Since \(52%\) equals the fraction \(\frac{52}{100},\) which equals \(\frac{4 \times 13}{4 \times 25},\) which reduces to \(\frac{13}{25},\) this means that the other piece of the vertical line under the yellow arrow is \(1 - \frac{13}{25} = \frac{12}{25} \) of the whole vertical side.

That's how we got this picture here:

Do you see that Angle Bisector Theorem here tells us that the ratio of the \( \textcolor{green}{\text{green}}\) and \(\textcolor{blue}{\text{blue}}\) sides is \( 13 : 12\) ?

We can't say for sure that the \( \textcolor{green}{\text{green}}\) side is \(13\) long and the \(\textcolor{blue}{\text{blue}}\) side is \(12\) long, but we just know that the length of the \( \textcolor{green}{\text{green}}\) side divided by the length of the \(\textcolor{blue}{\text{blue}}\) side is equal to \(\frac{13}{12},\) so we say that the \( \textcolor{green}{\text{green}}\) side has length \(13\) units of some amount, which is \(13a,\) and the \(\textcolor{blue}{\text{blue}}\) side has length \(12\) units of some amount, which is \(12a.\)

]]>Ratios are good way of comparing one number with another one.

Let's start with two numbers. The bigger number represents an older sister and the smaller number represents a little brother . When mother makes cookies, older sister eats \(4\) cookies but the little brother eats only \(2\) :cookies. How would you compare the amount of food that they each eat?

Would you say that the older sister eats \(2\) more cookies than the little brother ?

Should the older sister always eat \(2\) more of whatever food there is, compared with her little brother ?

But then what if mother serves rice -- should the older sister have exactly \(1\) more grain of rice compared with the little brother ?

No, that wouldn't make sense! If the older sister got just \(1\) more grain of rice compared with the little brother , then she would be eating almost the same amount of rice as the little brother, , because the little brother might eat \(1,000\) grains of rice (a cup of cooked rice has about \(3,000\) grains of rice in it. Then the older sister would only get \(1,001\) grains of rice.

What if the little brother eats \(\frac{1}{2}\) of a hamburger ? Would that mean that the older sister should eat \(2 + \frac{1}{2} = 2 \frac{1}{2}\) hamburgers?

Or, suppose the little brother can eat \(\frac{1}{12}\) of a birthday cake Does that mean that the older sister can eat \(2 \frac{1}{12}\) birthday cakes ?

This is why we use ratios, which describe how big numbers are compared with each other using multiplication and division rather than addition and subtraction.

The correct thing is to say that in order to get the older sister's serving, you take \(\frac{1}{2}\) of the little brother's amount and add it to what the little brother eats. So if the little brother eats \(1\) bowl of rice , the older sister eats \(1 + \frac{1}{2}\) bowls of rice . If the little brother eats \(\frac{1}{2}\) of a hamburger , the older sister eats \(\frac{1}{2} + \frac{1}{4} = \frac{3}{4}\) of a hamburger . If the little brothe eats \(\frac{1}{12}\) of a birthday cake , the older sister eats \(\frac{1}{12} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}\) of a birthday cake .

Another way to compare the older sister and little brother is to say that the older sister eats \(1.5\) times as much as the little brother. This is because she eats one-and-a-half of the little brother's serving.

This is why we use ratios!

We can say that the ratio between the little brother's serving and the older sisters's serving is \(1 \text{ to } 1.5,\) or \(1:1.5.\) If you multiply both numbers by \(2,\) you get \(2:3,\) which is the same ratio. As you get more experience with doing math problems, you might prefer using ratios to compare numbers, because it's easier to calculate things.

For your other question, Prof. Loh isn't actually switching around the subtraction (like you might switch around the terms in \(3 + 4 = 4 + 3),\) but he is simplifying

$$ \frac{2 \sqrt{3}}{\sqrt{3} + 2} $$

which is the same thing as

$$ \frac{2 \sqrt{3}}{2 + \sqrt{3} }. $$

He wants to "rationalize the denominator," so he multiplies the whole thing by a fraction equaling \(1\). This fraction is \(\frac{2 - \sqrt{3}}{2 - \sqrt{3}}\).

It's the same thing as what you do when you multiply a fraction by the same thing to the top and bottom in order to get a different denominator:

$$ \frac{2}{7} = \frac{2}{7} \times \frac{3}{3} = \frac{6}{21}$$

The reason he does this is so that we get a nice difference of squares in the denominator, which effectively gets rid of the square root.

$$ (a+b)(a-b) = a^2 - b^2 $$

$$ (2 + \sqrt{3} )(2 - \sqrt{3}) = 2^2 - \sqrt{3}^2 $$

Then, the bottom just becomes

$$ 4 - 3 $$

which equals \(1!\)

]]>Let's look at a different example of a simpler ratio:

Like in the Day 13: Your Turn question, the purple angles are equal, and the line going through the triangle is an angle bisector. The difference is that the lengths of the two sides which make up the bisected angle are simpler: 1 and 2. These sides are in the ratio 1 : 2, which we can illustrate with a pie chart:

The blue section corresponds to the "1" out of the total "1 + 2", so it takes up 1/3 not 1/2 of the whole pie. We would find that the segment labeled with a "?" takes up 1/3 of the length of x.

Now, back to our problem: Instead of 1 and 2, the lengths of the sides that touch the bisected angle are 2 x root(3) and 4, so the segment that we want, BD, takes up

$$ \frac{2\sqrt{3}}{2\sqrt{3} + 4} $$

of the length of the third triangle side. The length of BD is this ratio of the length of the third side, or

$$ \frac{2 \sqrt{3}}{2\sqrt{3} + 4} \times \text{ length of third side} $$

which equals:

$$ \frac{2\sqrt{3}}{2\sqrt{3} + 4} \times 2 $$

is because the total length of the third side (BD + CD) is equal to 2.

I hope this helps! It's a pleasure to help you with this, and congratulations on almost finishing the course. We really hope you have enjoyed learning with this problem-based approach and enjoy tackling challenges more and more.

Happy Learning!

The Daily Challenge Team

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