Missing Middle Digits

Today's question isn't long and isn't hard, but it makes you think a little. Tomorrow I'll have a nicer question, with an ellipse. Until then, you'll have to work your mind on digits.

In a two digit number, the units digit is bigger than the tens digit by 4. First, you add 1 between the two digits and call it x. Then, you add 5 to the right of the units digit of the original number and call it y. x + y = 692. What is the two digit number?

Let's take this step by step:
First, the original number is _ _.

x is _ 1 _
y is _ _ 5
x + y is 692

Now we can find the values of the units digit, since we know that the units digit + 5 gives 12 (it can't give 2, obviously). This makes the units digit 7.

x is _ 1 7
y is _ 7 5

As you can see, adding the two numbers will leave the hundreds digits untouched. As a result, the two hundreds digits, which are equal, must add up to 6. This makes the hundreds digit 3.

The two numbers are 317 and 375, and the original number is 37.

Short, nice, easy. Tomorrow will be a little harder, I promise.
Nadav

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Parametric Parabolas

It hasn't even been a week since we dealt with parabolas, and they come again, this time with actual numbers. The question itself isn't hard, but once again, some thinking is required.

A parabola goes through the points A(17, 3) B(34, z) C(50, -30). B is the maximum point of the parabola. Find point B and the parabola function in standard form.

As you know, a parabola can be written in two ways. One is standard form, with the a, b, and c coefficients, and one is the vertex form. To solve this question, we'll use the vertex form. This form says that the equation a parabola that has a vertex (h, k) is:
y = a(x - h)² + k

We know h and k, so let's plug them in:
y = a(x - 34)² + z

We have two other points, so let's use them:
3 = a(17 - 34)² + z
2 = a(50 - 34)² + z

3 = 289a + z
-30 = 256a + z

Subtract the second equation from the first:
33 = 33a
a = 1
3 = 289a + z
3 = 289 + z
z = -286

The function of the parabola is:
y = (x - 34)² - 286

Open parenthese and combine like terms to switch to standard form:
y = x² - 68x + 1156 - 286
y = x² - 68x + 870

Nice, isn't it?
Nadav

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Lawn Mowing Siblings

After yesterday's easy question I found something better. It doesn't deal with differential equations or have a fancy name, but it just deals with some thinking and setting up equations correctly.

A brother and a sister are mowing the lawn on their backyard. It takes the brother 15 minutes longer to mow the lawn than it takes his sister. Together they mow the lawn in 56 minutes. How long does it take each of them to mow the lawn?

To solve this, we need something, and we need it fast: variables. x sounds like a great variable. Let x be the time it takes the sister to mow the lawn.

Now, since it takes her x minutes to mow the lawn, she does 1/x of the lawn every minute. Her brother takes x + 15 minutes to mow the lawn, so he mows 1/(x + 15) every minute.

Since together they mow the lawn in 56 minutes, 56 times the mowing rate of the brother plus 56 times the mowing rate of the sister should give 1, which is 100% of the lawn. We finally have an equation:
56/x + 56/(x + 15) = 1

This equation is easily solved:
56(x + 15) + 56x = x(x + 15)
56x + 840 + 56x = x² + 15x
x² - 97x - 840 = 0

Now we can solve it by factoring or by the quadratic formula. Since the quadratic formula is boring, let's try to factor. The numbers here are big, so you can start from something like 10 and see how the difference between the factors grow. Finally, you will reach the correct factoring:
(x - 105)(x + 8) = 0

Since the sister can't mow the lawn in -8 minutes, it must be 105 minutes, or 1 hour and 45 minutes. This means the brother needs 120 minutes, or two whole hours to mow this lawn. It must be a big lawn.

To test that, check: 56/105 + 56/120 = 8/15 + 7/15 = 1

I really doubt this answer is correct. If it takes the sister alone almost two hours to mow the lawn, with her brother it will take them about 3-4 hours, and that's before they even start mowing.

There is a new proof added at super math tips, go check it out.

Have a great weekend,
Nadav

nadavs

Chairs, Tables, and Ratios

Today I have a very easy question, but once again, two solutions. They are different in their approach, and anyone can choose what works best for him.

The ratio of the price of a table to the price of a chair is 5:3. A table costs $500 more than a chair. How much does each cost?

First, here is one answer (not mine):

Let x be the price of chairs.
That means x + 500 is the price of tables. From the ratio, we can solve the following equation and get each price:
5/3 = (x + 500) / x
5x = 3x + 1500
2x = 1500
x = 750

That makes tables cost $1250. And indeed, when you figure out the ratio, 1250:750 = 25:15 = 5:3.

However, there is another way to answer this question (my way):
Since the ratio is 5:3, you can also call it 5x:3x, where 5x is the price of a table and 3x is the price of a chair. Now we can solve a simpler equation:
5x = 3x + 500
2x = 500
x = 250

Notice that x is not a price of a chair or a table, it's just the unit that gives the price of each. That means the price of a table is $250 * 5 = $1250 and chairs cost $250 * 3 = $750.

This is another way of solving simple math questions using creative thinking.

Hope you liked it,
Nadav

nadavs

Two Way Limits

It does not happen often, but when it does, it's simply beautiful: a question with two completely different yet right answers. I saw such question on Yahoo Answers today. Although I didn't answer it, the two different answers were so interesting and different in their approach, I had to bring it here.

The question is quite simple for people who know limits, but the two different answers are just amazing.

Find the limit:
lim_{x -> 4} (3 - (5 + x)^1/2) / (1 - (5 - x)^1/2)

As you can clearly see, plugging x = 4 gives a zero in the denominator, which is why the question asks for a limit. You can plug in numbers and get a rough estimation, but we want a definite answer.

The first method is using L'Hopital's rule. This rule says that in order to find lim_{x -> a} f(x)/g(x), you can also find lim_{x -> a} f'(x)/g'(x). All we need now is to differentiate the numerator and denominator and see what we get:
lim_{x -> 4} (-1/2 * (5 + x)^-1/2) / (1/2 * (5 - x)^-1/2)

The halves cancel, and by using the law that says aⁿ / bⁿ = (a/b)ⁿ, we can show that:
lim_{x -> 4} -((5 + x) / (5 - x))^-1/2

Now plug in 4 and you will get -((5 + 4)/(5-4))^-1/2
Which is -9^-1/2 = -1/3

This is one very good solution, and most people who know calculus would choose that. However, there is another solution to this problem. To do that, multiply the numerator and denominator by the conjugate of the denominator:
lim_{x -> 4} (3 - (5 + x)^1/2) / (1 - (5 - x)^1/2) * (1 + (5 - x)^1/2) / (1 + (5 - x)^1/2)

Using the law that says (a + b)(a - b) = a² - b² and the distributive property, we can conclude that:
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (1 - (5 - x))
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (x - 4)

Now multiply both parts of the fraction by the conjugate of the original numerator:
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (x - 4) * (3 + (5 + x)^1/2) / (3 + (5 + x)^1/2)
lim_{x -> 4} (9 - (5 + x)) * (1 + (5 - x)^1/2) / ((x - 4) * (3 + (5 + x)^1/2))
lim_{x -> 4} (4 - x) * (1 + (5 - x)^1/2) / ((x - 4) * (3 + (5 + x)^1/2) )

That is really nice. (4 - x) / (x - 4) = -1, so we can cancel out two terms and turn them into a nice little minus sign:
lim_{x -> 4} -(1 + (5 - x)^1/2) / (3 + (5 + x)^1/2)

Now we can safely plug 4 for x and we get:
-(1 + (5 - 4)^1/2) / (3 + (5 + 4)^1/2)
-(1 + 1^1/2) / (3 + 9^1/2)
-(1 + 1) / (3 + 3)
-1/3

Again, we get -1/3 as an answer.

Math can be very easy if you think outside the box. Try doing it as often as you can.
Nadav

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