Simplify the following:
15+32/8×7-80/(22-(10-8)×2+14-(1)/5×4)
(-1)/5×4 = (-4)/5:
15+32/8×7-80/(22-(10-8)×2+14+(-4)/5)
32/8×7 = (32×7)/8:
15+(32×7)/8-80/(22-(10-8)×2+14-(4)/5)
32/8 = (8×4)/8 = 4:
15+4×7-80/(22-(10-8)×2+14-(4)/5)
10-8 = 2:
15+4×7-80/(22-2×2+14-(4)/5)
-2×2 = -4:
15+4×7-80/(22+-4+14-(4)/5)
Put 22-4+14-4/5 over the common denominator 5. 22-4+14-4/5 = (5×22)/5+(5 (-4))/5+(5×14)/5-4/5:
15+4×7-80/(5×22)/5+(5 (-4))/5+(5×14)/5-4/5
5×22 = 110:
15+4×7-80/(110/5+(5 (-4))/5+(5×14)/5-4/5)
5 (-4) = -20:
15+4×7-80/(110/5+-20/5+(5×14)/5-4/5)
5×14 = 70:
15+4×7-80/(110/5-20/5+70/5-4/5)
110/5-(20)/5+70/5-(4)/5 = (110-20+70-4)/5:
15+4×7-80/(110-20+70-4)/5
110-20+70-4 = (110+70)-(20+4):
15+4×7-80/((110+70)-(20+4)/5)
| 1 | 1 | 0
+ | | 7 | 0
| 1 | 8 | 0:
15+4×7-80/((180-(20+4))/5)
20+4 = 24:
15+4×7-80/((180-24)/5)
| | 7 | 10
| 1 | 8 | 0
- | | 2 | 4
| 1 | 5 | 6:
15+4×7-80/(156/5)
Multiply the numerator of -80/(156/5) by the reciprocal of the denominator. -80/(156/5) = (-80×5)/156:
15+4×7+(-80×5)/156
The gcd of -80 and 156 is 4, so (-80×5)/156 = ((4 (-20)) 5)/(4×39) = 4/4×(-20×5)/39 = (-20×5)/39:
15+4×7+(-20×5)/39
-20×5 = -100:
15+4×7+-100/39
4×7 = 28:
15+28-(100)/39
Put 15+28-100/39 over the common denominator 39. 15+28-100/39 = (39×15)/39+(39×28)/39-100/39:
(39×15)/39+(39×28)/39-100/39
| 3 | 9
× | 1 | 5
1 | 9 | 5
3 | 9 | 0
5 | 8 | 5:
585/39+(39×28)/39-100/39
| | 3 | 9
× | | 2 | 8
| 3 | 1 | 2
| 7 | 8 | 0
1 | 0 | 9 | 2:
585/39+1092/39-100/39
585/39+1092/39-(100)/39 = (585+1092-100)/39:
(585+1092-100)/39
| | 1 | |
| 1 | 0 | 9 | 2
+ | | 5 | 8 | 5
| 1 | 6 | 7 | 7:
1677-100/39
| 1 | 6 | 7 | 7
- | | 1 | 0 | 0
| 1 | 5 | 7 | 7:
Answer: 1577/39=40 17/39
