Unit 13
Selesaikan Masalah
Harga sebuah komputer riba ialah 3 kali harga sebuah pencetak.
Harga pencetak itu RM 1 150.
Berapakah jumlah harga kedua-dua barang itu?
解决问题 = selesaikan masalah；倍 = kali；两步 = dua langkah
这一课学用四个步骤解应用题。题目说手提电脑的价钱是打印机的 3 倍，打印机 RM 1 150，问两样一共多少钱。这题要两步：先算电脑的价钱，再把两样加起来。很多人算出电脑的价钱就停了，那只做了一半。我们会用画长条的方法把“3 倍”画出来，就不容易漏步骤。
例子：书包是笔盒的 2 倍，笔盒 RM 15，两样一共 RM 15 × 2 + RM 15 = RM 45。
注意：题目问“jumlah”就是要两样加起来，不是只要电脑的价钱。
solving problems = selesaikan masalah; times = kali; two steps = dua langkah
In this unit we solve word problems in four steps. A laptop costs 3 times as much as a printer, the printer is RM 1 150, and we must find the cost of both together. That takes two steps: first the laptop price, then the two prices added. Many pupils stop after finding the laptop price, which is only half the work. Drawing bars to show the 3 times relationship helps us see both steps.
Example: a bag costs 2 times a pencil case at RM 15, so both cost RM 15 × 2 + RM 15 = RM 45.
Watch out: jumlah means both prices added, not the laptop price alone.
Empat langkah
Diberi → Dicari → Cara → Semak
Gunakan empat langkah ini untuk setiap masalah
1. Diberi
maklumat yang ada dalam soalan
2. Dicari
apa yang soalan mahu
3. Cara
lukis model → ayat matematik → kira
4. Semak
guna operasi songsang: tolak ↔ tambah, bahagi ↔ darab
Gariskan nombor dan kata kunci dalam soalan sebelum mula mengira
已知 = Diberi；所求 = Dicari；方法 = Cara；检查 = Semak；逆运算 = operasi songsang
四个步骤是固定的。Diberi：把题目告诉我们的数字和关系列出来。Dicari：弄清楚题目问什么。Cara：画长条模型、写算式、计算。Semak：用相反的运算检查，加法用减法查，乘法用除法查。做题前先把数字和关键词画线，就不会看漏条件。
例子：算出 3 450 + 1 150 = 4 600，检查用 4 600 − 1 150，得回 3 450 就对了。
注意：Semak 不是多余的，很多错误都是在检查时发现的。
given = Diberi; to find = Dicari; method = Cara; check = Semak; inverse operation = operasi songsang
The four steps are always the same. Diberi: list the numbers and relationships the question gives. Dicari: be clear about what the question asks. Cara: draw a bar model, write the number sentence and work it out. Semak: check with the opposite operation, subtraction for addition and division for multiplication. Underlining the numbers and key words before you start stops you missing a clue.
Example: after finding 3 450 + 1 150 = 4 600, check with 4 600 − 1 150, which gives back 3 450.
Watch out: Semak is not extra work; many mistakes are only caught at the checking step.
Diberi dan Dicari
Lukis model bar
3 kali = 3 bahagian yang sama panjang
Komputer riba = 3 kotak · Pencetak = 1 kotak · jumlah = 4 kotak
条形模型 = model bar；一格 = satu kotak；一样长 = sama panjang
先写 Diberi：打印机 RM 1 150，电脑是打印机的 3 倍。再写 Dicari：两样的总价。然后画模型：打印机画 1 格，电脑画 3 格一样长的，每格都是 RM 1 150。看图就知道电脑是 3 × RM 1 150 = RM 3 450，总价是 3 格加 1 格，一共 4 格。
例子：如果题目说“4 倍”，就画 4 格一样长的格子。
注意：“3 kali”是画 3 格，不是在打印机的价钱上加 3。
bar model = model bar; one box = satu kotak; the same length = sama panjang
First write Diberi: the printer is RM 1 150 and the laptop is 3 times the printer. Then Dicari: the total price of both. Now draw the model: one box for the printer and three boxes of the same length for the laptop, each worth RM 1 150. The picture shows the laptop is 3 × RM 1 150 = RM 3 450, and the total is three boxes plus one box, four boxes in all.
Example: if the question said 4 times, you would draw 4 boxes of the same length.
Watch out: 3 kali means drawing 3 boxes, not adding 3 to the printer price.
Cara dan Semak
Dua langkah pengiraan
Langkah 1: darab · Langkah 2: tambah · Semak: tolak
Langkah 1 · komputer
| 1 | 1 | 5 | 0 | |
| × | 3 | |||
| 3 | 4 | 5 | 0 |
Langkah 2 · jumlah
| 3 | 4 | 5 | 0 | |
| + | 1 | 1 | 5 | 0 |
| 4 | 6 | 0 | 0 |
Semak: RM 4 600 − RM 1 150 = RM 3 450 ✓ · RM 3 450 ÷ 3 = RM 1 150 ✓
第一步 = langkah 1；第二步 = langkah 2；检查 = semak
Cara 分两步写。第一步求电脑价钱：RM 1 150 × 3 = RM 3 450。第二步求总价：RM 3 450 + RM 1 150 = RM 4 600。最后 Semak：用总价减打印机，4 600 − 1 150 = 3 450，正是电脑的价钱；再用 3 450 ÷ 3 = 1 150，回到打印机的价钱。两个检查都对，答案 RM 4 600 可以放心写。
例子：答案如果写 RM 3 450，检查 3 450 − 1 150 = 2 300，不是 3 的倍数关系，就知道漏了一步。
注意：每一步都要写完整的算式，不能只写答案。
step 1 = langkah 1; step 2 = langkah 2; check = semak
Write Cara in two steps. Step one finds the laptop price: RM 1 150 × 3 = RM 3 450. Step two finds the total: RM 3 450 + RM 1 150 = RM 4 600. Then Semak: total minus printer, 4 600 − 1 150 = 3 450, which is exactly the laptop price; and 3 450 ÷ 3 = 1 150 brings back the printer price. Both checks work, so RM 4 600 is safe to write.
Example: if the answer were written as RM 3 450, checking 3 450 − 1 150 = 2 300 does not fit the 3 times relationship, showing a step was missed.
Watch out: write a full number sentence for every step, not just the answers.
Cuba
Harga sebuah basikal ialah 4 kali harga sebuah topi keledar. Harga topi keledar ialah RM 65. Berapakah jumlah harga kedua-dua barang?
四倍 = 4 kali；五格 = 5 kotak
画模型：头盔 1 格 RM 65，脚踏车 4 格一样长。第一步：脚踏车 RM 65 × 4 = RM 260。第二步：总价 RM 260 + RM 65 = RM 325。也可以直接看图：一共 5 格，RM 65 × 5 = RM 325。RM 260 是只算了脚踏车，RM 69 是把 4 加上去，都不对。
例子：桌子是椅子的 3 倍，椅子 RM 40，两样 RM 40 × 4 = RM 160。
注意：“4 倍”画 4 格，加上原来的 1 格，总共是 5 格。
4 times = 4 kali; 5 boxes = 5 kotak
Draw the model: the helmet is 1 box of RM 65 and the bicycle is 4 boxes of the same length. Step one: bicycle RM 65 × 4 = RM 260. Step two: total RM 260 + RM 65 = RM 325. You can also read the picture directly: 5 boxes in all, RM 65 × 5 = RM 325. RM 260 is the bicycle alone and RM 69 adds 4 to the price; both are wrong.
Example: a table costs 3 times a chair at RM 40, so both cost RM 40 × 4 = RM 160.
Watch out: 4 times means 4 boxes, and together with the helmet's 1 box the total is 5 boxes.
Jumlah dan baki
Beli dua barang, cari baki
Langkah 1: jumlah belanja · Langkah 2: baki
Langkah 1 · belanja
| 1 | 2 | 8 | 5 | |
| + | 3 | 4 | 9 | |
| 1 | 6 | 3 | 4 |
Langkah 2 · baki
| 2 | 0 | 0 | 0 | |
| − | 1 | 6 | 3 | 4 |
| 3 | 6 | 6 |
Semak: RM 366 + RM 1 634 = RM 2 000 ✓
花费 = belanja；余额 = baki；总数 = jumlah
这类题的模型是一条长条代表全部的钱 RM 2 000，里面分成三段：相机 RM 1 285、包 RM 349、剩下的钱。第一步先把花掉的两样加起来：1 285 + 349 = 1 634。第二步用全部的钱减去花掉的：2 000 − 1 634 = 366。检查：366 + 1 634 = 2 000，对。
例子：有 RM 100，买 RM 35 和 RM 42 的东西，余额 100 − 77 = RM 23。
注意：只减相机不减包，得到 RM 715，那是漏了一样东西。
spending = belanja; balance = baki; total = jumlah
For this type, the model is one long bar for all the money, RM 2 000, split into three parts: the camera RM 1 285, the bag RM 349, and the money left. Step one adds the two purchases: 1 285 + 349 = 1 634. Step two takes the spending away from the whole: 2 000 − 1 634 = 366. Check: 366 + 1 634 = 2 000, correct.
Example: with RM 100, buying items at RM 35 and RM 42 leaves 100 − 77 = RM 23.
Watch out: subtracting only the camera gives RM 715, which forgets one of the items.
Cuba
Encik Lim ada RM 500. Dia membeli sebuah kipas RM 189 dan sebuah seterika RM 76. Berapakah baki wangnya?
两样东西的余额 = baki selepas beli dua barang
第一步：风扇加熨斗，189 + 76 = 265，这是花掉的钱。第二步：500 − 265 = 235，这是余额。RM 311 是只减了风扇；RM 265 是花掉的总数，题目问的是剩下的，不是花掉的。检查：235 + 265 = 500。
例子：有 RM 60，买 RM 18 和 RM 25 的东西，余额 60 − 43 = RM 17。
注意：先把花掉的合起来，再从全部的钱里减掉。
balance after two purchases = baki selepas beli dua barang
Step one: fan plus iron, 189 + 76 = 265, which is the money spent. Step two: 500 − 265 = 235, the balance. RM 311 subtracts only the fan, and RM 265 is the spending itself, while the question asks for what is left. Check: 235 + 265 = 500.
Example: with RM 60, buying items at RM 18 and RM 25 leaves 60 − 43 = RM 17.
Watch out: combine the spending first, then take it away from the whole amount.
Kongsi sama banyak
RM 2 760 dikongsi 6 orang
Langkah 1: bahagi · Langkah 2: tolak
Langkah 1: RM 2 760 ÷ 6 = RM 460 · Langkah 2: RM 460 − RM 300 = RM 160
Semak: RM 160 + RM 300 = RM 460 ✓ · RM 460 × 6 = RM 2 760 ✓
平均分 = kongsi sama banyak；每人的份 = bahagian setiap orang；还要付 = perlu bayar lagi
长条代表全部费用 RM 2 760，平均分成 6 段，每段就是每个学生要付的份：2 760 ÷ 6 = 460。但题目问的是“还要付多少”，所以第二步：每人已付 300，还差 460 − 300 = 160。检查：160 + 300 = 460，460 × 6 = 2 760，两步都对。
例子：RM 900 平分给 3 人，每人 300；已付 100，还要付 200。
注意：算出每人 RM 460 不是答案，题目问的是“还要付”。
shared equally = kongsi sama banyak; each person's share = bahagian setiap orang; still to pay = perlu bayar lagi
The bar stands for the whole cost, RM 2 760, cut into 6 equal parts, and each part is one pupil's share: 2 760 ÷ 6 = 460. But the question asks how much more each pupil must pay, so step two: each has paid 300, leaving 460 − 300 = 160. Check: 160 + 300 = 460 and 460 × 6 = 2 760, so both steps hold.
Example: RM 900 shared by 3 people is 300 each; after paying 100, each still owes 200.
Watch out: RM 460 per pupil is not the answer; the question asks for the amount still to pay.
Cuba
Wang hadiah RM 1 800 dikongsi sama banyak oleh 4 orang. Setiap orang menyimpan RM 200 dan membelanjakan bakinya. Berapakah yang dibelanjakan oleh setiap orang?
先分再减 = bahagi dahulu, kemudian tolak
第一步：1 800 ÷ 4 = 450，每人分到 RM 450。第二步：每人存起 200，花掉的是 450 − 200 = 250。RM 450 是每人分到的，还没减储蓄；RM 400 是先减 200 再除以 4，但 200 是每个人各自存的，不是全部只存一次 200。检查：250 + 200 = 450，450 × 4 = 1 800。
例子：RM 600 平分 3 人各 200，各存 50，各花 150。
注意：看清楚“setiap orang menyimpan RM 200”，是每个人都存。
divide first, then subtract = bahagi dahulu, kemudian tolak
Step one: 1 800 ÷ 4 = 450, so each person receives RM 450. Step two: each keeps 200, so each spends 450 − 200 = 250. RM 450 is the share before saving, and RM 400 comes from subtracting 200 once and then dividing by 4, but the 200 is saved by every person, not once in total. Check: 250 + 200 = 450 and 450 × 4 = 1 800.
Example: RM 600 shared by 3 is 200 each; saving 50 each leaves 150 each to spend.
Watch out: read setiap orang menyimpan RM 200 carefully; every person saves that amount.
Wang baki
Bayar RM 300 untuk barang RM 235
Langkah 1: wang baki · Langkah 2: susun wang kertas
1. Diberi
harga RM 235; bayar 3 keping RM 100
2. Dicari
wang baki; berapa keping RM 10 dan RM 5
3. Cara
3 × RM 100 = RM 300
RM 300 − RM 235 = RM 65
RM 65 = 6 × RM 10 + 1 × RM 5
RM 300 − RM 235 = RM 65
RM 65 = 6 × RM 10 + 1 × RM 5
4. Semak
RM 235 + RM 65 = RM 300 ✓
RM 60 + RM 5 = RM 65 ✓
RM 60 + RM 5 = RM 65 ✓
找零 = wang baki；张 = keping；纸币 = wang kertas
找零的题有两步。第一步：先算付了多少，3 张 RM 100 是 RM 300；再算找回多少，300 − 235 = 65。第二步：把 RM 65 换成题目要的纸币：RM 10 的拿 6 张是 RM 60，再加 1 张 RM 5，刚好 RM 65。检查：235 + 65 = 300，60 + 5 = 65。
例子：买 RM 42 的东西付 RM 50，找 RM 8 = 1 张 RM 5 + 3 张 RM 1。
注意：算出 RM 65 还没完，题目还问几张 RM 10、几张 RM 5。
change = wang baki; piece = keping; banknote = wang kertas
A change problem has two steps. Step one: find how much was paid, 3 notes of RM 100 make RM 300, then the change, 300 − 235 = 65. Step two: turn RM 65 into the notes the question asks for: 6 notes of RM 10 make RM 60, plus 1 note of RM 5 gives exactly RM 65. Check: 235 + 65 = 300 and 60 + 5 = 65.
Example: paying RM 50 for an item costing RM 42 gives RM 8 change, which is 1 note of RM 5 and 3 notes of RM 1.
Watch out: finding RM 65 is not the end; the question also asks how many RM 10 and RM 5 notes.
Cuba
Mak Cik Rose membeli sebuah periuk berharga RM 78. Dia membayar dengan sekeping wang kertas RM 100. Wang bakinya diberi dalam wang kertas RM 10 dan RM 1. Berapa keping setiap satu?
找零换纸币 = tukar wang baki kepada wang kertas
第一步：100 − 78 = 22，找回 RM 22。第二步：用 RM 10 和 RM 1 凑 RM 22：2 张 RM 10 是 RM 20，再加 2 张 RM 1，刚好 RM 22。另外两个选项用了 RM 5，题目没有说可以用 RM 5，而且 20 + 5 = 25、10 + 10 = 20 都不是 22。检查：78 + 22 = 100。
例子：找 RM 31 用 RM 10 和 RM 1：3 张 RM 10 加 1 张 RM 1。
注意：先看题目允许哪几种纸币，再凑数。
turning change into notes = tukar wang baki kepada wang kertas
Step one: 100 − 78 = 22, so the change is RM 22. Step two: make RM 22 with RM 10 and RM 1 notes: 2 notes of RM 10 give RM 20, plus 2 notes of RM 1 make exactly RM 22. The other two options use RM 5, which the question does not allow, and 20 + 5 = 25 and 10 + 10 = 20 are not 22 anyway. Check: 78 + 22 = 100.
Example: RM 31 of change in RM 10 and RM 1 notes is 3 notes of RM 10 and 1 note of RM 1.
Watch out: check which notes the question allows before building the amount.
Padankan
Masalah dan ayat matematik
题目配算式 = padankan masalah dengan ayat matematik
“2 倍”配 RM 90 × 2 + RM 90：先乘再加。“买两样求余额”配 RM 400 − (RM 120 + RM 95)：先加再减。“平分后已付部分”配 RM 600 ÷ 3 − RM 50：先除再减。“付 RM 100 买 RM 78 的东西”只有一步，配 RM 100 − RM 78。先找关键词决定运算，再数步骤。
例子：看到 kongsi sama banyak，一定有 ÷。
注意：括号里的要先算，RM 400 − (RM 120 + RM 95) 是先把两样加起来。
matching problems to number sentences = padankan masalah dengan ayat matematik
2 times goes with RM 90 × 2 + RM 90: multiply, then add. Buying two items and finding the balance goes with RM 400 − (RM 120 + RM 95): add, then subtract. Sharing equally after a part payment goes with RM 600 ÷ 3 − RM 50: divide, then subtract. Paying RM 100 for a RM 78 item is a single step, RM 100 − RM 78. Find the key words to choose the operation, then count the steps.
Example: kongsi sama banyak always brings a ÷.
Watch out: brackets are worked out first, so RM 400 − (RM 120 + RM 95) adds the two items before subtracting.
Ringkasan
Syabas!
Diberi → Dicari → Cara → Semak
Baca soalan, gariskan nombor. Lukis model bar: kali = kotak sama panjang; baki = jumlah tolak belanja; kongsi = bahagi. Masalah dua langkah: siapkan kedua-dua langkah. Semak dengan operasi songsang.
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