Park Forest Math Team. Meet #2. Self-study Packet

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1 Park Forest Math Team Meet #2 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. : Order of operations; mean, median, mode; rounding; statistics 5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities

2 Meet #2 Ideas you should know: Multiplying fractions: = Cancel before x for speed Dividing Fractions: 2? Flip, then x " Writing fractions of fractions Reciprocal: Multiplicative Inverse. Reciprocal of 3 = 1/3. Reciprocal of 2/7 = 7/2. Divide by Y = Multiply by reciprocal of Y Adding Fractions common denominator Meet #2, 2 10/24/2005

3 What do you mean by of? of 24 = Of = Times Fraction in Lowest Terms Repeating decimal equivalent: 103/999 =? =? = =? 1/9 = /90 = = 1/2 + 1/90 = 45/90 + 1/90 = 46/90 = 23/45 Improper Fraction Mixed Numeral 2 Mike is 50% taller than Bob: This means he s 1.5 times as tall, not as tall I ate 50% as much as you = half as much. I ate 50% more than you = 1.5 times as much I ate 100% as much as you = same I ate 100% more than you = twice as much I ate 200% more than you = 3 times as much I ate 50% of you = well, nevermind. The price is 1/3 higher: The price is 1+1/3 as high. If the original price was $30, then 1/3 higher means it s $40. Meet #2, 3 10/24/2005

4 What fraction is this repeating decimal? Another way to figure it out: If digits before the repeating pattern: 15 th digit in the decimal expansion of problems What is the 15 th digit of the decimal expansion of 1/7? 1/7 = You could just write it out and count digits. Another way is to say digit 3 is 2, and every 6 th digit after that is also a 2, and 15=3+6x2, so it s also 2. What is the 601 st digit of the decimal expansion of 2/7=? Answer: It s 600 digits past the 1 st, so it s the same as the 1 st, or 2. What is 2/3 of 25% of 3/7 of 4/9 of 81 problems These are simply multiplication with a lot of cancellation usually. Rewrite 25/100 as 1/4, cancel 3 s and 4 s: or and also cancel 9s from 1/9 and 81, and so we get 2x9/7 or 18/7 or 2 4/7. Adding or subtracting repeating decimals If you have plus you get which makes sense if you look at them as fractions: 3/9 + 1/9 = 4/9. It s tricky if the two repeating patterns have a different length: From the 1999 meet: What is? Answer: Write 0.2 as 0.22 and then it s 51/99+22/99 = 73/99 or Dividing repeating decimals This seems harder, but you can often do it in your head using fractions: What is Answer: Meet #2, 4 10/24/2005

5 Category 4 Meet #2, November What is 35% of 0.4 of 9 14 of 8 9 of 625? 2. Find the fraction in lowest terms that is equivalent to the repeating decimal What is the 39 th digit in the repeating decimal equivalent of 5 14?

6 Solutions to Category 4 Average team got points, or 1.1 questions correct Meet #2, November It is helpful to rewrite all quantities in fraction form. The percent can be written as 35 7, which reduces to The decimal 0.4 can be written as 4, which reduces to Since the word of means multiply, we can rewrite 5 the expression as Now we can 1 reduce by cancelling common factors, until we get 2 25 = We use a clever bit of algebra to convert a repeating decimal to a fraction. Let s say x = Then 10x = 2.7 and 100x = Now we subtract one equation from another as shown below: 100x = x = x = 25 Dividing both sides of the equation by 90 and simplifying, we get: x = = After the digit 3 in the tenths place, the decimal expansion of 5/14 has a six-digit repeating cycle, as shown at left. This means that the 39 th digit in the repeating decimal will actually be the 38 th digit in the repeating part. Since 38 = , it will be the 2 nd digit in the cycle, which is a 7.

7 Category 4 Meet #2, December What is 20% of 5 8 of 4 9 of 162? 2. Find the value of Express your answer in simplest form The repeating decimal for 1 is What digit will appear 17 in the twenty-third decimal place of the decimal equivalent for ?

8 Solutions to Category 4 Meet #2, December Students with correct answer in a cluster of 6 schools: 1. 31/ /36 (the most difficult of this meet) 3. 23/ It helps to convert the percent to a fraction: 20% = = 1. The word of generally means multiply, so 5 the value we are looking for is = 9 18 = To convert each repeating decimal to a fraction, we use a little algebra as follows. If x = 0.83, then 10x = Subtracting the first equation from the second, we get 10x x = or 9x = 7.5. Dividing both sides of the equation by 9, we find that x = = = 5. Similarly, if y = 0.416, then 6 10y = Again subtracting the first equation from the second, we get 10y y = or 9y = Dividing both sides by 9, we find that y = = = 5. Finally, the value of our original 12 fraction must be = 6 = = = The repeating decimal pattern for 1 is 16 digits 17 long. The same 16-digit pattern occurs in the decimal equivalent of = 16, but it starts with the 9 instead 17 of the 0. Thus we have 16 = The 17 twenty-third digit in the pattern will be the same as the seventh digit in the pattern, since = 7. The digit is 4.

9 Category 4 Meet #2, December What is the fraction that is equivalent to the decimal.3125? Express your answer as a common fraction. (A common fraction is written in the form where cannot be simplified. A mixed number is not a common " " fraction.) 2. How much larger is #$ % ' of 96 than ( of 84? & %& 3. What is the 80 th digit to the right of the decimal point in the repeating decimal equivalent of ) *+,

10 Solutions to Category 4 Meet #2, December "#$% & '()* (++++ &,)* )+++ & ()* -++ & )*.+ & * (, 1. * (, "/ ( 0 of 96 = ' 1 23 & "3 ). * of 84 = * 1 45 & "% () () 36 is 1 larger than 35 ' 3. & 775$4%/# The repeating part of the decimal has 6 digits, but due to the 6+ 0 at the beginning, each set of 6 repeating digits would end at the 7 th, 13 th, 19 th, 67 th, 73 rd, and 79 th place after the decimal. So the 79 th place after the decimal is 1 and the 80 th place after the decimal is 4.

11 Category 4 ""#$%&$$$$$$'"(")*"+$&,-,$ "# $%&'())*+(,(-./01 0)0-2//234'0-*.23# 564'0-*.2324*+(42'/ *8()./&1.4.(,9# :# $%&'())*+(4'0-*.23 0)0,(-./01# ;)(80'32*0*.23*232*('(&(0*.3<,.<.*)# =# >./&?*011+.))0@.3<).3*+(A.'0-1(803B# # 64*('*+()(-23,C(0'D+.)80103-(<'(78C030,,.*.2301 D03,70)327 EFG/2'(*+03+.)2'.<.301,(&2).*# H27/?-+/23(C,2()>./+0@( )0--2?3*I """#$%&'%#()*

12 Solutions to Category 4 - " # # $" % # ""#$%&$$$$$$'"(")*"+$&,-,$ 3., then we can write the information as follows: Replacing percents with numbers and aggregating: which we can aggregate further into: or. The balance now is $180 (or 180) """#$%&'%#()*

13 Meet #2 December 2010 Category 4 1. Express the decimal as a common fraction. [A fraction of the form which cannot be simplified]. 2. Express the fraction as a decimal. Use bar notation to note repeating digits. 3. Tim put all his savings in the Miracle bank. After one year, his account s balance grew by. After the second year, his balance grew by an additional $60 more than his original deposit. How much money does Tim have now in his account?, and was now

14 Solutions to Category 4-1. Meet #2 December $180 (or 180) 3. If we call his original deposit s amount, then we can write the information as follows: Replacing percents with numbers and aggregating: which we can aggregate further into: or. The balance now is

15 Category 4 Meet #2, November/December What number is 20% less than 9 2 7? Express your answer as a mixed number in lowest terms. 2. Simplify the expression below to a common fraction What is the 202 nd digit to the right of the decimal point in the decimal 25 equivalent of 202?

16 Solutions to Category 4 Meet #2, November/December Twenty percent less than all of a number is 80% or 4/5 of that number. Multiplying 4/5 by , we get 7 2. We can simplify the expression as follows: We have to do long division to find the repeating decimal pattern for 25/202, as shown at right. The first digit to the right of the decimal point is not part of the repeating pattern. Then we get a repeating pattern of four digits. We know it repeats because we get a repeat remainder of 48. The 202 nd digit to the right of the decimal point is the 201 st digit in the pattern. Since 201 is 1 more than a multiple of 4, we get the first digit in the pattern, which is

Fractions. There are several terms that are commonly used when working with fractions.

$Fractions. There are several terms that are commonly used when working with fractions.$ Chapter 0 Review of Arithmetic Fractions There are several terms that are commonly used when working with fractions. Fraction: The ratio of two numbers. We use a division bar to show this ratio. The number