Article: We use cross multiplication for math problems where we know the fractions are equivalent, but one of the numbers has been replaced with a variable (typically x) for which we must solve. In cases like this, we know these fractions are equivalent because they're the sole terms on opposite sides of an equal sign, but it's often not obvious how to solve for the variable. Luckily, with cross multiplication, solving these types of problems is easy. In other words, you multiply the numerator of one fraction by the denominator of the other and vice versa, then set these two answers equal to each other and solve. Take our two examples of 4/8 and 8/16. These two don't contain a variable, but we can prove the concept since we already know they're equivalent. By cross multiplying, we get 4 x 16 = 8 x 8, or 64 = 64, which is obviously true. If the two numbers are not the same, then the fractions are not equivalent. Since cross multiplication is the easiest way to determine equivalent fractions when you must solve for a variable, let's add a variable. For example, let's consider the equation 2/x = 10/13. To cross multiply, we multiply 2 by 13 and 10 by x, then set our answers equal to each other: 2 13 = 26 10 x = 10x 10x = 26. From here, getting an answer for our variable is a matter of simple algebra. x = 26/10 = 2.6, making the initial equivalent fractions 2/2.6 = 10/13. One of the best things about cross multiplication is that it works in essentially the same way whether you're dealing with two simple fractions (as above) or with more complex fractions. For instance, if both fractions contain variables, you just have to eliminate these variables at the end during the solving process. Similarly, if the numerators or denominators of your fractions contain variable expressions (such as x + 1), simply "multiply through"by using the distributive property and solve as you normally would. For instance, let's consider the equation ((x + 3)/2) = ((x + 1)/4). In this case, as above, we'll solve by cross multiplying: (x + 3) 4 = 4x + 12 (x + 1) 2 = 2x + 2 2x + 2 = 4x + 12, then we can simplify the equation by subtracting 2x from both sides 2 = 2x + 12, then we should isolate the variable by subtracting 12 from both sides -10 = 2x, and divide by 2 to solve for x -5 = x
Question: What is a summary of what this article is about?
Set the two fractions equal to one another. Take the two equivalent fractions and multiply across the equals sign in an "X" shape. Introduce a variable. Use cross multiplication for equations with multiple variables or variable expressions.
Article: If your leather is not responding to conventional methods, vinegar and olive oil can be used instead. Mix one part vinegar and one part extra virgin olive oil. Then, dab onto the leather with a soft cleaning cloth, using circular motions to apply. Let sit for 10 minutes before buffing the chair with a dry cloth until it shines. Baby wipes can be used to remove any gunk or built up residue stuck on a leather couch. If you're hesitant to use baby wipes on furniture, they make wipes specifically designed for removing gunk from leather as well. You can buy these wipes online or at a local furniture store. If you have young children, you may end up with marker stains on your leather chair. These can be removed with hairspray. Use aerosol hairspray. Spray it over the stained areas. Then, rub it off gently and see if the stain comes out. Eucalyptus oil can also be used to remove marker stains if hairspray does not work.
Question: What is a summary of what this article is about?
Try vinegar and olive oil. Use baby wipes. Try hairspray.
Article: Once you feel comfortable with straightforward rhythms, it is time to start playing with shorter, but more complex, phrases. Duplets are simply two sounds thrown quickly together ("da-DA!"), and triplets are three sounds ("BEEP-da-BOP"). Instead of using straight quarter notes, where you have one sound per beat (1, 2, 3, 4), start stringing together these other phrases, leaving spaces in between them for a bouncy, swinging feel. Hold some notes for three beats, cram 10 notes into two beats, and then leave some silence before launching back in. Rhythmic variety is about playing with the beat to create tension and surprise. Alternating different types of rhythms is a great way to craft a complex, enjoyable scat solo without crazy notes or vocal range. Check out both Louis Armstrong and Ella Fitzgerald, for example, on "Stomping at the Savoy" for a master class in rhythmic variation. An extension of rhythmic variation, this is when you transcend the "written" rhythm and get into the song with improvisational fervor. Most scatting is over a swing feel, where the 2nd and 4th beats are emphasized. Think of counting out your "1 and, 2 and, 3 and, 4 and" with a little extra emphasis on these two beats. If you're going to hit a high note, or pause and come back in, make it on the swinging beats. Great scat singers, like any great soloist or musician, are locked into the underlying chords and melody of the song they sing over. They know when chord changes are coming, and adjust their melody right in time with the rest of the band. Listen to the song several times until the chords' feel natural, and you know exactly what your band is doing behind you. If you're hoping to play professionally, there are a few progressions you should know: 12-Bar Blues-- the most common progression in Western Music. No matter what the key, the chords will always change in the same order, meaning you can quickly scat to any 12-bar blues once you know the form. I Got Rhythm -- known as the most covered chord progression in jazz, these changes are found in hundreds of songs, including popular music. Listen to versions from Duke Ellington to Django Reinhardt If you are classically trained, there is no reason to abandon your skills just to start scatting. Run through your exercises, scales, and warm-ups with improvised syllables and sounds, and start putting your vocal scales to music. As a warm up, read music but ignore the lyrics, trying to sound out just the notes of vocal, brass, and woodwind sheet music. Listen to Ella Fitzgerald closing out "Tenderly," a dear, calm, and loving song, with a deep and almost monstrous scat sound. And yet, it fits, complementing her smooth tone with an unexpected burst of passion and power. Scatting isn't about sounding "human." So, the further you can modulate and adjust your voice to sound more instrumental the better scat singer you'll become.
Question: What is a summary of what this article is about?
Introduce a little rhythmic variety with duplets and triplets. Get a little swing behind your scatting. Learn the chord progressions to improvise like a jazz singer. Scat through your vocal solos to bridge classical singing and scatting. Get a little weird with tone, voice, and coloring.