AMC 8 L1: Calculation Tricks

Complete problem set with solutions and individual problem pages

Problem 1 Easy

There is an arithmetic sequence 2, 4, 6, 8, \cdot \cdot \cdot. The 20^{th} number is            .

Answer:40

Solution 1: 2+(20-1)\times2=40

Solution 2: 2\times20=40

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Problem 2 Easy

On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days? (2015 AMC 8 Problem, Question #9)

  • A.

    39

  • B.

    40

  • C.

    210

  • D.

    400

  • E.

    401

Answer:D

Solution 1: 1+3+5+\cdots +\left[ \left( 20-1 \right)\times 2+1 \right] =1+3+5+\cdots +39 =\left( 1+39 \right)\times 20\div 2 =400.

Solution 2: The sum is just the sum of the first 20 old integers, which is {{20}^{2}}=\text{D} 400.

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Problem 3 Easy

James read a story book. On the first day, he read 4 pages. From the second day, he read 4 more pages each day than he read on the previous day. On the last day, he read 40 pages. How many pages did he read in total?

  • A.

    210

  • B.

    220

  • C.

    230

  • D.

    240

Answer:B

(40-4)\div4+1=10, (4+40)\times10\div2=220

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Problem 4 Easy

Calculate: 26-25+24-23+\cdots +4-3+2-1=            

Answer:13

(26-25)+(24-23)+\cdots +(4-3)+(2-1)=13\times1=13

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Problem 5 Easy

What is the value of 1+3+5+\cdots +2017+2019-2-4-6-\cdots -2016-2018? (2018 AMC 8 Problem, Question #5)

  • A.

    -1010

  • B.

    -1009

  • C.

    1008

  • D.

    1009

  • E.

    1010

Answer:E

Solution 1: (1+3+5+\cdots +2017+2019)-(2+4+6+\cdots +2018)

=1010\times1010-(2+2018)\times1009\div 2=1010

Solution 2: 1+(3-2)+(5-4)+\cdots +(2019-2018)=1+1+1+\cdots +1=1010

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Problem 6 Easy

Find the value of expression 100-98+96-94+92-90+\cdots +8-6+4-2. (2016 AMC 8 Problem, Question #8)

  • A.

    20

  • B.

    40

  • C.

    50

  • D.

    80

  • E.

    100

Answer:C

100-98+96-94+92-90+\cdots +8-6+4-2

=(100-98)+(96-94)+(92-90)+\cdots +(8-6)+(4-2)

=2\times25

=50

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Problem 7 Easy

Calculate: 100+99-98-97+96+95-94-93+\cdots +4+3-2-1=            

Answer:100

100+99-98-97+96+95-94-93+\cdots +4+3-2-1

=(100+99-98-97)+(96+95-94-93)+\cdots +(4+3-2-1)

=4\times25

=100

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Problem 8 Easy

Calculate: \frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}=            

Answer:\frac{1}{2}

\frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}=\frac12

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Problem 9 Easy

What is the product of \frac32 \times \frac43 \times \frac54\times\cdots \times \frac{2006}{2005}? (2006 AMC 8 Problem, Question #9)

  • A.

    1

  • B.

    1002

  • C.

    1003

  • D.

    2005

  • E.

    2006

Answer:C

\frac32 \times \frac43 \times \frac54\times\cdots \times \frac{2006}{2005}=\frac{2006}2=1003

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Problem 10 Easy

What is the value of the product (1+\frac11)·(1+\frac12)·(1+\frac13)·(1+\frac14)·(1+\frac15)·(1+\frac16)? (2018 AMC 8 Problem, Question #2)

  • A.

    \frac76

  • B.

    \frac43

  • C.

    \frac72

  • D.

    7

  • E.

    8

Answer:D

(1+\frac11)·(1+\frac12)·(1+\frac13)·(1+\frac14)·(1+\frac15)·(1+\frac16)=\frac21\times \frac32 \times \frac 43 \times \frac54 \times \frac65\times \frac76=7

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Problem 11 Easy

Calculate: \left( 1-\frac{1}{2} \right)\times \left( 1-\frac{1}{3} \right)\times \left( 1-\frac{1}{4} \right)\times \left( 1-\frac{1}{5} \right)=            

Answer:\frac{1}{5}

\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}=\frac{1}{5}

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Problem 12 Easy

Calculate: 1 \frac{1}{2} \times 1 \frac{1}{3} \times 1 \frac{1}{4} \times 1 \frac{1}{5} \times 1 \frac{1}{6} \times 1 \frac{1}{7} \times 1 \frac{1}{8}=            

Answer:\frac92

\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} \times \frac{7}{6} \times \frac{8}{7} \times \frac{9}{8}=\frac92

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Problem 13 Easy

If a@b=\frac{a\times b}{a+b} for a, b positive integers, then what is 5@10? (2010 AMC8 Problem, Question #2)

  • A.

    \frac3{10}

  • B.

    1

  • C.

    2

  • D.

    \frac{10}3

  • E.

    50

Answer:D

Substitude a=5 and b=10 into the expression for a@b to get:

5@10=\frac{5\times10}{5+10}=\frac{50}{15}=\frac{10}{3}.

Thus, answer choice D is correct.

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Problem 14 Easy

Suppose that a*b means 3a-b. What is the value of x if 2*(5*x)=1? (2016 AMC8 Problem, Question #10)

  • A.

    \frac{1}{10}

  • B.

    2

  • C.

    \frac{10}{3}

  • D.

    10

  • E.

    14

Answer:D

Suppose 5*x=y, then 2*y=3\times 2-y=1\Rightarrowy=5.

So 5*x=5, 5\times 3-x=5\Rightarrowx=10.

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Problem 15 Easy

Suppose that a\otimes b=ab-a-b. If (3\otimes x)\otimes3=11, then x=            .

Answer:5

Suppose 3\otimes x=y, then y\otimes3=11\Rightarrow 3y-y-3=11\Rightarrow y=7.

3\otimes x=7\Rightarrow 3x-3-x=7\Rightarrow x=5.

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Problem 16 Easy

Calculate: 1+3+5+7+9+11+\cdots +29=            

Answer:225

~~~~~1+3+5+7+9+11+\cdots +29

=\left( 1+29 \right)\times 15\div 2

=225

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Problem 17 Easy

Calculate: 100-99+98-97+96-95+\cdots +2-1=            

Answer:50

~~~~100-99+98-97+96-95+\cdots +2-1

=1+1+1+1+\cdots+1

=1\times50

=50

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Problem 18 Easy

Calculate: \frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{11}{12}=            

Answer:\frac1{12}

\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{11}{12}=\frac1{12}

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Problem 19 Medium

Suppose that A☆B means 3A-2B. What is the value of x if x☆ (4☆1) =7?

  • A.

    7

  • B.

    8

  • C.

    9

  • D.

    10

Answer:C

x☆ (4☆1) =x☆ (3\times 4-2\times 1) =x☆10=3x-20=7 \Rightarrowx=9.

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Related Paper
2015 AMC 8 2014 AMC 8 AMC 8 L1: Assignment AMC 8 L2: Basic Geometry
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