (ml)

Let a be the side of the square, then \(\dfrac{1}{3}a;\dfrac{2}{3}a\) are the lengths of the legs of each right triangle. The answer is :

\(\dfrac{\dfrac{\dfrac{1}{3}a.\dfrac{2}{3}a}{2}.4.100}{a^2}\%=44.\left(4\right)\%\approx44\%\)

The number of dimes and nickels in his pocket is : 18 - 3 = 15

The number of nickels in his pocket is : 15 : (2 + 1) = 5

The number of dimes in his pocket is : 5 x 2 = 10

The answer is : 10 x 10 + 5 x 5 = 125 (cents) (= $ 1.25)

The answer is : \(\left(\dfrac{-2+3}{2};\dfrac{4-3}{2}\right)=\left(0.5;0.5\right)\)

The diagonals of the regular hexagon as shown divide the hexagon into 6 equilateral triangles whose side is 2 in long.

The area of each equilateral is : \(\dfrac{2^2.\sqrt{3}}{4}=\sqrt{3}\) (cm²)

The area of the hexagon is : \(6\sqrt{3}\) (cm²)

The area of the circle is : \(2^2\pi=4\pi\) (cm²)

The answer is : \(4\pi-6\sqrt{3}\approx2.17\) (cm²)

Dao Trong Luan, I advise you to round the result at the end. (I can't explain it in English). Your result is luckily correct.

Let x be the distance form Sam's home to work in miles. To get to work on time, it takes Sam :

\(\dfrac{x}{30}-\dfrac{3}{10}=\dfrac{x}{45}+\dfrac{2}{15}\) (hours)

\(\Leftrightarrow\dfrac{x}{30}-\dfrac{x}{45}=\dfrac{2}{15}+\dfrac{3}{10}\Leftrightarrow\dfrac{x}{90}=\dfrac{13}{30}\Leftrightarrow x=39\)

So, the answer is 39 miles

\(\widehat{BCE}\) is the exterior angle of \(\Delta ABC\), so \(\widehat{CAB}=\widehat{BCE}-\widehat{B}=105^0-75^0=30^0\)

Since AB // CD, \(\widehat{BAD}+\widehat{D}=180^0\)(2 same-side interior angles)

\(\Rightarrow\widehat{BAD}=180^0-115^0=65^0\)

So, \(\widehat{DAC}=65^0-30^0=35^0\)

There are 26 letters in the English alphabet

There are 2 choices to choose the first letter

There are 24 choices to choose each of the last 3 letters

The answer is : 2 x 24³ = 27648

\(a=GCF\left(72;48\right)=24;b=GCF\left(108;144\right)=36\)

\(LCM\left(a;b\right)=72\)

Let 21a and 32a be the number of seventh-graders and eighth-graders at the dance \(\left(a\in Z^+\right)\), then :

The number of seventh-graders is : \(21a:30\%=70a\)

The number of eighth-graders is : \(32a:40\%=80a\)

The number of seventh-graders that didn't go to the dance is :

\(70a-21a=49a\)

The number of eighth-graders that didn't go to the dance is :

\(80a-32a=48a\)

The answer is : 49 : 48