25
The triangle given by 9, unknown, and 15 and the triangle given by 15, unknown, and x are similar triangles and therefore 9:15 = 15:x
X is 25 in this case
<span>There are several ways to do this problem. One of them is to realize that there's only 14 possible calendars for any year (a year may start on any of 7 days, and a year may be either a leap year, or a non-leap year. So 7*2 = 14 possible calendars for any year). And since there's only 14 different possibilities, it's quite easy to perform an exhaustive search to prove that any year has between 1 and 3 Friday the 13ths.
Let's first deal with non-leap years. Initially, I'll determine what day of the week the 13th falls for each month for a year that starts on Sunday.
Jan - Friday
Feb - Monday
Mar - Monday
Apr - Thursday
May - Saturday
Jun - Tuesday
Jul - Thursday
Aug - Sunday
Sep - Wednesday
Oct - Friday
Nov - Monday
Dec - Wednesday
Now let's count how many times for each weekday, the 13th falls there.
Sunday - 1
Monday - 3
Tuesday - 1
Wednesday - 2
Thursday - 2
Friday - 2
Saturday - 1
The key thing to notice is that there is that the number of times the 13th falls upon a weekday is always in the range of 1 to 3 days. And if the non-leap year were to start on any other day of the week, the numbers would simply rotate to the next days. The above list is generated for a year where January 1st falls on a Sunday. If instead it were to fall on a Monday, then the value above for Sunday would be the value for Monday. The value above for Monday would be the value for Tuesday, etc.
So we've handled all possible non-leap years. Let's do that again for a leap year starting on a Sunday. We get:
Jan - Friday
Feb - Monday
Mar - Tuesday
Apr - Friday
May - Sunday
Jun - Wednesday
Jul - Friday
Aug - Monday
Sep - Thursday
Oct - Saturday
Nov - Tuesday
Dec - Thursday
And the weekday totals are:
Sunday - 1
Monday - 2
Tuesday - 2
Wednesday - 1
Thursday - 2
Friday - 3
Saturday - 1
And once again, for every weekday, the total is between 1 and 3. And the same argument applies for every leap year.
And since we've covered both leap and non-leap years. Then we've demonstrated that for every possible year, Friday the 13th will happen at least once, and no more than 3 times.</span>
? can be any number except for 3 or -2 as it is already used in (3,4), (-2,-5)
8.454 is 3 tenths more than 8.154
3.625 is 2 and 2 tenths more than 1.425
Place values after the decimal point are tenths, hundredths, thousandths, ten thousandths...so on.
So 3 tenths looks like this 0.30 ⇒ 3 is in the tenths place. When you say 3 tenths more, it is a clue that the operation to be done is addition. thus,
8.154
<u>+ 0.30
</u> 8.454 * in addition or subtraction, the decimal points must be aligned to avoid confusion.
2 and 2 tenths is like this 2.20 ⇒ 1st 2 is in the ones or units place and the 2nd 2 is in the tenths place. Again it states more than so addition must be done.
1.425
<u>+ 2.20</u>
3.625
<u>
</u>
