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Simple operation of doctoral mathematics
1.(1/69+2/7 1) * 23+25 divided by 71.

2.( 1+ 1/2)*( 1- 1/2)* 1+ 1/3)*( 1- 1/3)*........*( 1+ 1/9)*( 1- 1/9)

3.1/2 * 3+1/3 * 4+...+119 times 20.

4. 1/27+505/2727+ 13 13 13/272727

5. 100-98+96-94+....+4-2

1.(1/69+2/7 1) times 23+25 divided by 71.

= 1/69*23+2/7 1*23+25/7 1

= 1/3+(46/7 1+25/7 1)

= 1/3+ 1

=4/3

2.( 1+ 1/2) times (1-kloc-0//2) times (1+ 1/3) times (1-kloc)

=3/2* 1/2*4/3*2/3...* 10/9*8/9

= 1/2* 10/9

=5/9

3. 1/2 times 3+ 1/3 times 4+...+119 times 20.

= 1/2- 1/3+ 1/3- 1/4+...+ 1/ 19- 1/20

= 1/2- 1/20

=9/20

4. 1/27+505/2727+ 13 13 13/272727

= 1/27+5* 10 1/(27* 10 1)+ 13* 10 10 1/(27* 10 10 1)

= 1/27+5/27+ 13/27

= 19/27

5. 100-98+96-94+....+4-2

=( 100-98)+(96-94)+...(4-2)

= 2+2+...+2 (* * has 25)

=2*25

=50

10-2.65=

0.9×0.08=

528-349=

6+ 14.4=

24÷0.04=

12.34-2.3=

0÷3.8=

0.77+0.33=

7÷ 1.4=

67.5+0.25=

7.2÷8×4=

5- 1.4- 1.6=

400÷ 125÷8=

1.9×4×0.5=

80×0. 125=

2×5-6+2=

3.2×7÷3.2×7=

4 +3.2+5 +6.8 25×(8×0.4)× 1.25 7

1 125-997 998+ 1246+9989 (8700+870+87)÷87

125×8.8 1.3+4.25+3.7+3.75 17. 15-(3.5-2.85)

3.4×99+3.4 4.8× 1.0 1 0.4×(2.5÷73)

( 1.6+ 1.6+ 1.6+ 1.6)×25

12.3-2.45-5.7-4.55 2 + 0. 125×0.25×64

64.2×87+0.642× 1300 78×36+7.8×74 1-7

0. 125× +0.5 2.42 +4.58 -43×0

25÷ 100 4.25-3 -(2 - 1 )

1.25* 17.6+36. 1/0.8+2.36* 12.5

=(5/4)* 17.6+36. 1*(5/4)+23.6*(50/4)

= 176/8+36 1/8+236/8

=773/8=96.625

7.5*2.3+ 1.9*2.5

=7.5*( 1.9+0.4)+ 1.9*2.5

=(7.5+2.5)* 1.9+7.5*0.4

= 19+3 =22

2004/2003*2005

=(2004/2003)*(2003+2)

=2004+4008/2003

276*543-267/276+543*275

=543*(276+275)-267/276

=543*55 1-267/276

17/5 1+ (68/ 1+5 1/2)* 17

Fifty-two and twenty-one× 79.45+159× 47.56+79 and twenty-one× 52.44

=52.44×79.45+ 159×47.56+79.55×52.44

=52.44×(79.45+79.55)+ 159×47.56

=52.44× 159+ 159×47.56

= 159×(52.44+47.56)

= 159× 100

= 15900

2002+200 1-2000- 1999+ 1998+ 1997- 1996- 1995+……+2+ 1

=(2002-2000)+(200 1- 1999)+( 1998- 1996)+( 1997- 1995)+……+(6-4)+(5-3)+2+ 1

= 2+2+2+2+...+2+2 (from 3-2002 * * 2000, with 1000 binary) +2+ 1.

= 1000×2+2+ 1

=2003

5 Both questions use a conversion formula1/(a× b) =1/(b-a )× (1/a-1/b).

Such as115 =1(3× 5) =1(5-3 )× (1/3-1/5) =/kloc-0.

( 1×2) +(2×3) +(3×4)+...+( 10× 1 1)

= 1/( 1×2)+ 1/(2×3)+ 1/(3×4)+……+ 1/( 10× 1 1)

=( 1- 1/2)+( 1/2 - 1/3)+( 1/3 - 1/4)+……+( 1/ 10 - 1/ 1 1)

= 1- 1/ 1 1

= 10/ 1 1

One third+one fifteenth+one thirty-fifth+one sixty-third+one ninety-ninth.

= 1/( 1×3)+ 1/(3×5)+ 1/(5×7)+ 1/(7×9)+ 1/(9× 1 1)

= 1/2×( 1- 1/3)+ 1/2×( 1/3- 1/5)+ 1/2×( 1/5- 1/7)+ 1/2×( 1/7- 1/9) + 1/2×( 1/9- 1/ 1 1)

= 1/2×( 1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/ 1 1)

= 1/2×( 1- 1/ 1 1)

= 1/2× 10/ 1 1

=5/ 1 1

One and a half+seventy-two-ninety-twenty+eleven-thirty-thirteen-forty-two+fifty-six.

(According to the prompt, 1 and 1/2= 1+ 1/2,+1/3 = 5/6 ...)

=( 1+ 1/2)-( 1/2+ 1/3)+( 1/3+ 1/4)-( 1/4+ 1/5)+( 1/5+ 1/6)-( 1/6+ 1/7)+( 1/7+ 1/8)

= 1+ 1/2 - 1/2 - 1/3 + 1/3 + 1/4 - 1/4 - 1/5 + 1/5 + 1/6 - 1/6 - 1/7 + 1/7 + 1/8

= 1+ 1/8

=9/8

1.7+ 150 +3.98 17.625-(4.4+58 ) 3.35×6.47×2+6.7×3.6

125×(8+0.8+0.08) 9 10 ×(56 -49 ÷45 )

2.8×0.4+ 10.2÷3.4 35 ÷〔78 -(25 +38 )〕 18.7-3.375-6.625

(9+92+93)×0.0 1 13.5×[ 1.5×( 1.07+ 1.93)]