Assume that \(\widehat{B}=\widehat{D}=30^0\). Draw \(AH\perp BC\). Let E be the point symmetric to A with respect to H.

\(\Delta AHB\) right at H has : \(\widehat{A_1}=90^0-30^0=60^0\)

The median BH of \(\Delta ABE\) is also the altitude, so \(\Delta ABH\) isosceles at B. Moreover, \(\widehat{A_1}=60^0\). Hence, \(\Delta ABH\) is equilateral

\(\Rightarrow AH=\dfrac{AE}{2}=\dfrac{AB}{2}=\dfrac{4}{2}=2\) (cm)

\(\Rightarrow S_{ABCD}=AH.BC=2.4=8\) (cm²)

15 yards = 540 inches

The answer is : 540 : 20 = 27 (inches)

We have: 15 yd = 540 inches

The answer is: 540 : 20 = 27 (inches)

Let q and d be the number of quarter and dimes in the box respectively, then the total value of the money in the box is 25q + 10d cents. We have :

\(\left(1+10\%\right)\times25q+10d=\left(1+7.5\%\right)\left(25q+10d\right)\)

\(\Leftrightarrow110\%\times25q+10d=107.5\%\left(25q+10d\right)\)

\(\Leftrightarrow110\%\times25q-107.5\%\times25q=107.5\%\times10d-10d\)

\(\Leftrightarrow2.5\%\times25q=7.5\%\times10d\)

\(\Leftrightarrow\dfrac{q}{d}=\dfrac{7.5\%\times10}{2.5\%\times25}=\dfrac{3\times2}{5}=\dfrac{6}{5}\)

So, the answer is \(\dfrac{6}{5}\)

The exact number of weeks of Martian year each year is: \(668:7=95\dfrac{3}{7}\left(weeks\right)\)

So the fraction of the years are leap years is: \(95\dfrac{3}{7}-95=\dfrac{3}{7}\) 

1 + 4 + 16 + ... + 1024 

= 2⁰ + 2² + 2⁴ + ... + 2¹⁰

= 2⁰ + 2² + 2⁴ + 2⁶ + 2⁸ + 2¹⁰

= 1 + 4 + 16 + 64 + 256 + 1024 

= 1365

1 + 4 + 16 + ... + 1024 

= 20 + 22 + 24 + ... + 210

= 20 + 22 + 24 + 26 + 28 + 210

= 1 + 4 + 16 + 64 + 256 + 1024 

= 1365

Call two integers is a and b ( \(a,b\in Z\))

Because a and b have the same role, suppose a > b

So we have:

a - b = -18

a + b = 2 

=> a = ( -18 + 2) / 2 = -8

=> b = -8 -2 = -10

So their product is: -8 x (-10) = 80 

No, b just be: 2 - ( -8 ) = 10 and the product is -80, Oh :( 

The time for the express train: \(\dfrac{18}{72}=\dfrac{1}{4}\left(h\right)=15minutes\)
The time for the local train to travel (without the stops): \(\dfrac{18}{54}=\dfrac{1}{3}\left(h\right)=20minutes\)
And now counting the stops: \(20+6\cdot1.5=29\left(minutes\right)\)
So obviously, the local train takes \(29-15=14\left(minutes\right)\)more than the express train

We can also put dot (.): 6.7