Math Fun

-1. ABCD is a square with the length of 20 inches on each side. 4 quarter circumferences are tangented at the center of the square. How many square inches are in the total shaded area?




-2. A five digit number is represented by ABCDE. If we add the number 1 in front of ABCDE, then multiply by 3, the result number will be ABCDE appended by the number 1 (as shown below). What is this five digit number?

1ABCDE x 3 = ABCDE1

-3. In the equation shown, A, B, C, D, and E are 5 consecutive positive integers. What are they?

A^2 + B^2 + C^2 = D^2 + E^2

-4. John, Joe and Jack are having lunch together. John bought 3 pieces of corn bread. Joe bought 2 pieces and Jack bought nothing. John, Joe and Jack decided to share all the bread and had equal amounts of bread for lunch. After lunch, Jack paid 10 dollars to John and Joe. How much money should John and Joe get?

3. 10, 11, 12, 13, 14

3: b = a+1, c = a+2, d=a+3, e=a+4

a^2 + (a+1)^2 + (a+2)^2 = (a+3)^2 + (a+4)^2
a^2 + a^2+2a+1 + a^2+4a+4 = a^2+6a+9 + a^2+8a+14
Simplifies to:
a^2 - 8a - 20 = 0
Factoring:
(a - 10)(a + 2) = 0
a = 10 or a = -2 ; since positive, a = 10

So 10,11,12,13,14

Both correct! Dennis gets extra credit for showing his work.

Since Jack's cost is $10, then total cost = $30.
(assumes all 3 guys paid same amount)

Since total cost = $30 and 5 pieces purchased,then
each piece = $6: so John paid $18, Joe paid $12.

So John got $8: 18 - 8 = 10
and Joe got $2: 12 - 2 = 10

Thassa cute li'l problem, Sue...

Well done, Denis. Very clear.

1.

Area of a quarter circle = (pi/4)[sqrt(10^2 + 10^2)]^2
= (pi)(50) = 157.08 square inches

Total Area of the shaded region = 4(Area of a quarter circle) - (Area of the square)
= 4(157.08) - 20^2
= 628.32 - 400 = 228.32 square inches

Spot on, CCb!

I must have missed a "simple/quick" way;
I did it this rather "loooooong" way:
I substituted e = 7 since only 7 is possible
to produce a "1" after multiplied by 3:7*3=21.
Initial equation:
3(100000 + 10000a + 1000b + 100c + 10d + 7) =
100000a + 10000b + 1000c + 100d + 70 + 1

Doing the Jailhouse Rock and Peppermint Twist on that monster led to:
70000a + 7000b + 700c + 70d = 299950
Dividing by 70:
1000a + 100b + 10c + d = 4285
Since e = 7 : 42857
With 1 at start and multiplying by 3:
142857 * 3 = 428571

PHEWWWW!!

Here's an easier way, Denis...

1ABCDE x 3 = ABCDE1

(ABCDE+100000)3 = 10(ABCDE)+1
7(ABCDE) = 299999
ABCDE = 42857

Denis did the Jailhouse Rock and the Peppermint Twist while CCb waltzed through it. Well done, guys!

Bewteefull CCb!

Very elegant CCbomber