Problem of the Day: 3/10/13

Carlo needs two cans of paint to coat every three walls in his house. How many cans does he need to paint two coats on each of his eight four-walled rooms?

Solution to yesterday’s problem:

Let’s find how many different ways we can subdivide one beat. First of all, we can have one quarter note. Then we can subdivide that into two eighth notes. That’s two different ways so far. Next, we can subdivide either of those two eighth notes into sixteenth notes, which adds two more ways. Finally we can subdivide both eighth notes into sixteenth notes, adding one more way. So there are five possible ways to subdivide a beat in this fashion. Using the blanks method to find how many possible ways there are for four beats:

__5__ __5__ __5__ __5__

Multiplying these numbers together gives a total of 625 ways.

Problem of the Day: 3/5/13

For every red flower in a garden, there are three blue flowers. For every two blue flowers, there are five purple flowers. For every flower of these colors, there are six yellow flowers. If there are fifteen purple flowers, how many flowers are there in all?

Solution to yesterday’s problem:

To solve this problem we need to know that for any triangle, the sum of two sides must be greater than the length of the other side. So if we call x the length of the missing side, we can write these inequalities: $10+16>x$ or $x16$ or $x>6$

If we know that x is between 6 and 26, and not including 6 and 26, we can add all these values together to find the answer. Here’s a quick way to do 7 + 8 + 9 + … + 24 + 25:

Notice that the sum of the first and last terms, 32, is the same as the sum of the second and second-last terms. Also notice that we are adding 19 terms overall (26-6 and minus one for including 26). So to get the sum of the terms, we multiply 32 by half of the number of terms: 32 x 9.5 = 304.

Problem of the Day: 2/28/13

A bathtub can fill to the top in 30 minutes. The drain can empty the tub in 45 minutes. If the tap is on and the drain open, how long will the bathtub take to fill?

Solution to yesterday’s problem:

Let’s list the factors of 60 to consider some of the possible integers.

1,60
2,30
3,20
4,15
5,12
6,10

We can say that two of the three integers must have a product of one of the factors because they are part of the product to make 60. We might reason that if the factors are closer together, then the sum will be smaller. So for (6,10), we can form (6,5,2) and (3,2,10) as two possible sets of factors. The sum of (6,5,2) is less: only 13. However, we’d better check some of the other factors because some of the larger numbers may have small factors as well. After a little trial and error, we find that (3,5,4) has a sum of 12. So the least possible sum of the factors is 12.

Problem of the Day: 2/20/13

Kelly walks her dog around her block in 20 minutes. She knows that 3 of these blocks is half a mile. In how many minutes can she walk her dog around her friend’s block, two of which constitute a mile?

Solution to yesterday’s problem:

Let’s represent the problem with two linear equations. The first one, the supply equation/curve, has a slope of 100,000 because that’s how many is added for each $1 increase. So we can write that equation as $y=100000x$ The demand equation has a y-intercept of 1,000,000 because that’s how many people would get the product at price zero. The slope is -300,000 because 300,000 less people would get it for each$1 increase. So the demand equation is $y=-300000x+1000000$

The point where supply and demand are equal is the intersection of these two lines. We can find this point by setting the two equations equal to each other: $100000x=-300000x+1000000$

Solving this equation, $400000x=1000000$

(Dividing by 100,000 to make the numbers a bit easier) $4x=10$ $x=2.5$

So Company X should theoretically price Product X at exactly \$2.50.

Problem of the Day: 2/12/13

A 6-foot-tall man stands near a tall tree. His shadow is 8 feet long, and the tree’s shadow is 48 feet long. How tall is the tree?

Solution to yesterday’s problem:

First let’s substitute in the expression in parentheses. 13 ☆ 12 = $\sqrt{13^2-12^2}$, which you may recognize as two components of a Pythagorean triple. The expression simplifies to 5. Then, 5 ☆ 4 = $\sqrt{5^2-4^2}$, which you may recognize as two components of a different Pythagorean triple. What a coincidence! The final expression simplifies to give an answer of 3.

Note: I apologize for changing these math problems when I give the solution, but you can imagine how creating a unique, creative, and solvable math problem in the ten minutes before school can lead to interesting mishaps! Anyway, I hope these problems are useful for you.

Problem of the Day: 1/16/13

The ratio of Lauren’s current age to her age in two years is $\frac{5}{6}$. How old is she now?

Solution to yesterday’s problem:

Let’s use my favorite blanks method to solve this problem. Since the number is four digits long, we can write it like this:

____ ____ ____ ____

Multiples of 11 have a special characteristic: the sum of the odd-numbered digits equals the sum of the even-numbered digits. For instance, 286 is a multiple of 11 because 2+6=8. So in this problem, the first and third digits must add up to the same as the second and fourth digits. Also, the sum of all the digits must be 10. We can say that the (sum of the first and third digits) x 2 = 10. Dividing by 2, the sum of the first and third (and that of the second and fourth) digits must be 5. What are the possible digits that could add up to 5?

0, 5; 1, 4; 2, 3

Now let’s fill in the blanks. For the first blank, the number could be any of the numbers above except zero (it would then be three digits). That makes 5 possible digits:

__5__ ____ ____ ____

For the second blank, the number could be any of the numbers above, or 6 possible digits:

__5__ __6__ ____ ____

The digit in the third blank must make 5 when added to the first digit, so it can ONLY be the other number in the pair from which the first digit came. Same goes for the fourth blank.

__5__ __6__ __1__ __1__

Multiplying these numbers together, we find that there are 30 numbers that satisfy these conditions.

Problem of the Day: 1/5/13

The Jones family wants to paint one of the bedrooms in the Jones house. The bedroom has 240 square feet of wall to be painted, after excluding doors and windows. Daddy Jones can paint 4 square feet in one minute, Mommy Jones can paint 5 square feet in two minutes, and Baby Jones can paint 3 square feet in two minutes. If all of them paint continuously together, how long will it take the Jones family to lay three coats of paint on the bedroom walls?

Solution to yesterday’s problem:
We know there are 50 coins, and 16 of them are green on both sides. That means that 34 of the coins are green on at most one side. We also know that 6 of the coins are not green on either side, which leaves 28 coins green on exactly one side. Since it is mentioned that the number with green heads and the number with green tails are equal, we know that 14 coins have a green head, and 14 have a green tail. The probability of picking a coin with green tails is $\frac{14}{50}$, or after reducing, $\frac{7}{25}$.