算数【基本】ばらす・まとめる計算

問1

分配法則を使って、式をまとめてみましょう。

例)2 × 5 + 2 × 7 = 2 × (5 + 7)

(1)5 × 82 + 48 × 5 – 5 × 30 =

(2)18 + 18 × 9 =

(3)25 × 28 – 25 × 27 =

(4)12 × 63 + 3 × 4 × 37 =

(5)96 – 3 × 8 + 4 × 9 =

答え
(1)5 × (82 + 48 – 30)
(2)18 × (1 + 9)
(3)25 × (28 – 27)
(4)12 × (63 + 37)
(5)12 × (8 – 2 + 3)
解き方
(1)5 × 82 + 48 × 55 × 30 = 5 × 82 + 5 × 485 × 30 = 5 × (82 + 48 – 30)
※48 × 5 = 5 × 48(交換法則)
(2)18 + 18 × 9 = 18 × 1 + 18 × 9 = 18 × (1 + 9)
(3)25 × 28 – 25 × 27 = 25 × (28 – 27)
(4)12 × 63 + 3 × 4 × 37 = 12 × 63 + 12 × 37 = 12 × (63 + 37)
(5)963 × 8 + 4 × 9 = 12 × 83 × 4 × 2 + 4 × 3 × 3 = 12 × 8 – 12 × 2 + 12 × 3 = 12 × (8 – 2 + 3)

※まとめることで、複雑な計算が簡単な計算になる場合があります。

問2

次の計算をしなさい。

(1)777 × 20 + 777 × 80 =

(2)13 × 1331 – 13 × 331 =

(3)37 × 76 + 37 × 68 – 44 × 37 =

(4)89 × 101 – 89 =

(5)9 × 22 + 9 × 78 =

答え
(1)77700
(2)13000
(3)3700
(4)8900
(5)900
解き方
(1)777 × 20 + 777 × 80 = 777 × (20 + 80) = 777 × 100 = 77700
(2)13 × 1331 – 13 × 331 = 13 × (1331 – 331) = 13 × 1000 = 13000
(3)37 × 76 + 37 × 68 – 44 × 37 = 37 × (76 + 68 – 44) = 37 × 100 = 3700
(4)89 × 101 – 89 = 89 × (101 – 1) = 89 × 100 = 8900
(5)9 × 22 + 9 × 78 = 9 × (22 + 78) = 9 × 100 = 900

問3

次の計算をしなさい。

(1)625 × 16 =

(2)75 × 32 – 25 × 36 =

(3)625 × 8 – 125 × 16 + 250 × 4 =

(4)25 × 32 × 625 =

(5)125 × 35 – 375 × 9 =

ヒント
数をばらして計算しやすい順番に並べかえたり、計算しやすいようにまとめましょう!
答え
(1)10000
(2)1500
(3)4000
(4)500000
(5)1000
解き方
(1)625 × 16 = 25 × 25 × 4 × 4 = 100 × 100 = 10000
(2)75 × 32 – 25 × 36 = 3 × 25 × 4 × 8 – 25 × 4 × 9 = 24 × 100 – 100 × 9 = 1500
(3)625 × 8 – 125 × 16 + 250 × 4 = 125 × 5 × 8 – 125 × 8 × 2 + 1000 = 1000 × (5 – 2) + 1000 = 4000
(4)25 × 32 × 625 = 25 × 4 × 8 × 125 × 5 = 100 × 1000 × 5 = 500000
(5)125 × 35 – 375 × 9 = 125 × 35 – 125 × 3 × 9 = 125 × (35 – 27) = 125 × 8 = 1000

※25 × 4 = 100、125 × 8 = 1000はもちろん、その他の25の倍数、125の倍数も覚えておきましょう!
125 = 25 × 5
625 = 25 × 25 = 125 × 5

問4

47 × 48 + 53 × 48 =

答え
4800
解き方
47 × 48 + 53 × 48
= 48 × ( 47 + 53 )
= 48 × 100 = 4800

問5

25 × 70 + 250 + 23 =

答え
2023
解き方
25 × 70 + 250 + 23
= 250 × 7 + 250 + 23
= 250 × ( 7 + 1 ) + 23
= 250 × 8 + 23
= 2000 + 23 = 2023

問6

225 – { 90 – 3 × ( 31 – 6 ) } × 7 =

答え
120
解き方
225 – { 90 – 3 × ( 31 – 6 ) } × 7
= 225 – ( 90 – 3 × 25 ) × 7
= 15 × 15 – 15 × 7
= 15 × ( 15 – 7 )
= 15 × 8 = 120

問7

2 + ( 5 × 5 + 3 × 5 ) ÷ 5 =

答え
10
解き方
2 + ( 5 × 5 + 3 × 5 ) ÷ 5
= 2 + { ( 5 + 3 ) × 5 } ÷ 5
= 2 + 8 = 10

問8

( 90+91+92+93+94+95+96+97+98+99 ) ÷ 45 =
答え
21
解き方
( 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 ) ÷ 45
= { 90 × 10 + ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 ) } ÷ 45
= ( 45 × 2 × 10 + 45 ) ÷ 45
= ( 20 + 1 ) × 45 ÷ 45 = 21

問9

43 × 7 – 43 × 4 – 43 × 1 =

答え
86
解き方
43 × 7 – 43 × 4 – 43 × 1
= 43 × ( 7 – 4 – 1 )
= 43 × 2 = 86

問10

31 × 7 – 5 × 31 + 2 × 31 × 9 =

答え
620
解き方
31 × 7 – 5 × 31 + 2 × 31 × 9
= 31 × ( 7 – 5 + 18 )
= 31 × 20 = 620

問11

138 – ( 18 + 3 ) × 6 =

答え
12
解き方
138 – ( 18 + 3 ) × 6
= 6 × 23 – 21 × 6
= 6 × ( 23 – 21 )
= 6 × 2 = 12

問12

250 × 9 – 25 × 50 + 23 =

答え
1023
解き方
250 × 9 – 25 × 50 + 23
= 250 × 9 – 250 × 5 + 23
= 250 × ( 9 – 5 ) + 23
= 250 × 4 + 23
= 1000 + 23 = 1023

問13

( 1 + 12 + 123 + 1234 ) ÷ ( 1 + 2 + 3 + 4 ) =

答え
137
解き方
( 1 + 12 + 123 + 1234 ) ÷ ( 1 + 2 + 3 + 4 )
= { ( 1 + 2 + 3 + 4 ) + ( 10 + 120 + 1230 ) } ÷ 10
= { 10 + 10 × ( 1 + 12 + 123 ) } ÷ 10
= 1 + 136 = 137

問14

921 × 1515 + 921 × 1517 – 32 × 921 =

答え
2763000
解き方
921 × 1515 + 921 × 1517 – 32 × 921
= 921 × ( 1515 + 1517 – 32 )
= 921 × 3000 = 2763000

問15

3021 × 10 + 3021 × 7 + 3021 × 13 =

答え
90630
解き方
3021 × 10 + 3021 × 7 + 3021 × 13
= 3021 × ( 10 + 7 + 13 )
= 3021 × 30 = 90630

問16

4017 × 16 – 25 × 16 + 8 × 16 =

答え
64000
解き方
4017 × 16 – 25 × 16 + 8 × 16
= 16 × ( 4017 – 25 + 8 )
= 16 × 4000 = 64000

問17

374 × 374 + 374 × 626 =

答え
374000
解き方
374 × 374 + 374 × 626
= 374 × ( 374 + 626 )
= 374 × 1000 = 374000

問18

51 × 3 – 255 ÷ 5 =

答え
102
解き方
51 × 3 – 255 ÷ 5
= 51 × 3 – 51
= 51 × ( 3 – 1 )
= 51 × 2 = 102

問19

21 ÷ 4 × 8 – 21 =

答え
21
解き方
21 ÷ 4 × 8 – 21
= 21 × 2 – 21
= 21 × ( 2 – 1 ) = 21

問20

52 × 3.14 + 48 × 3.14 =

答え
314
解き方
52 × 3.14 + 48 × 3.14
= ( 52 + 48 ) × 3.14
= 100 × 3.14 = 314

問21

47 × 6.87 – 47 × 3.23 – 47 × 2.64 =

答え
47
解き方
47 × 6.87 – 47 × 3.23 – 47 × 2.64
= 47 × ( 6.87 – 3.23 – 2.64 )
= 47 × 1 = 47

問22

\(\dfrac{1}{74}\)×\(\dfrac{1}{2}\) + \(\dfrac{1}{74}\)×\(\dfrac{1}{3}\) – \(\dfrac{1}{74}\)×\(\dfrac{1}{4}\) + \(\dfrac{1}{74}\)×\(\dfrac{1}{5}\) – \(\dfrac{1}{74}\)×\(\dfrac{1}{6}\) =
答え
\(\dfrac{1}{120}\)
解き方
\(\dfrac{1}{74}\) × \(\dfrac{1}{2}\) + \(\dfrac{1}{74}\) × \(\dfrac{1}{3}\) – \(\dfrac{1}{74}\) × \(\dfrac{1}{4}\) + \(\dfrac{1}{74}\) × \(\dfrac{1}{5}\) – \(\dfrac{1}{74}\) × \(\dfrac{1}{6}\)
= \(\dfrac{1}{74}\) × ( \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) – \(\dfrac{1}{4}\) + \(\dfrac{1}{5}\) – \(\dfrac{1}{6}\) )
= \(\dfrac{1}{74}\) × ( \(\dfrac{1}{2}\) + \(\dfrac{1}{12}\) + \(\dfrac{1}{30}\) )
= \(\dfrac{1}{74}\) × \(\dfrac{30\ +\ 5\ +\ 2}{60}\)
= \(\dfrac{1}{74}\) × \(\dfrac{37}{60}\) = \(\dfrac{1}{120}\)

問23

\(\dfrac{1}{3\ ×\ 4}\) + \(\dfrac{1}{4\ ×\ 5}\) + \(\dfrac{1}{5\ ×\ 6}\) =

答え
\(\dfrac{1}{6}\)
解き方
\(\dfrac{1}{3\ ×\ 4}\) + \(\dfrac{1}{4\ ×\ 5}\) + \(\dfrac{1}{5\ ×\ 6}\)
= \(\dfrac{1}{3}\) – \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) – \(\dfrac{1}{5}\) + \(\dfrac{1}{5}\) – \(\dfrac{1}{6}\)
= \(\dfrac{1}{3}\) – \(\dfrac{1}{6}\)
= \(\dfrac{2\ -\ 1}{6}\) = \(\dfrac{1}{6}\)

問24

91 × 46 + 37 × 107 – 37 × 7 + 46 × 9 =

答え
8300
解き方
91 × 46 + 37 × 107 – 37 × 7 + 46 × 9
= 46 × ( 91 + 9 ) + 37 × ( 107 – 7 )
= 46 × 100 + 37 × 100
= ( 46 + 37 ) × 100
= 83 × 100 = 8300

問25

3.14 × 5.26 + 31.4 × 0.472 + 314 × 0.0002 =

答え
31.4
解き方
3.14 × 5.26 + 31.4 × 0.472 + 314 × 0.0002
= 3.14 × 5.26 + 3.14 × 4.72 + 3.14 × 0.02
= 3.14 × ( 5.26 + 4.72 + 0.02 )
= 3.14 × 10 = 31.4