ํ้The problem was solved as shown in the diagram below

$\angle a + \angle b + \angle c + \angle d + \angle e = 180^\circ $

Attachment 7896

Let Emma is ab years old now

There for a+b = 5 ...(1)

7 year from now Emma is 10a+b+7 years

The reverse of the digits of her age is ba = 10b+a

So 10a+b+7 +2 = 10b+a

9a +9 = 9b

b - a = 1 (2)

From (1) and (2) a =2, b =3

Now Emma is 23 years old

At start, there are x cups of pure mango juice in 2x cups punch in the bowl.
(pure mango x cups and the other mixed x cups in the bowl)

When add 6 cups of pure mango juice, there are (x+6) cups of pure mango juice in 2x+6 cups of punch

$x+6 = \frac{2}{3}(2x+6)$

$x = 6$

So, at start there are 2x or 12 cups of punch in the bowl


12 cups Ans.

mean = 88
median = 89
mode = 93

the two lowest test score are x, y

Then x, y, 89, 93, 93

x+y = (5 x 88) - (59+93+93) = 165

The sum of the two lowest test scores is 165 Ans.

2011 is prime number.

So, the possibly integer of the lenght and the breadth of the rectangle are 2011 and 1.

The perimeter is 2(1+2011) = 4024 cm.

there are 18 paths as shown in the diagram below

Attachment 7898

18 paths Ans.

Attachment 7899

BC = 5 +12 = 17

AC = 12 + 5 = 17

$AB^2 = AC^2 + BC^2 = 17^2 +17^2$

$AB = 17\sqrt{2} $

$ \binom{6}{3} = 20 $

But A,B,C and D,E,F are arrange in linears, they can not formed any triangle.

So there are only 18 triangles can be formed

18 triangles Ans.

Let m, n denode the area as shown in the diagram

$m = \frac{1}{2}(m+n) \ \ \to \ n = m$ ....(*)

$m+m+n = 96$

$3m = 96$

$m = 32$


Triangle MNA'
$\frac{1}{2} x^2 = 32 $

$2x^2 = 128 = MN^2$

$MN = 8\sqrt{2} $cm

Attachment 7901

7 Ans.

Total surfaces = $4 \times \frac{\sqrt{3} }{4} \times 5^2 = 25\sqrt{3} \ cm^2 \ $Ans.

The six-digit number is even number

802802 is not valid

804804 is divisible by 12

The smallest last two digits are 04 Ans.

Attachment 7902

math + MTV = 17 + 80 = 97

But neither both = 4

Include both = 96

So overlapse 97 - 96 = 1

Between 2.30 p.m. and 4.00 p.m. = 90 minutes

She will miss $\frac{40}{90} = \frac{4}{9}$

Let x ounces to be added

Start at 29 ounces of Greg's mixture


Mothor's mixture $ = \frac{200}{7} \ $%

Greg's mixture $ = \frac{400}{29} \ $%


$ \frac{400}{29} \times 29 + 100x = \frac{200}{7} \times (29+x)$

$x = 6$

6 ounces Ans.