練習問題

1章 正の数と負の数

4.四則の混じった計算


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(問1)
(1) (−4)×(+3)−(−3)
(2) −12+(−2)×(−6)
(3) (−6)×3−(−2)×8
(4) 16÷(−4)−3
(5) −14−(−21)÷7
(6) (−15)÷3−(−24)÷2


(1) (−4)×(+3)−(−3)
 =−12+3
 =−9

(2) −12+(−2)×(−6)
 =−12+12
 =0

(3) (−6)×3−(−2)×8
 =−18−(−16)
 =−18+16
 =−2

(4) 16÷(−4)−3
 =−4−3
 =−7

(5) −14−(−21)÷7
 =−14−(−3)
 =−14+3
 =−11

(6) (−15)÷3−(−24)÷2
 =−5−(−12)
 =−5+12
 =7


(問2)
(1) \(\left( -\dfrac {2} {3} \right) \times \left( -\dfrac {5} {9} \right) -\dfrac {7} {27}\)
(2) \(\left( -\dfrac {3} {4} \right) \times \dfrac {5} {6}  -\dfrac {1} {4}\)
(3) \(\left( -\dfrac {1} {2} \right) \times \left( -\dfrac {6} {5} \right) -1.3\)
(4) \( -\dfrac {1} {6}  - \left( -\dfrac {3} {10} \right) \times 5\)
(5) \( -3  \times \dfrac {7} {12}  -\dfrac {2} {3} \times \left( -\dfrac {1} {4} \right) \)
(6) \( \dfrac {1} {2}  \times \left( -\dfrac {1} {3} \right) -\dfrac {2} {3} \times \left( -\dfrac {1} {4} \right) \)


(1) \(\left( -\dfrac {2} {3} \right) \times \left( -\dfrac {5} {9} \right) -\dfrac {7} {27}\)
 \(=\dfrac {10} {27} -\dfrac {7} {27} \)
 \(=\dfrac {3} {27} \)
 \(=\dfrac {1} {9} \)

(2) \(\left( -\dfrac {3} {4} \right) \times \dfrac {5} {6}  -\dfrac {1} {4}\)
 \(=\left( -\dfrac {1} {4} \right) \times \dfrac {5} {2}  -\dfrac {1} {4}\)
 \(-\dfrac {5} {8} -\dfrac {2} {8}\)
 \(-\dfrac {7} {8}\)

(3) \(\left( -\dfrac {1} {2} \right) \times \left( -\dfrac {6} {5} \right) -1.3\)
 \(=\dfrac {6} {10} -\dfrac {13} {10}\)
 \(=-\dfrac {7} {10}\)

(4) \( -\dfrac {1} {6}  - \left( -\dfrac {3} {10} \right) \times 5\)
 \(=-\dfrac {1} {6} +\dfrac {3} {2}\)
 \(=-\dfrac {1} {6} +\dfrac {9} {6}\)
 \(=\dfrac {8} {6}\)
 \(=\dfrac {4} {3}\)

(5) \( -3  \times \dfrac {7} {12}  -\dfrac {2} {3} \times \left( -\dfrac {1} {4} \right) \)
 \(=-\dfrac {21} {12} +\dfrac {2} {12}\)
 \(=-\dfrac {19} {12}\)

(6) \( \dfrac {1} {2}  \times \left( -\dfrac {1} {3} \right) -\dfrac {2} {3} \times \left( -\dfrac {1} {4} \right) \)
 \(=-\dfrac {1} {6} +\dfrac {1} {6}\)
 \(=0\)



(問3)
(1) \(\left( -\dfrac {1} {5} \right) \div \dfrac {4} {25}  +\dfrac {3} {4}\)
(2) \( -\dfrac {2} {3} -\left( -\dfrac {1} {2}  \right) \div \left( -\dfrac {1} {3} \right) \)
(3) \( \dfrac {2} {3} \div \left( -\dfrac {1} {2}  \right) -\dfrac {1} {4} \div \left( -\dfrac {3} {8} \right) \)
(4) \( \left( -3 \right) \div \dfrac {3} {5} + \left( -\dfrac {3} {4}  \right) \div \left( -\dfrac {5} {12} \right) \)
(5) \( -\dfrac {2} {3}  \div  2 + \dfrac {8} {9} \times \left( -\dfrac {1} {2}  \right) \)
(6) \( \dfrac {3} {5}  \times \dfrac {3} {4} - \left( -\dfrac {2} {7}  \right)\div \left( -\dfrac {8} {21}  \right) \)


(1) \(\left( -\dfrac {1} {5} \right) \div \dfrac {4} {25}  +\dfrac {3} {4}\)
 \(=\left( -\dfrac {1} {5} \right) \times \dfrac {25} {4}  +\dfrac {3} {4}\)
 \(=-\dfrac {5} {4}  +\dfrac {3} {4}\)
 \(=-\dfrac {2} {4}\)
 \(=-\dfrac {1} {2}\)

(2) \( -\dfrac {2} {3} -\left( -\dfrac {1} {2}  \right) \div \left( -\dfrac {1} {3} \right) \)
 \( =-\dfrac {2} {3} -\left( -\dfrac {1} {2}  \right) \times \left( -3\right) \)
 \( =-\dfrac {2} {3} -\dfrac {3} {2}  \)
 \( =-\dfrac {4} {6} -\dfrac {9} {6}  \)
 \( =-\dfrac {13} {6} \)

(3) \( \dfrac {2} {3} \div \left( -\dfrac {1} {2}  \right) -\dfrac {1} {4} \div \left( -\dfrac {3} {8} \right) \)
 \( =\dfrac {2} {3} \times \left( -2 \right) -\dfrac {1} {4} \times \left( -\dfrac {8} {3} \right) \)
 \( =-\dfrac {4} {3}  +\dfrac {2} {3} \)
 \( =-\dfrac {2} {3} \)

(4) \( \left( -3 \right) \div \dfrac {3} {5} + \left( -\dfrac {3} {4}  \right) \div \left( -\dfrac {5} {12} \right) \)
 \( =\left( -3 \right) \times \dfrac {5} {3} + \left( -\dfrac {3} {4}  \right) \times \left( -\dfrac {12} {5} \right) \)
 \(=-5+\dfrac {9} {5}\)
 \(=-\dfrac {25} {5}+\dfrac {9} {5}\)
 \(=-\dfrac {16} {5}\)
(5) \( -\dfrac {2} {3}  \div  2 + \dfrac {8} {9} \times \left( -\dfrac {1} {2}  \right) \)
 \( =-\dfrac {2} {3}  \times \dfrac {1} {2}  -\dfrac {4} {9} \)
 \( =-\dfrac {1} {3}  -\dfrac {4} {9} \)
 \( =-\dfrac {3} {9}  -\dfrac {4} {9} \)
 \( =-\dfrac {7} {9} \)

(6) \( \dfrac {3} {5}  \times \dfrac {3} {4} - \left( -\dfrac {2} {7}  \right)\div \left( -\dfrac {8} {21}  \right) \)
 \( =\dfrac {9} {20}  - \left( -\dfrac {2} {7}  \right)\times \left( -\dfrac {21} {8}  \right) \)
 \( =\dfrac {9} {20}  -\dfrac {3} {4} \)
 \( =\dfrac {9} {20}  -\dfrac {15} {20} \)
 \( =-\dfrac {6} {20} \)
 \( =-\dfrac {3} {10} \)


(問4)
(1) \(5\times \{ -4-\left( 3-11\right) \} \)
(2) \(16\div \{ -3+\left( 4-9\right) \} \)
(3) \(-14\div \left(-7-5\right)+ \left(-12\right)\div \left(-4-2\right) \)
(4) \(-3 \times \left( \dfrac {2} {7} -\dfrac {2} {3} \right) \)
(5) \(1.1 \div \left( \dfrac {2} {7} -\dfrac {3} {5} \right) \)
(6) \( \left( -\dfrac {1} {8} +\dfrac {3} {4} \right) \div \left( -\dfrac {5} {12} \right) \)


(1) \(5\times \{ -4-\left( 3-11\right) \} \)
 \(=5\times \{ -4-\left( -8\right) \} \)
 \(=5\times \left( -4+8\right) \)
 \( =5\times 4\)
 \( =20\)

(2) \(16\div \{ -3+\left( 4-9\right) \} \)
 \( =16\div \{ -3+\left( -5\right) \} \)
 \( =16\div \left( -8\right) \)
 \( =-2\)

(3) \(-14\div \left(-7-5\right)+ \left(-12\right)\div \left(-4-2\right) \)
 \(=-14 \div \left(-12\right)+ \left(-12\right)\div \left(-6\right) \)
 \( =\dfrac {7} {6}  + 2\)
 \( =\dfrac {7} {6}  + \dfrac {12} {6}\)
 \( =\dfrac {19} {6}\)

(4) \(-3 \times \left( \dfrac {2} {7} -\dfrac {2} {3} \right) \)
 \(=-3 \times \left( \dfrac {6} {21} -\dfrac {14} {21} \right) \)
 \(=-3 \times \left( -\dfrac {8} {21} \right) \)
 \( =\dfrac {24} {21}\)
 \( =\dfrac {8} {7}\)

(5) \(1.1 \div \left( \dfrac {2} {7} -\dfrac {3} {5} \right) \)
 \( =\dfrac {11} {10}  \div \left( \dfrac {10} {35} -\dfrac {21} {35} \right) \)
 \( =\dfrac {11} {10}  \div \left( -\dfrac {11} {35} \right) \)
 \( =\dfrac {11} {10}  \times \left( -\dfrac {35} {11} \right) \)
 \( =-\dfrac {7} {2}\)

(6) \( \left( -\dfrac {1} {8} +\dfrac {3} {4} \right) \div \left( -\dfrac {5} {12} \right) \)
 \( =\left( -\dfrac {1} {8} +\dfrac {6} {8} \right) \times \left( -\dfrac {12} {5} \right) \)
 \( =\dfrac {5} {8}  \times \left( -\dfrac {12} {5} \right) \)
 \( =-\dfrac {3} {2}\)


(問5)
(1) \( -4^{2}  \div 8 - \left(-3\right) \)
(2) \( 3^{2} +8\times \left(-2^{2} \right)\)
(3) \(\left(-2\right)^{4} +\left(-12\right)\div \dfrac {6} {5}\)
(4) \(\left(-2^{2} \right)-\left(-3\right)^{2}  \times \left(-2\right)\)
(5) \(7^{2} +2\times \left(-5^{2} \right)\)
(6) \(3^{3} +\left(-3\right)^2 \times \left(-2\right)^3\)


(1) \( -4^{2}  \div 8 - \left(-3\right) \)
 \(=-16\div 8+3\)
 \(=-2+3\)
 \(=1\)

(2) \( 3^{2} +8\times \left(-2^{2} \right)\)
 \(=9+8\times \left(-4\right)\)
 \(=9+\left(-32\right)\)
 \(=-23\)

(3) \(\left(-2\right)^{4} +\left(-12\right)\div \dfrac {6} {5}\)
 \(=16+\left(-12\right)\times \dfrac {5} {6}\)
 \(=16+\left(-10\right)\)
 \(=6\)

(4) \(\left(-2^{2} \right)-\left(-3\right)^{2}  \times \left(-2\right)\)
 \(=-4-9\times \left(-2\right)\)
 \(=-4+18\)
 \(=14\)

(5) \(7^{2} +2\times \left(-5^{2} \right)\)
 \(=49+2\times \left(-25\right)\)
 \(=49-50\)
 \(=-1\)

(6) \(3^{3} +\left(-3\right)^2 \times \left(-2\right)^3\)
 \(=27+9\times \left(-8\right)\)
 \(=27-72\)
 \(=-45\)


(問6)
(1) \(\left(\dfrac {1} {2} \right)^3 -\left(-\dfrac {1} {4} \right)^2 \)
(2)  \(2-\left(-\dfrac {4} {3} \right)^2 \div \dfrac {2} {3} \)
(3) \(\left(-4\right)^3 \div \dfrac {4} {3} -3 \times \left(-2^4 \right) \)
(4) \(\left(-\dfrac {8} {9} \right)\times \left(-0.75\right)^2 -0.25^2 \)
(5) \(\left(-0.125\right)\div \left(-0.5\right)^3 -1.25^2 \)
(6) \(\left(-0.375\right)^2 \div 0.25^2 -0.75^2 \)


(1) \(\left(\dfrac {1} {2} \right)^3 -\left(-\dfrac {1} {4} \right)^2 \)
 \(=\dfrac {1} {8} -\dfrac {1} {16} \)
 \(=\dfrac {2} {16} -\dfrac {1} {16} \)
 \(=\dfrac {1} {16} \)

(2)  \(2-\left(-\dfrac {4} {3} \right)^2 \div \dfrac {2} {3} \)
 \(=2-\dfrac {16} {9} \times \dfrac {3} {2} \)
 \(=2-\dfrac {8} {3}\)
 \(=\dfrac {6} {3} -\dfrac {8} {3}\)
 \(=-\dfrac {2} {3}\)

(3) \(\left(-4\right)^3 \div \dfrac {4} {3} -3\times \left(-2^4 \right) \)
 \(\left(-4\right)^3 \times \dfrac {3} {4} -3\times \left(-16\right)\)
 \(=-48+48\)
 \(=0\)

(4) \(\left(-\dfrac {8} {9} \right)\times \left(-0.75\right)^2 -0.25^2 \)
 \(=\left(-\dfrac {8} {9} \right)\times \left(-\dfrac {3} {4} \right)^2-\left(\dfrac {1} {4} \right)^2\)
 \(=\left(-\dfrac {8} {9} \right)\times \dfrac {9} {16} -\dfrac {1} {16} \)
 \(=-\dfrac {1} {2} -\dfrac {1} {16} \)
 \(=-\dfrac {8} {16} -\dfrac {1} {16} \)
 \(=-\dfrac {9} {16} \)

(5) \(\left(-0.125\right)\div \left(-0.5\right)^3 -1.25^2 \)
 \(=\left(-\dfrac {1} {8} \right) \div \left(-\dfrac {1} {2} \right)^3 -\left(\dfrac {5} {4} \right)^2\)
 \(=\left(-\dfrac {1} {8} \right) \times \left(-2\right)^3 -\dfrac {25} {16} \)
 \(=\left(-\dfrac {1} {8} \right) \times \left(-8\right) -\dfrac {25} {16} \)
 \(=1-\dfrac {25} {16} \)
 \(=-\dfrac {9} {16} \)

(6) \(\left(-0.375\right)^2 \div 0.25^2 -0.75^2 \)
 \(=\left(-\dfrac {3} {8} \right)^2 \div \left(\dfrac {1} {4} \right)^2-\left(\dfrac {3} {4} \right)^2\)
 \(=\left(-\dfrac {3} {8} \right)^2 \times 4^2 -\left(\dfrac {3} {4} \right)^2\)
 \(=\dfrac {9} {64} \times 16-\dfrac {9} {16}\)
 \(=\dfrac {9} {4} -\dfrac {9} {16}\)
 \(=\dfrac {36} {16} -\dfrac {9} {16}\)
 \(=\dfrac {27} {16}\)


(問7)
(1) \(\left(-12\right)\times \left(\dfrac {7} {6} -\dfrac {7} {4} \right)\)
(2) \(24\times \left(\dfrac {1} {6} +\dfrac {3} {2} -\dfrac {9} {8} \right)\)
(3) \(\left(-7\right)\times 26+\left(-7\right)\times \left(-6\right)\)
(4) \(19\times 3.1415-3.1415\times 9\)


(1) \(\left(-12\right)\times \left(\dfrac {7} {6} -\dfrac {7} {4} \right)\)
 \(=\left(-12\right)\times \dfrac {7} {6} -\left(-12\right)\times \dfrac {7} {4}\)
 \(=-14-\left(-21\right)\)
 \(=-14+21\)
 \(=7\)

(2) \(24\times \left(\dfrac {1} {6} +\dfrac {3} {2} -\dfrac {9} {8} \right)\)
 \(=24\times \dfrac {1} {6} +24\times \dfrac {3} {2} -24\times \dfrac {9} {8} \)
 \(=4+36-27\)
 \(=13\)

(3) \(\left(-7\right)\times 26+\left(-7\right)\times \left(-6\right)\)
 \(=\left(-7\right)\times \{26+ \left(-6\right)\} \)
 \(=\left(-7\right)\times 20\)
 \(=-140\)

(4) \(19\times 3.1415-3.1415\times 9\)
 \(=3.1415\times \left(19-9\right)\)
 \(=3.1415\times 10\)
 \(=31.415\)



   





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