Let CP of 1 article be x

CP of 80 articles=80x

Let SP of 1 article=y

SP of 100 articles=100y

80x=100y

Y=0.8x

SP of 80 articles = 64x

SP<CP. So, loss occurs

Loss percentage = 16x×100/80x

=20%

Let the whole goods be worth x

CP of 1/5^th of goods=x/5

SP of 1/5^th of goods=1.5×x/5

=0.3x

CP of Remaining goods =4x/5

SP of them=0.9×4x/5

=0.72x

Total SP=1.02x

CP=1x

So, profit=2%

Let the number of boys be x.

Number of girls=50-x

Average weight =Total weight/Number of people

SO, total weight of class=25.26×50 = 1263

25.5×x+25(50-x) = 1263

0.5x + 1250 = 1263

0.5x = 13

X = 26

Number of girls = 50 – 26 = 24

Ratio of boys to girls = 13 :  12

Let the two numbers be x and y

x/y = 11/7

Applying componendo dividendo,

(X+y)/(x-y) = (11+7)/(11-7)

=18/4

=9/2

Required ratio is the reciprocal of this one

=2 : 9

a²+b²+c² – 2a + 2b – 2c + 3 = 0

a² – 2a + 1 + b² + 2b + 1 + c² – 2c + 1 = 0

(a-1)² + (b + 1)² + (c – 1)² = 0

So, a = 1, b = -1, c = 1

a + 2b + c = 1 – 2 + 1 = 0

Area of equilateral triangle = (1/2) ×a × (√3a/2)= √3a²/4(where √3a/2 is the altitude of the triangle)

In an equilateral triangle, altitude = median

So, median= √3a/2

√3a²/4=441√3

Solving, we get a=(441×4)^0.5

A=21×2=42

So, median length=√3×42/2 = 21√3 units

Let quantity of in initial solution = x

Water =40 – x

55% of the container is full after the removal as said above

So, 45% of the solution has been removed.

45% of 40 = 18 l

0.6x+0.4(40 – x) = 18

0.2x + 16 = 18

0.2x = 2

X = 10

So, milk in initial solution = 10 l

Water in initial solution = 30 l

Ratio = 1 : 3

5 men can finish work in 40 days

So, 1 man can finish in 200 days

Similarly, 1 woman can finish in 160 days

2 men and 4 women work together.

Work done by them in one day=2/200+4/160=1/100+1/40

=2/200+5/200

=7/200

So, they can finish a tool in 200/7 days.

So, 7 tools can be finished in 200 days.

Put x=90°, √3 ×0+1=1

Opposite angles of a cyclic quadrilateral are supplementary.

A+C=180°

C=117°

B+D=180°

D=88°

C-D=29°