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# KCC's Quizzes AQQ231 about Birthdays and Cake Candles

Mary celebrates her birthday every year from the very first year.

The number of candles on her cake used reflects her age : 1 at age of 1, 2 at age of 2, 3 at age of 3 etc…

Up to now, she has blow out a total of 946 candles.

Questions:

1. How old is Mary today?
2. In an other family, there are 2 sisters, Ann and Helen. Helen is 5 years younger than Ann. They both follows the same scenario for the candles as Mary. The cumulated candled used by the 2 sisters is 1687. How old are Ann and Helen?

P.S. If you think there are still colleagues or friends (internal or external) you think who can be interested in those quizzes, please let let them know....

Kuo-Chang

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[edited by: emassa at 1:54 PM (GMT -4) on 29 Mar 2023]
• 1. Let's call Mary's age as n. Sum of n consecutive integers = n(n+1)/2 = 946. That gives us n=43. Mary is 43 years old today.

2. Let's call Helen's age as n and Ann's age as n+5. n(n+1)/2 + (n+5)(n+6)/2 = 1687. Solving that gives us, Helen's age as 38, and Ann's age as 43.

• 1. n(n+1)/2=946 --> n=43 (sum of first consecutive integers)

2. If m is the age of Ann, the age of Helen is m-5. So m(m+1)/2 + (m-5)(m-4)/2 = 1687. By solving the equation you get m= 43, so Anne is 43 and Helen is 38.

• You can also calculate this easily in Excel. Just make a column (column A) with numbers 1,2,3, etc. In the column next to it calculate the number of candles. The first cell (B1) gets: SUM(\$A\$1:A1) and drag this down. You will see that at age 43 946 candles have been blown out. Now copy both columns with an offset of 5 downwards and you will get the results for a second person 5 years younger. Now make a new column with the sum of the candles of each sister and you will get the total number of candles blown out by both sisters. You will easily find that for 1687 sister 1 is 43 and sister 2 is 38.

• 1 + 2 + 3 ... + n = n(n+1)/2  is the series.

1.   Mary has blown 946 candles.
n(n+1)/2 = 946
solving for n we get  n=86/2 = 43
Mary now is 43 years  old.

2.  If Ann is A years and Helen is A-5 years of age.
A(A+1)/2  +  [A-5](([A-5]+1)/2 = 1687

solving for A , we get.  A= 43 so H =43-5=38

Ann is 43 years of age  and Helen is 38 years of age.

• Mary and Ann age will be 43 and Helen will be of 48.
As no of candles has blow is 1+2+3+4.......n that is n(n+1) /2 and after calculation it will be 43.
For Ann and Helen if their age is n and n+5 then no of candles blow will be (1+2+3+4..... n) +(1+2+3+4..... n+5) so after calculation value of n will be 38 so n+5= 43

• 1. Mary is 43 years old.
2. Ann is 43 years old and Helen is 38 years old.
The result can be easily found using Excel:

 Age Candles Mary/Ann Helen Mary /Ann Helen Ann+Helen 1 1 2 3 3 6 4 10 5 15 6 1 21 1 22 7 2 28 3 31 ... ... ... ... ... 39 34 780 595 1375 40 35 820 630 1450 41 36 861 666 1527 42 37 903 703 1606 43 38 946 741 1687 44 39 990 780 1770 45 40 1035 820 1855 46 41 1081 861 1942