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Katie L.

Student at Princeton University

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Pre-Algebra

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Question:

Determine which fraction is greater: 6/17 or 4/10?

Katie L.

Answer:

First, we would need to find a common denominator, and then compare the two fractions again. Right now, it seems like we are comparing apples to oranges here, and by using a common denominator, all we have to do is compare numerators! So now we have denominators of 17 and 10. To find a common denominator, we can multiply (in this case), or find the Least Common Multiple, a number both can divide into. So, 17 * 10 = 170, and we rewrite both of our fractions with this denominator. For 6/17, we multiply both the top and the bottom by 10, to get 60/170. For 4/10, we multiple both the top and bottom by 17, to get 68/170. By comparing 60 and 68, we know that 68 is greater, and therefore the fraction 4/10 is greater than 6/17.

Basic Math

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Question:

Add 124092348 and 1034909.

Katie L.

Answer:

Although counting on your fingers may be an option, that would take a whole lot of time! A helpful method for adding large numbers is as follows: Align the each digit in a vertical column. In other words, stack your numbers on top of each other, with the right end of each lining up with each other! 124092348 + 1034909 ----------------- Now, start on the farthest right column, and add the two numbers, 8 and 9. We get 17! However, it is a two digit number, and there is only space in this column for one. Write the "7", or rightmost digit under this column, and then carry the 1. In other words, add that to your next column. The next column is 4 + 0, with 1 added from the previous carry-on, which gives five. Now we move on to the next column until the end, and then we get the answer! So, \begin{align*} 124092348\\ \underline{+\hspace{0.2cm} 1034909}\\ 125127257 \end{align*} \end{document}

Music Theory

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Question:

What exactly is 4-part writing and how do you write a "good" progression?

Katie L.

Answer:

Essentially, your progression is a series of roman numerals in a specified key, and each phrase ends in a cadence, which is comprised of two chords. Like its name, there are four voices in each chord - the soprano, alto, tenor, and bass. Roman numerals are centered around each specific key and are numbered from 1 to 7. So, in C major, the I chord would be C E G and the rest of the chords are as follows: ii: D F A III: E G B IV: F A C V: G B D vi: A C E vii(dim): B D F With each numeral representing a scale degree and its chord. The uppercase and lowercase (or diminished) signify either a major or minor triad. There are many rules in 4-part writing, including but not limited to: - Raise the leading tone (in the V or vii chord, the seventh scale degree should go to the tonic, or first scale degree) - NO parallel 5th or octaves (or two 5ths in a row for two voices) - Since there are 4 voices, try and double the root of the chord. - Stay within a certain range for each voice You write progressions by keeping in mind where you want your progression to go, and where it ends up. For instance, cadences end a progression, and types of cadences include a Perfect Authentic Cadence (V - I), Deceptive Cadence (V - vi), Plagal Cadence (IV - I) and vice versa. Generally, appropriate progressions are as follows: iii --> vi --> (IV or ii) --> (vii or V) --> I Where iii can also go to IV, IV can go to I, V can go to vi, ii to vii, IV to V, and I to any other chord.

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