Park Forest Math Team
Meet #2
Arithmetic
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. Arithmetic: Order of operations; mean, median, mode; rounding; statistics
5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown
including identities
Meet #2 – Arithmetic
Ideas you should know:
Multiplying fractions:
=
Cancel before x for speed
Dividing Fractions:
÷? Flip, then x
2
!
"
2
3
1
6
2
3
1
6
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, Arithmetic
2
10/24/2005
What do you mean by “of” ?
Of = Times
of 24 =
!
Fraction in Lowest Terms
Repeating decimal equivalent:
103/999 =
?
0.51111… =
0.17171717… =
=
?
?
1/9 = 0.11111…
1/90 = 0.011111…
0.51111…. = 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
“I
“I
“I
“I
“I
ate
ate
ate
ate
ate
ate
50% as much as you” = half as much.
50% more than you” = 1.5 times as much
100% as much as you” = same
100% more than you” = twice as much
200% more than you” = 3 times as much
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, Arithmetic
3
10/24/2005
“What fraction is this repeating decimal?”
Another way to figure it out:
If digits before the repeating pattern:
“15th digit in the decimal expansion of” problems
What is the 15th 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 6th digit after that is also a 2, and 15=3+6x2, so it’s also 2.
What is the 601st digit of the decimal expansion of 2/7=
?
st
st
Answer: It’s 600 digits past the 1 , so it’s the same as the 1 , 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 0.33333… plus 0.11111… you get 0.44444… 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 0.737373…
Dividing repeating decimals
This seems harder, but you can often do it in your head using fractions:
What is
Meet #2, Arithmetic
Answer:
4
10/24/2005
Category 4
Arithmetic
Meet #2, November 2004
1. What is 35% of 0.4 of
8
9
of of 625?
9
14
2. Find the fraction in lowest terms that is equivalent to the repeating
decimal 0.27 .
3. What is the 39th digit in the repeating decimal equivalent of
Answers
1. _______________
2. _______________
3. _______________
www.Imlem.org
5
?
14
Solutions to Category 4 Average team got 12.91 points, or 1.1 questions correct
Arithmetic
Meet #2, November 2004
Answers
1. 50
5
18
2.
3. 7
0.3571428
14 5.0000000
− 42
80
− 70
100
− 98
20
− 14
60
− 56
40
− 28
120
− 112
80
1. It is helpful to rewrite all quantities in fraction form.
35
7
The percent can be written as
, which reduces to
.
100
20
4
The decimal 0.4 can be written as , which reduces to
10
2
. Since the word “of” means multiply, we can rewrite
5
7 2 9 8 625
the expression as
⋅ ⋅ ⋅ ⋅
. Now we can
20 5 14 9 1
reduce by cancelling common factors, until we get 2 × 25
= 50.
2. We use a clever bit of algebra to convert a repeating
decimal to a fraction. Let’s say x = 0.27 . Then
10 x = 2.7 and 100 x = 27.7 . Now we subtract one
equation from another as shown below:
100x = 27.7
− 10x = − 2.7
90x = 25
Dividing both sides of the equation by 90 and
simplifying, we get:
25 5
x=
=
90 18
3. 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 39th digit in the
repeating decimal will actually be the 38th digit in the
repeating part. Since 38 = 6 × 6 + 2, it will be the 2nd
digit in the cycle, which is a 7.
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Category 4
Arithmetic
Meet #2, December 2006
1. What is 20% of
5
4
of of 162?
8
9
2. Find the value of
0.83
. Express your answer in simplest form.
0.416
1
is 0.0588235294117647. What digit will appear
17
1
in the twenty-third decimal place of the decimal equivalent for 1 − ?
17
3. The repeating decimal for
Answers
1. _______________
2. _______________
3. _______________
www.imlem.org
Solutions to Category 4
Arithmetic
Meet #2, December 2006
Answers
1. 9
2. 2
3. 4
Students with
correct answer in a
cluster of 6 schools:
1. 31/36
2. 19/36!
(the most difficult of
this meet)
3. 23/36
.
1. It helps to convert the percent to a fraction: 20% =
20 1
= . The word “of” generally means multiply, so
100 5
1 5 4
162
the value we are looking for is × × × 162 =
= 9.
5 8 9
18
2. To convert each repeating decimal to a fraction, we
use a little algebra as follows. If x = 0.83 , then
10 x = 8.33 . Subtracting the first equation from the
second, we get 10 x − x = 8.33 − 0.83 or 9 x = 7.5 .
Dividing both sides of the equation by 9, we find that
7.5 75 5
x=
=
= . Similarly, if y = 0.416 , then
9
90 6
10 y = 4.166 . Again subtracting the first equation from
the second, we get 10 y − y = 4.166 − 0.416 or 9 y = 3.75 .
Dividing both sides by 9, we find that
3.75 375 5
y=
=
= . Finally, the value of our original
9
900 12
5
0.83
5 5 5 12
fraction must be
= 65 = ÷ = × = 2 .
0.416 12 6 12 6 5
1
is 16 digits
17
long. The same 16-digit pattern occurs in the decimal
1 16
equivalent of 1 − = , but it starts with the 9 instead
17 17
16
of the 0. Thus we have
= 0.9411764705882352. The
17
twenty-third digit in the pattern will be the same as the
seventh digit in the pattern, since 23 – 16 = 7. The digit
is 4.
3. The repeating decimal pattern for
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Category 4
Arithmetic
Meet #2, December 2008
1.
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
3.
What is the 80th digit to the right of the decimal point in the repeating
)
decimal equivalent of ,
*+
Answers
1. _______________
2. _______________
3. _______________
of 84?
Solutions to Category 4
Arithmetic
Meet #2, December 2008
Answers
1.
! "#$% &
'()*
(++++
&
,)*
)+++
&
()*
-++
&
)*
.+
&
*
(,
*
1.
(,
2.
1
3.
4
2.
(
'
)
.
"/ 0 of 96 = 1 23 & "3
*
()
of 84 =
*
()
1 45 & "%
36 is 1 larger than 35
'
8888888888. The repeating part of the decimal has 6 digits, but due to the
3.
& 7!75$4%/#
6+
0 at the beginning, each set of 6 repeating digits would end at the 7th, 13th, 19th,
……67th, 73rd, and 79th place after the decimal. So the 79th place after the decimal
is 1 and the 80th place after the decimal is 4.
!""#$%&$$$$$$'"(")*"+$&,-,$
!
Category 4 ± Arithmetic
!
"# $%&'())!*+(!,(-./01!ͲǤͶʹͷ!0)!0!-2//23!4'0-*.23#!
௠
56!4'0-*.23!24!*+(!42'/! !7+.-+!-0332*!8(!)./&1.4.(,9#!
௡
!
!
!
:# $%&'())!*+(!4'0-*.23!
଻
ସ଼
!0)!0!,(-./01#!!
;)(!80'!32*0*.23!*2!32*(!'(&(0*.3<!,.<.*)#!
!
!
!
=# !>./!&?*!011!+.)!)[email protected]<)!.3!*+(!A.'0-1(!803B#!
$IWHURQH\HDUKLVDFFRXQW¶VEDODQFHJUHZE\ʹͲΨ#!
64*('!*+(!)(-23,!C(0'D!+.)!80103-(!<'(7!8C!03!0,,.*.2301!ʹͷΨD!03,!70)!327!
EFG!/2'(!*+03!+.)!2'.<.301!,(&2).*#!
H27!/?-+!/23(C!,2()!>./!+0@(!327!.3!+.)!0--2?3*I!
!
Answers
1. _______________
2. _______________
3. _______________
"""#$%&'%#()*!
!
!""#$%&$$$$$$'"(")*"+$&,-,$
!
Solutions to Category 4 - Arithmetic!
!" ͲǤͶʹͷ ൌ
ସଶହ
ଵ଴଴଴
ൌ
଼ହ
ଶ଴଴
ൌ
Answers
ଵ଻
#
1.
ସ଴
#
$" ͲǤͳͶͷͺ͵ത!
2.
തതതതതതത
͹ȁͶͺ!
%!
ଵ଻
ସ଴
ͲǤͳͶͷͺ͵ത
3. $180 (or 180)
͹Ͳ!
Ͷͺ!
ʹʹͲ!
ͳͻʹ!
ʹͺͲ!
ʹͶͲ!
ͶͲͲ!
͵ͺͶ!
ͳ͸Ͳ!
ͳͶͶ!
ͳ͸Ͳ ǥ!
#
3. ,IZHFDOOKLVRULJLQDOGHSRVLW¶VDPRXQW‫ܦ‬, then we can
write the information as follows:
‫ ܦ‬൅ ʹͲΨ ȉ ‫ ܦ‬൅ ʹͷΨ ȉ ሺ‫ ܦ‬൅ ʹͲΨ ȉ ‫ܦ‬ሻ ൌ ‫ ܦ‬൅ ̈́͸Ͳ
Replacing percents with numbers and aggregating:
‫ ܦ‬ȉ ሺͳ ൅ ͲǤʹ ൅ ͲǤʹͷ ‫ͳ כ‬Ǥʹሻ ൌ ‫ ܦ‬൅ ̈́͸Ͳ which we can aggregate further into:
ͳǤͷ ‫ ܦ כ‬ൌ ‫ ܦ‬൅ ̈́͸Ͳ or ‫ ܦ‬ൌ ̈́ͳʹͲ.
The balance now is ͳǤͷ ‫ ܦ כ‬ൌ ̈́ͳͺͲ
"""#$%&'%#()*!
!
Meet #2
December 2010
Category 4 – Arithmetic
1. Express the decimal
[A fraction of the form
2. Express the fraction
as a common fraction.
which cannot be simplified].
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?
Answers
1. _______________
2. _______________
3. _______________
www.imlem.org
, and was now
Meet #2
December 2010
Solutions to Category 4 - Arithmetic
Answers
1.
1.
2.
2.
0
3. $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
www.imlem.org
Category 4
Arithmetic
Meet #2, November/December 2012
2
1. What number is 20% less than 9 ?
7
Express your answer as a mixed number in lowest terms.

2. Simplify the expression below to a common fraction.
3
 0.24
11
4
 0.41
9

3. What is the 202nd digit to the right of the decimal point in the decimal
25
equivalent of
?
202

Answers
1. _____________________
2. _____________________
3. _____________________
Solutions to Category 4
Arithmetic
Meet #2, November/December 2012
Answers
1.
1. Twenty percent less than all of a number is 80% or 4/5
2
of that number. Multiplying 4/5 by 9 , we get

7
4
2 4 65 4 13 52
3
9   

7 .
5
7 5 7
7
7
7



2.
7
3
7
3
5
3. 2
2. We can simplify the expression as follows:
3
3 24
27  24
51
 0.24

51 3
11
99
 11 99 
 99  
4
4 41
44  41
85
85 5
 0.41

9
9 99
99
99
3. 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 202nd digit to the right of the
decimal point is the 201st digit in the pattern. Since 201
is 1 more than a multiple of 4, we get the first digit in the
pattern, which is 2.

0.12376
202 25.00000
202
480
404
760
 606
1540
1414
1260
1212
48
