Unit 22
Darab
Sebuah kedai alat tulis ada 18 ikat pensel. Setiap ikat ada 10 batang pensel.
Berapakah jumlah pensel?
乘法 = darab；积 = hasil darab
每盒数量一样多、有好几盒时，用乘法比一次一次相加快得多。这个单元要学乘以 10、100、1 000，一位数乘五位数，两位数乘两位数、三位数和四位数，积不超过 100 000。
例子：6 盒、每盒 10 个 → 6 × 10 = 60 个。
注意：setiap（每）只告诉我们每组一样多；要看题目是求总数（用乘法），还是求每组有多少（用除法）。
multiplication = darab; product = hasil darab
When there are several groups of the same size, multiplying is much quicker than adding again and again. In this unit we multiply by 10, 100 and 1 000, a one-digit number by a five-digit number, and a two-digit number by two-, three- and four-digit numbers, with products up to 100 000.
Example: 6 boxes of 10 each make 6 × 10 = 60.
Watch out: setiap only tells us every group is the same size; decide whether the question wants the total (multiply) or the size of each group (divide).
Darab dengan 10
18 × 10
Cara 1: blok · Cara 2: tambah satu sifar
Cara 1
18 puluh = 100 + 80
= 180
= 180
Cara 2
18 × 1 = 18
18 × 10 = 180
18 × 10 = 180
乘以 10 = darab dengan 10
一个数乘 10，就是有这么多个“十”：18 × 10 = 18 个十 = 180。每个数字都往左移一位，个位补上一个 0。所以快速的方法是：在原来的数后面写一个 0。
例子：318 × 10 = 3 180。
注意：补的 0 放在个位，不是放在最前面。
multiply by 10 = darab dengan 10
Multiplying by 10 gives that many tens: 18 × 10 = 18 tens = 180. Each digit moves one place to the left and a 0 fills the ones place. So the quick way is to write the number and put one 0 after it.
Example: 318 × 10 = 3 180.
Watch out: the 0 goes in the ones place, at the end, not at the front.
Darab dengan gandaan 10
30 × 26
Cara 1: bentuk lazim · Cara 2: × 10, kemudian × 3
Cara 1
| 2 | 6 | ||
| × | 3 | 0 | |
| 0 | |||
| + | 7 | 8 | 0 |
| 7 | 8 | 0 |
Cara 2
30 = 10 × 3
26 × 10 = 260
260 × 3 = 780
26 × 10 = 260
260 × 3 = 780
乘整十数 = darab dengan gandaan 10
30 可以看成 10 × 3。方法一用竖式，先乘个位 0 得 0，再乘十位 3 得 780，两行相加。方法二先把 26 乘 10 得 260，再乘 3 得 780。两种方法一样。
例子：40 × 17 → 17 × 10 = 170，170 × 4 = 680。
注意：乘整十数时，答案的个位一定是 0。
multiply by a multiple of 10 = darab dengan gandaan 10
30 can be seen as 10 × 3. Method 1 uses the column method: multiply by the ones digit 0, then by the tens digit 3 to get 780, and add the rows. Method 2 multiplies 26 by 10 to get 260, then by 3 to get 780. Both give the same answer.
Example: 40 × 17: 17 × 10 = 170, then 170 × 4 = 680.
Watch out: when multiplying by a multiple of 10, the answer always ends in 0.
Darab dengan 100 dan 1 000
Lihat polanya
72 × 10 · 72 × 100 · 72 × 1 000
72 × 10 = 720
72 × 100 = 7 200
72 × 1 000 = 72 000
72 × 100 = 7 200
72 × 1 000 = 72 000
乘 100 和 1 000 = darab dengan 100 dan 1 000
乘 10 补一个 0，乘 100 补两个 0，乘 1 000 补三个 0。乘数里有几个 0，就在原来的数后面补几个 0。
例子：356 × 100 = 35 600；48 × 1 000 = 48 000。
注意：原来的数末尾本来就有 0 的，也要照样再补：60 × 100 = 6 000。
multiply by 100 and 1 000 = darab dengan 100 dan 1 000
Times 10 adds one 0, times 100 adds two 0s and times 1 000 adds three 0s. Count the zeros in the multiplier and write that many after the number.
Example: 356 × 100 = 35 600; 48 × 1 000 = 48 000.
Watch out: if the number already ends in 0, still add the zeros: 60 × 100 = 6 000.
Peta pemikiran
Cari nombor yang tidak diketahui
? × 10 = ? × 100 = ? × 1 000 = 72 000
7 200 × 10 = 72 000
720 × 100 = 72 000
72 × 1 000 = 72 000
720 × 100 = 72 000
72 × 1 000 = 72 000
倒推乘法 = cari nombor yang didarab
已知积是 72 000，乘 10 前的数要少一个 0，乘 100 前的数要少两个 0，乘 1 000 前的数要少三个 0。这样可以很快找出空格里的数。
例子：□ × 100 = 5 300 → □ = 53。
注意：只能去掉末尾的 0，中间的 0 不能动。
find the number that was multiplied = cari nombor yang didarab
The product is 72 000. Before multiplying by 10 the number had one 0 fewer; before × 100, two 0s fewer; before × 1 000, three 0s fewer. This finds the missing numbers quickly.
Example: □ × 100 = 5 300, so □ = 53.
Watch out: only remove zeros from the end; zeros in the middle stay.
Satu digit × lima digit
3 × 12 406
Cara 1: tambah berulang · Cara 2: bentuk lazim
Cara 1
| 1 | 1 | ||||
| 1 | 2 | 4 | 0 | 6 | |
| 1 | 2 | 4 | 0 | 6 | |
| + | 1 | 2 | 4 | 0 | 6 |
| 3 | 7 | 2 | 1 | 8 |
Cara 2
| 1 | 1 | ||||
| 1 | 2 | 4 | 0 | 6 | |
| × | 3 | ||||
| 3 | 7 | 2 | 1 | 8 |
一位数乘五位数 = darab satu digit dengan lima digit
乘法是重复加法，3 × 12 406 就是三个 12 406 相加。竖式乘法更快：从个位乘起，6 × 3 = 18 写 8 进 1；十位 0 × 3 + 1 = 1；百位 4 × 3 = 12 写 2 进 1；依次乘到万位。
例子：4 × 21 305 = 85 220。
注意：进上来的数要在乘完以后再加。
one digit times five digits = darab satu digit dengan lima digit
Multiplication is repeated addition: 3 × 12 406 is three lots of 12 406 added together. The column method is quicker: start at the ones, 6 × 3 = 18, write 8 carry 1; tens 0 × 3 + 1 = 1; hundreds 4 × 3 = 12, write 2 carry 1; and so on to the ten thousands.
Example: 4 × 21 305 = 85 220.
Watch out: add the carried number after you multiply, not before.
Dua digit × dua digit
36 × 57
Cara 1: darab sa, darab puluh, jumlahkan
两位数乘两位数 = darab dua digit dengan dua digit
先用个位 6 去乘 57，得 342；再用十位 3（其实是 30）去乘 57，得 1 710，所以第二行末尾有 0。最后把两行相加：342 + 1 710 = 2 052。
例子：26 × 14 → 26 × 4 = 104，26 × 10 = 260，104 + 260 = 364。
注意：第二行一定要补 0，因为乘的是几十。
two digits times two digits = darab dua digit dengan dua digit
First multiply 57 by the ones digit 6 to get 342. Then multiply 57 by the tens digit, which is really 30, to get 1 710, so the second row ends in 0. Add the rows: 342 + 1 710 = 2 052.
Example: 26 × 14: 26 × 4 = 104, 26 × 10 = 260, and 104 + 260 = 364.
Watch out: the second row must end in 0, because you are multiplying by tens.
Cara 2
Empat hasil darab separa
57 × 36 = (50 + 7) × (30 + 6)
| × | 50 | 7 |
|---|---|---|
| 30 | 1 500 | 210 |
| 6 | 300 | 42 |
42 + 300 + 210 + 1 500 = 2 052
部分积 = hasil darab separa
把两个两位数都按数位拆开：57 = 50 + 7，36 = 30 + 6。每一部分都要和另一个数的每一部分相乘，一共四个部分积：7 × 6 = 42，50 × 6 = 300，7 × 30 = 210，50 × 30 = 1 500。四个部分积加起来是 2 052，和竖式的结果一样。
例子：23 × 14 → 3 × 4、20 × 4、3 × 10、20 × 10，即 12 + 80 + 30 + 200 = 322。
注意：四个部分积一个都不能漏，最容易漏掉的是十位乘十位那一个。
partial products = hasil darab separa
Split both numbers by place value: 57 = 50 + 7 and 36 = 30 + 6. Every part is multiplied by every part of the other number, giving four partial products: 7 × 6 = 42, 50 × 6 = 300, 7 × 30 = 210 and 50 × 30 = 1 500. Their total, 2 052, matches the column method.
Example: 23 × 14 uses 3 × 4, 20 × 4, 3 × 10 and 20 × 10, which are 12 + 80 + 30 + 200 = 322.
Watch out: do not miss any of the four; the one most often forgotten is tens times tens.
Cara lain
Pecahkan kepada faktor
36 × 57 = 57 × 4 × 9
36 = 4 × 9
| 2 | |||
| 5 | 7 | ||
| × | 4 | ||
| 2 | 2 | 8 |
× 9
| 2 | 7 | |||
| 2 | 2 | 8 | ||
| × | 9 | |||
| 2 | 0 | 5 | 2 |
用因数分解 = gunakan faktor
36 = 4 × 9，所以乘 36 可以分成先乘 4、再乘 9 两步，每一步都只是乘一位数，比较容易。57 × 4 = 228，228 × 9 = 2 052，和竖式的答案一样。
例子：25 × 12 = 25 × 4 × 3 = 100 × 3 = 300。
注意：两个因数相乘要刚好等于原来的数：4 × 9 = 36。
use factors = gunakan faktor
36 = 4 × 9, so multiplying by 36 can be done in two steps: multiply by 4, then by 9. Each step multiplies by one digit only, which is easier. 57 × 4 = 228 and 228 × 9 = 2 052, the same as the column method.
Example: 25 × 12 = 25 × 4 × 3 = 100 × 3 = 300.
Watch out: the two factors must multiply to the number you split: 4 × 9 = 36.
Cari kesilapan
Di manakah kesilapannya?
43 × 26
✘ Salah
| 4 | 3 | ||
| × | 2 | 6 | |
| 2 | 5 | 8 | |
| + | 8 | 6 | |
| 3 | 4 | 4 |
✔ Betul
| 4 | 3 | |||
| × | 2 | 6 | ||
| 2 | 5 | 8 | ||
| + | 8 | 6 | 0 | |
| 1 | 1 | 1 | 8 |
找错 = cari kesilapan
左边的算法第二行写成 86，是把十位的 2 当成 2 来乘了。其实十位的 2 代表 20，43 × 20 = 860，要写在十位对齐的位置、末尾补 0。所以正确答案是 258 + 860 = 1 118。
例子：15 × 21 → 15 × 1 = 15，15 × 20 = 300，答案 315。
注意：第二行漏了 0 是最常见的错误。
find the mistake = cari kesilapan
The working on the left writes 86 in the second row, treating the tens digit 2 as just 2. That 2 is really 20, and 43 × 20 = 860, which ends in 0. So the correct answer is 258 + 860 = 1 118.
Example: 15 × 21: 15 × 1 = 15, 15 × 20 = 300, answer 315.
Watch out: forgetting the 0 in the second row is the most common mistake.
Tiga digit × dua digit
Kaedah petak
214 × 35
方格法 = kaedah petak
把两个数都按数位拆开：214 = 200 + 10 + 4，35 = 30 + 5。画一个方格，每一格是一对数相乘，例如 200 × 30 = 6 000。最后把所有格子的积加起来，就是答案。
例子：123 × 21 → 100×20、20×20、3×20、100×1、20×1、3×1，总和 2 583。
注意：每一格都要算到，不能漏。
box method = kaedah petak
Split both numbers by place value: 214 = 200 + 10 + 4 and 35 = 30 + 5. Draw a grid; each box multiplies one pair, for example 200 × 30 = 6 000. Add up all the boxes to get the answer.
Example: 123 × 21 uses 100 × 20, 20 × 20, 3 × 20, 100 × 1, 20 × 1 and 3 × 1, which total 2 583.
Watch out: work out every box; do not skip any.
Tiga digit × dua digit
Bentuk lazim
214 × 35
| 2 | 1 | 4 | ||
| × | 3 | 5 | ||
| 1 | 0 | 7 | 0 | |
| + | 6 | 4 | 2 | 0 |
| 7 | 4 | 9 | 0 |
214 × 5 = 1 070 · 214 × 30 = 6 420
竖式乘法 = darab bentuk lazim
三位数乘两位数，和两位数乘两位数的步骤一样：先用个位 5 乘 214，得 1 070；再用十位（30）乘 214，得 6 420，末尾有 0；两行相加得 7 490。答案和方格法相同。
例子：132 × 23 → 396 + 2 640 = 3 036。
注意：两行的数位要对齐。
column multiplication = darab bentuk lazim
Three digits times two digits follows the same steps: multiply 214 by the ones digit 5 to get 1 070, then by the tens (30) to get 6 420, which ends in 0, and add the rows to get 7 490. The answer matches the box method.
Example: 132 × 23: 396 + 2 640 = 3 036.
Watch out: keep the place values of both rows lined up.
Anggar, kemudian kekisi
29 × 3 104
Anggaran: 30 × 3 000 = 90 000
29 × 3 104 = 90 016 ≈ 90 000 ✔
格子乘法 = kaedah kekisi
先估计：29 约是 30，3 104 约是 3 000，积大约 90 000。再画格子：上面写 3 104，右边写 2 和 9。每一格写两个数字相乘的积，十位在左上、个位在右下。最后沿着斜线从右下往左上相加，满十进位，读出答案 90 016。
例子：12 × 34 用格子法得 408。
注意：答案要和估计值接近才合理。
lattice method = kaedah kekisi
Estimate first: 29 is about 30 and 3 104 is about 3 000, so the product is about 90 000. Then draw the grid: write 3 104 along the top and 2 and 9 down the right. In each box write the product of its two digits, tens in the upper left and ones in the lower right. Add along each diagonal from the bottom right, carrying when needed, and read the answer 90 016.
Example: 12 × 34 by the lattice method gives 408.
Watch out: the answer should be close to the estimate.
Kekisi P atau Q?
Kekisi manakah betul bagi 476 × 38?
Kekisi P
Kekisi Q
检查格子乘法 = semak kaedah kekisi
两份格子的每一格都一样，差别在斜线相加这一步。检查时按三步：一、每一格是不是上面和右边两个数字的积（十位在左上，个位在右下）；二、沿每条斜线从右下往左上相加，和满十时只写个位，把 1 进到下一条斜线；三、用估计看读出来的数是否合理。
例子：27 × 13 的格子，斜线和从右下起是 1、15、2；15 写 5 进 1，第三条就变成 3，读出 351。
注意：两份格子的数字只差一点点，要逐条斜线核对进位，不能只看大概。
check the lattice = semak kaedah kekisi
Every box in the two lattices is the same; the difference is in adding along the diagonals. Check in three steps: 1 each box holds the product of the digits above and on the right (tens upper left, ones lower right); 2 add along each diagonal from the bottom right, and when a sum reaches ten write only the ones and carry 1 to the next diagonal; 3 compare the result with an estimate.
Example: for 27 × 13 the diagonal sums from the bottom right are 1, 15 and 2; write 5 and carry 1, so the third becomes 3 and the answer reads 351.
Watch out: the two lattices differ only slightly, so check the carrying on every diagonal.
Kira cepat
407 × 100 = ?
乘 100 = darab dengan 100
这题要把一个中间有 0 的三位数乘 100。乘 100 就是有这么多个“百”，每个数字都向左移两个数位，个位和十位空出来，用两个 0 补上。原来夹在中间的 0 要留在原位。
例子：802 × 100 = 80 200。
注意：乘 10 补一个 0，乘 100 补两个，乘 1 000 补三个；先数清楚乘数有几个 0 再写。
multiply by 100 = darab dengan 100
This question multiplies a three-digit number with a 0 in the middle by 100. Times 100 means that many hundreds, so every digit moves two places to the left and two zeros fill the tens and ones. The 0 already inside the number stays where it is.
Example: 802 × 100 = 80 200.
Watch out: times 10 adds one zero, times 100 adds two and times 1 000 adds three, so count the zeros in the multiplier first.
Cuba
57 × 84 = ?
两位数乘两位数 = darab dua digit dengan dua digit
先用乘数的个位去乘，写出第一行；再用乘数的十位去乘，这一位代表几十，所以第二行的个位先写 0；最后把两行对齐相加。做完把两个数都取成整十数相乘，估计一下答案大约多大。
例子：63 × 25 → 63 × 5 = 315，63 × 20 = 1 260，315 + 1 260 = 1 575。
注意：第二行忘了补 0，得数会小很多，只有几百，和估计值差得很远。
two digits times two digits = darab dua digit dengan dua digit
Multiply by the ones digit of the multiplier to get the first row. Then multiply by the tens digit; it stands for tens, so start the second row with a 0 in the ones place. Line the rows up and add. Finally round both numbers to tens and multiply to estimate the size of the answer.
Example: 63 × 25: 63 × 5 = 315, 63 × 20 = 1 260, and 315 + 1 260 = 1 575.
Watch out: without the 0 in the second row the result is far too small, only a few hundred, and far from the estimate.
Cuba
128 × 46 = ?
三位数乘两位数 = darab tiga digit dengan dua digit
步骤和两位数乘两位数一样：第一行是被乘数乘个位，第二行是被乘数乘十位（个位先写 0），然后两行对齐相加。算完后用估计检查位数对不对。
例子：213 × 32 → 213 × 2 = 426，213 × 30 = 6 390，426 + 6 390 = 6 816。
注意：只算出一行就停下来是常见的错误；两行都要算，还要把它们加起来。
three digits times two digits = darab tiga digit dengan dua digit
The steps are the same as for two digits: the first row is the number times the ones digit, the second row is the number times the tens (start it with a 0), and then add the two rows. Estimate afterwards to check the number of digits.
Example: 213 × 32: 213 × 2 = 426, 213 × 30 = 6 390, and 426 + 6 390 = 6 816.
Watch out: stopping after only one row is a common mistake; work out both rows and add them.
Ringkasan