(From 28.8:16.2, which is 28 4/5 : 16 1/5, which converted to an improper fraction is 144/5 : 81/5, and when reduced is 16:9).
POTW #21 is below:
Thursday, February 4, 2016
POTW #21 - Check it out!
Last week's POTW elicited some great responses. The numbers were a bit tricky (with decimals and improper fractions and mixed numbers), but the correct answer was in fact that the final ratio of pure apple juice to water in Steve's apple juice is 16:9.
(From 28.8:16.2, which is 28 4/5 : 16 1/5, which converted to an improper fraction is 144/5 : 81/5, and when reduced is 16:9).
POTW #21 is below:
(From 28.8:16.2, which is 28 4/5 : 16 1/5, which converted to an improper fraction is 144/5 : 81/5, and when reduced is 16:9).
POTW #21 is below:
Subscribe to:
Post Comments (Atom)
So here's how I solved it:
ReplyDeleteI first did what I could:
A = 2 + 5 + P + Q + 3 + 7 = 17 + P + Q
B: 8 * 2 = 16
3 * 2 = 6
9 * 2 = 18
9 * 2 = 18
6 * 2 = 12
B = ( 1 + 6 ) + 6 ( 1 + 8 ) + ( 1 + 8 ) + ( 1 + 2 ) =
B = 7 + 6 + 9 + 9 + 3 = 18 + 16 = 34
B = 34
C:
34 + 17 = 51 < without P and Q
Possible Cs: 60
P + Q = 9 < can't be 19 to get 70 because highest one digit number is 9 and not even 9 + 9 equals 19 it equals 18.
Possible P and Qs:
P:4 Q:5
P:3 Q:6
P:1 Q:8
P:0 Q:9
P:2 Q:7
Those are all the possible numbers P and Q could be.
I forgot to reverse the digits but luckily for this set of numbers it doesn't matter if you reverse it or not. You will get the same answer anyways.
ReplyDelete2853 P9Q9 367 = 763 9Q9P 3582
ReplyDelete7 + 3 + Q + P + 5 + 2 = 17 (without Q and P)
2 x 6 = 12
2 x 9 = 18
2 x 9 = 18
2 x 3 = 6
2 x 8 = 16
(1+2) + (1+8) + (1+8) + 6 + (1+6) = 34
17 + 34 = 51
60 - 51 = 9
P = 0
Q = 9
P = 1
Q = 8
P = 2
Q = 7
P = 3
Q = 6
P = 4
Q = 5
Hard copy
ReplyDeleteSo first, I calculated what I could.
ReplyDelete763 9Q9P 3582 (backwards). A = 7 + 3 + Q + P + 5 + 2. A = 17 (without Q and P). I did the rest of the problem without Q or P.
2 * 6 = 12
2 * 9 = 18
2 * 9 = 18
2 * 3 = 6
2 * 8 = 16
B = (1 + 2) + (1 + 8) + (1 + 8) + 6 + (1 + 6). B = 3 + 9 + 9 + 6 + 7. B = 34.
C = 17 + 34. C = 51. It doesn't end with a 0. The next possible C would be 60, so you need Q + P to equal 9. Therefore, the possible values for P and Q are:
P = 0 + Q = 9
P = 1 + Q = 8
P = 2 + Q = 7
P = 3 + Q = 6
P = 4 + Q = 5
At this point you cannot continue or P would be a bigger number than Q.
Hardcopy
ReplyDeleteanswers: P can be 0,1,2,3,4 and Q can be 5,6,7,8,9
To start off, I found the reverse...
ReplyDeleteReverse = 763 9Q9P 3582
A = 7 + 3 + Q + P + 5 + 2
All of the numbers above = 17 (excluding the variables Q and P)
Moving on to the next step... Multiplying the left over numbers by 2...
2 * 6 = 12
2 * 9 = 18
2 * 9 = 18
2 * 3 = 6
2 * 8 = 16
B = (1 + 2) + (1 + 8) + (1 + 8) + 6 + (1 + 6). B = 3 + 9 + 9 + 6 + 7...
So, B is equal to 34.
A(17) + B(34) = 51
The next number to end with a 0, is 60. The difference between 51 and 60 is 9.
Therefore, Q + P = 9
Since, Q has to be greater than P, here are the possible combinations...
P = 0
Q = 9
P = 1
Q = 8
P = 2
Q = 7
P = 3
Q = 6
P = 4
Q = 5
All in all, P may equal 0-4 inclusive and Q may equal 5-9 inclusive.
Provided credit card number: 2853 P9Q9 367.
ReplyDeleteReversal: 763 9Q9P 3582
Numbers in odd positions: 7, 3, Q, P, 5, 2
Expression: 7 + 3 + Q + P + 5 + 2
Total: 17 + P + Q
Doubled Numbers:
2 x 6 = 12
2 x 9 = 18(2)
2 x 3 = 6
2 x 8 = 16
Expression: (1+2) + (1+8) + (1+8) + 6 + (1+6)
Equation: 3 + 9 + 9 + 6 + 7 = 34
B = 34
A + B = 17 + 34 = 51
Closest number ending in zero: 60
60 - 51 = 9
Therefore, P + Q, must equal 9.
As P must be smaller than Q, the possible combinations could be:
P = 0, Q = 9
P = 1, Q = 8
P = 2, Q = 7
P = 3, Q = 6
P = 4, Q = 5
P can equal any number ranging from 0 to 4, and Q can equal any number from 5 to 9.
The total amount of possibilities are 5. P is a single digit number that is less than Q, so P is the smaller number in these ratios. 0:9, 1:8, 2:7, 3:6, 4:5.
ReplyDeleteExplanation:
The credit card number is 2853 P9Q9 367, the second step of the Luhn Algorithm is reversing the order of the digits, making it 763 9Q9P 3582.
The third step is to find A, add up all the digits in an odd column, so the numbers are: 7, 3, Q, P, 5, and 2. Adding it up would equal to 17+Q+P. I added all the numbers I could, and left the letters because nothing can be done about it right now.
Now, to calculate B, double the remaining numbers, 6,9,9,3, and 8 become 12,18,18,6, and 16. Now add up the digits of each doubled number, meaning I have to answer these expressions: (1+2), (1+8), (1+8), (0+6), and (1+6). This equals to, 3,9,9,6, and 7 in that order.
One of the last steps is to calculate C, adding A and B. A is actually (A-Q-P) because Q and P could literally not be found earlier. So (A-P-Q)+B=51+Q+P. Since it does not end with 0, it is an invalid number, so the closest number that ends with a 0 is 60. So P and Q has to add up to a number that when added to 51, equals 60, that means P and Q adds up to 9 because 60-51 is 9. So the possible solutions are 2 whole one digit numbers that add up to 9, the smaller number is P because P is less than Q. Those possible numbers were already said in the beginning of the answer.