INPUT ARTICLE: Article: Make 4 chain stitches. Next, make a slip stitch into the first chain to form a circle. Use a size I/9 or 5.5mm crochet hook and medium worsted weight yarn. Start off with 2 chain stitches. Next, do 3 double crochet stitches in each space. Close your round with a slip stitch. You will end up with 12 stitches total. Mark the end of your round with a stitch marker. If you don't have one, you can use a safety pin or even a piece of yarn in a contrasting color. Start off with 2 chain stitches. Do 2 double crochet stitches in the first stitch. Follow up with 1 double crochet. Repeat the last 3 stitches (2 double crochets in 1 stitch, followed by 1 double crochet) until you reach the end of the round. You should have a total of 18 stitches. Do not turn your project. You might want to move your stitch marker to the end of your round, however. Start with 2 chain stitches. Then, do 2 double crochets in the same stitch. Follow up with 2 double crochets over the next 2 stitches. Repeat the last 4 stitches (2 double crochets in 1 stitch, 2 double crochets over the next 2 stitches) until you reach the end of the round. Close the round off with a slip stitch. You'll end up with 24 stitches. Remember to move your stitch marker, if you are using one. Start each row with 2 chain stitches and 2 double crochets in the first stitch. Do x-number of stitches, then repeat (2 double crochets in one stitch, followed by x-number of double crochets) until you reach the end of the row. Close each round off with a slip stitch. How many stitches you do for "x" depends on what round you are on. The number "x" increases with each round. For example:  Round 5: 2 double crochet in 1 stitch, 3 double crochets over the next 3 stitches. 30 stitches total. Round 6: 2 double crochet in 1 stitch, 4 double crochets over the next 4 stitches. 36 stitches total. Round 7: 2 double crochet in 1 stitch, 5 double crochets over the next 5 stitches. 42 stitches total. Round 8: 2 double crochet in 1 stitch, 6 double crochets over the next 6 stitches. 48 stitches total. If you have a very large head, you can add more rounds with similar increases. If you have a smaller head, you can do fewer rounds. Start off with 2 chain stitches. Next, do a double crochet in each space until you finish the round. Use a slip stitch to close the round. Start each round with 2 chain stitches, then do 1 double crochet in each space all around. You aren't adding any increases to these rounds, so your hat will start to form a tube shape. Start of with 1 chain stitch, then do a single crochet in each space. Do this for 2 rows, then finish off with a slip stitch. Your beanie is now done and ready to wear! If you'd like, you can turn the bottom up by a few inches/centimeters) to create a brim, or even add a pompom to the top!

SUMMARY: Start with a chain circle. Do your second round. Start your first increasing round. Close the third found off with a slip stitch. Do your second increasing row, then close it off with a slip stitch. Continue doing your increases for rounds 5 to 8. Start doing the body of your hat. Continue doing the body of your hat until it is the desired length. Finish off with a border using single crochets. Tie the end off, then weave it into the brim of your beanie using a yarn needle.


INPUT ARTICLE: Article: Often, lenders require that you make monthly or quarterly payments. Therefore, it is more useful to know what the monthly or quarterly payment is, rather than simply the annual payment. Fortunately, the same formula is used, with some minor revisions. For the sake of this example, assume the new loan is the same as previously-discussed one, with the only change being you are now required to make monthly payments for the two year period. Although the formula is largely the same as that for annual payments, a few minor changes occur to reflect the fact that there are now more payments. Again, the standard formula is: Payment=(r(P))(1−(1+r)−n){\displaystyle Payment={\frac {(r(P))}{(1-(1+r)^{-n})}}}  First, the amount of periods in the loan, or "n", would change. Instead of being 2 (representing two years before, or two annual payments), it is now 24 for monthly payments (representing 1 payment a month for 2 years) and 8 for quarterly payments (representing one payment each quarter for the two years). Second, the annual interest rate would need to change to reflect the fact there are more payments. To determine an interest rate for periodic payments, divide the annual interest rate by the number of payments required within a year. For example, a 9% annual interest rate is equivalent to a .0075 or .75% monthly interest rate (.09/12). The new formula, with all the example numbers plugged in looks like this: Payment=(0.0912($10,000))(1−(1+0.0912)−24){\displaystyle Payment={\frac {({\frac {0.09}{12}}(\$10,000))}{(1-(1+{\frac {0.09}{12}})^{-24})}}} Start by simplifying the rate by solving for the monthly interest rate. This is done by dividing the annual rate of 9% by 12, as in the equation, to get 0.0075. After you do so, your equation should look like this: Payment=(0.0075($10,000))(1−(1+0.0075)−24){\displaystyle Payment={\frac {(0.0075(\$10,000))}{(1-(1+0.0075)^{-24})}}} Continue by solving the numerator (the top part of the equation). Multiply the two numbers (rate and principal) together to solve this step. Your equation should now look like this: Payment=($75)(1−(1+0.0075)−24){\displaystyle Payment={\frac {(\$75)}{(1-(1+0.0075)^{-24})}}} Next, simplify the denominator (bottom of the equation) by adding the rate to 1. This comes to 1.0075 in our example. The equation now looks like this: Payment=($75)(1−(1.0075)−24){\displaystyle Payment={\frac {(\$75)}{(1-(1.0075)^{-24})}}} Next, solve the exponent in the equation by raising the (rate +1) found in the last step to the power of -24. This comes to 0.8358. The equation now looks like this: Payment=$751−(0.8358){\displaystyle Payment={\frac {\$75}{1-(0.8358)}}} Simplify by subtracting your result in the last step from one. In our example, this would be 1−0.8358{\displaystyle 1-0.8358}, which yields 0.1642. At this point, the equation looks like this: Payment=$750.1642{\displaystyle Payment={\frac {\$75}{0.1642}}} Finally, divide the top part of the equation by the bottom to get your monthly payment. In this case, Payment=$456.76{\displaystyle Payment=\$456.76} If necessary, you can convert your monthly payment to an annual total by multiplying it by 12. Here, 12∗473.78=$5,481.12{\displaystyle 12*473.78=\$5,481.12}. Once again, keep in mind that there are plenty of online calculators available to calculate this online, without ever calculating the payment yourself.

SUMMARY:
Understand the reason to calculate periodic payments on a loan. Learn the formula for calculating periodic payments on a loan. Fill in the equation with your values. Begin to calculate the periodic payments on the loan. Solve the numerator. Simplify the denominator. Solve the exponent. Simplify the denominator again. Solve for your monthly payment. Convert your answer to an annual payment total. Use an online calculator to confirm results.