A fair number cube is rolled. What are the odds in favor of rolling a 6

We will set a variable, d, to represent the day of the week that January starts on. For instance, if it started on Monday, d + 1 would be Tuesday, d + 2 would be Wednesday, etc. up to d + 6 to represent the last day of the week (in our example, Sunday). The next week would start over at d, and the month would continue. For non-leap years:
If January starts on d, February will start 31 days later. Following our pattern above, this will put it at d + 3 (28 days would be back at d; 29 would be d+1, 30 would be d+2, and 31 is at d+3). In a non-leap year, February has 28 days, so March will start at d+3 also. April will start 31 days after that, so that puts us at d+3+3=d+6. May starts 30 days after that, so d+6+2=d+8. However, since we only have 7 days in the week, this is actually back to d+1. June starts 31 days after that, so d+1+3=d+4. July starts 30 days after that, so d+4+2=d+6. August starts 31 days after that, so d+6+3=d+9, but again, we only have 7 days in our week, so this is d+2. September starts 31 days after that, so d+2+3=d+5. October starts 30 days after that, so d+5+2=d+7, which is just d. November starts 31 days after that, so d+3. December starts 30 days after that, so d+5. Remember that each one of these expressions represents a day of the week. Going back through the list (in numerical order, and listing duplicates), we have d, d, d+1, d+2, d+3, d+3, d+3, d+4, d+5, d+5, d+6 and d+6. This means we have every day of the week covered, therefore there is a Friday the 13th at least once a year (if every day of the week can begin a month, then every day of the week can happy for any number in the month).
For leap years, every month after February would change, so we have (in the order of the months) d, d+3, d+4, d, d+2, d+5, d, d+3, d+6, d+1, d+4, and d+6. We still have every day of the week represented, so there is a Friday the 13th at least once. Additionally, none of the days of the week appear more than 3 times, so there is never a year with more than 3 Friday the 13ths.

5 0

3 years ago

The frosty ice cream shop sells sundaes for $2 and banana splits for $3. On a hot day the shoot sold 8 more sundaes than banana

sashaice [31]

Let’s designate the number of banana splits sold as x. That means the number of sundaes sold is x + 8.

Now if we put this into an equation, also keeping in mind their costs, we get:
3(x) + 2(x + 8) = 156

Now to solve for x:
3x + 2(x + 8) = 156
3x + 2x + 16 = 156
5x = 140
x = 28

That means you’ve sold 28 banana splits and 36 sundaes! :)

7 0

3 years ago

Pentagon CDEFG is shown on the coordinate plane. Pentagon CDEFG is translated 7 units up and 5 units left, resulting in C'D'E'F'

Sati [7]

Answer:

The length of line segment FG is equal to the length of F'G'

The perimeter of pentagon CDEFG is equal to the perimeter of pentagon C'D'E'F'G.

Step-by-step explanation:

Given

CDEFG and C'D'E'F'G

Translation: 7 units up and 5 units left

Solving (a): Segment FG and F'G'

When a shape is translated, the resulting image will have the same lengths as the original image (i.e, translation does not change measurements)

Hence:

$FG = F'G'$

Solving (b): Perimeter CDEFG and C'D'EF'G'

In (a), we established that lengths do not change during translation;

Hence:

The perimeter of the CDEFG and C'D'EF'G' will remain the same

3 0

3 years ago