Two names, two sentences
A word problem becomes a system the moment two unknown quantities appear and the passage gives two independent facts about them. The work is mechanical if it is done in this order.
- Name the two unknowns with a letter and a unit: “$a=$ number of adult tickets”, not just “$a=$ adults”.
- Find the two facts. Each fact is one sentence, and each becomes one equation.
- Check the units inside each equation. Every term of an equation must carry the same unit — tickets with tickets, dollars with dollars.
- Solve, then re-read the last line of the question.
Almost every two-variable story on the test is built from a count equation and a value equation: \[ x+y=\text{(total number of items)},\qquad px+qy=\text{(total value)} . \] The first counts things; the second weighs them. They can never be added together.
Cái làm học sinh Việt Nam mất điểm ở dạng này gần như không bao giờ là đại số — mà là bước đặt ẩn. Hai mẹo giữ cho bước đó sạch: (1) viết đơn vị ngay cạnh chữ cái, (2) đọc lại từng phương trình thành lời. Nếu đọc lên nghe vô nghĩa (“240 cái vé bằng 2432 đô”), phương trình sai — dừng lại sửa, đừng giải tiếp.
Count-and-value pairs
A theater sells adult tickets for $12 and student tickets for $8. One evening it sold $240$ tickets and collected $2,432. How many student tickets were sold?
Let $a$ and $s$ be the numbers of adult and student tickets. \[ a+s=240 \quad(\text{tickets}),\qquad 12a+8s=2432 \quad(\text{dollars}).\] Substitute $a=240-s$: $\;12(240-s)+8s=2432$, so $2880-4s=2432$ and $4s=448$, giving $\boxed{s=112}$ (with $a=128$). Check: $128\cdot 12+112\cdot 8=1536+896=2432$.
Notebooks cost $3.00 each and pens cost $1.50 each. Riley bought $22$ items in total and spent $49.50. How many notebooks did Riley buy?
With $n$ notebooks and $p$ pens: $n+p=22$ and $3n+1.5p=49.5$. Substituting $p=22-n$ gives $3n+1.5(22-n)=49.5$, i.e.\ $1.5n+33=49.5$, so $1.5n=16.5$ and $\boxed{n=11}$. Then $p=11$ as well, and $33+16.5=49.5$ confirms it.
Cộng nhầm hai phương trình khác đơn vị. “$240$ vé” và “$2432” không bao giờ đứng chung một phương trình. Nếu trong một dòng vừa có số lượng vừa có tiền mà không có hệ số giá đi kèm, dòng đó chắc chắn sai.
Percent and mixture
Concentration problems look different but obey the same count–value shape: volume plays the role of count, and amount of substance plays the role of value. \[ \underbrace{x+y=V}_{\text{total volume}},\qquad \underbrace{c_1x+c_2y=c\,V}_{\text{total amount of the substance}} . \]
Quy tắc một dòng cho mọi bài pha trộn: nồng độ $\times$ thể tích $=$ lượng chất. Lượng chất cộng được, còn nồng độ thì không — trộn dung dịch $20\%$ với dung dịch $50\%$ không ra $70\%$, và cũng không nhất thiết ra $35\%$. Hai bài toán tiền lãi theo lãi suất khác nhau chạy đúng cùng công thức này, chỉ đổi tên: lãi suất $\times$ vốn $=$ tiền lãi.
A chemist mixes a $20\%$ acid solution with a $50\%$ acid solution to obtain $60$ mL of a $30\%$ acid solution. How many millilitres of the $50\%$ solution are used?
Let $x$ and $y$ be the millilitres of the $20\%$ and $50\%$ solutions. \[ x+y=60,\qquad 0.20x+0.50y=0.30(60)=18 .\] Substituting $x=60-y$: $\;0.20(60-y)+0.50y=18$, so $12+0.30y=18$ and $0.30y=6$, giving $\boxed{y=20}$ mL. Check: $0.2(40)+0.5(20)=8+10=18$.
Rate, distance, and two-part trips
Two motion facts turn into two equations through $\text{distance}=\text{rate}\times\text{time}$. Decide first what is shared: two travellers meeting share the time; a single traveller splitting a trip shares neither rate nor time but shares the totals.
Two cars start $300$ miles apart and drive toward each other along the same road. They meet after $2.5$ hours. One car travels $12$ mph faster than the other. Find the slower speed.
Let the slower speed be $r$; the faster is $r+12$. The two distances add to $300$: \[ 2.5r+2.5(r+12)=300 \;\Longrightarrow\; r+(r+12)=120 \;\Longrightarrow\; 2r=108 .\] So $\boxed{r=54}$ mph and the other car travels $66$ mph. Check: $2.5(54)+2.5(66)=135+165=300$.
A technician is paid $15 per hour on weekdays and $22 per hour on weekends. Last week she worked $34$ hours and earned $622. How many weekend hours did she work?
With $x$ weekday hours and $y$ weekend hours: $x+y=34$ and $15x+22y=622$. Substituting $x=34-y$: $\;15(34-y)+22y=622$, so $510+7y=622$, giving $7y=112$ and $\boxed{y=16}$ (with $x=18$). Check: $18(15)+16(22)=270+352=622$.
When the context rejects the algebra
Every unknown in these stories is a physical quantity, so it inherits restrictions the equations know nothing about: a number of crates must be a non-negative integer, a length must be positive, a percentage must sit between $0$ and $100$.
Small crates weigh $10$ kg each and large crates weigh $15$ kg each. A truck is reported to be carrying $40$ crates with a total weight of $700$ kg. How many large crates are on the truck?
With $s$ small and $\ell$ large crates: $s+\ell=40$ and $10s+15\ell=700$. Substituting $s=40-\ell$ gives $10(40-\ell)+15\ell=700$, so $400+5\ell=700$ and $\ell=60$ — which forces $s=40-60=-20$. A negative number of crates is impossible, so no such load exists; the report is inconsistent. The largest weight $40$ crates can reach is $40(15)=600$ kg.
- The solution is negative where only non-negative values make sense.
- The solution is a fraction where only whole items make sense.
- The solution exists but violates a stated cap (“at most $60$ items”, “fewer than $10$ hours”).
In each case the correct response is not “recompute” — it is to report that the described situation cannot occur, or to pick the nearest value the context does allow.
Bài ra nghiệm không nguyên và học sinh làm tròn cho “đẹp”. Nếu đề hỏi số vật thể thì nghiệm lẻ nghĩa là đọc sai đề hoặc dữ kiện mâu thuẫn — không phải chỗ để làm tròn. Chỉ được làm tròn khi chính đề nói “tối đa” hay “tối thiểu”.
Reading the question that is actually asked
The system is set up to find $x$ and $y$, but the final line very often asks for something built out of them: the difference, the sum, one of them expressed in other units, or the value of a third quantity.
A theater charges $9 per adult ticket and $6 per child ticket. On Tuesday it sold $180$ tickets and collected $1,395. How many more adult tickets than child tickets were sold?
$a+c=180$ and $9a+6c=1395$. Substituting $c=180-a$: $\;9a+6(180-a)=1395$, so $3a+1080=1395$, giving $3a=315$ and $a=105$, hence $c=75$. The question asks for the difference: $105-75=\boxed{30}$.
The sum of the digits of a two-digit number is $11$. Reversing its digits produces a number that is $27$ greater than the original. What is the original number?
Let $t$ be the tens digit and $u$ the units digit, so the number is $10t+u$. \[ t+u=11,\qquad (10u+t)-(10t+u)=27 \;\Longrightarrow\; 9(u-t)=27 \;\Longrightarrow\; u-t=3 .\] Adding $t+u=11$ and $u-t=3$ gives $2u=14$, so $u=7$ and $t=4$. The number is $\boxed{47}$, and indeed $74-47=27$.
Trước khi khoanh, gạch chân đúng cụm cuối cùng của đề: how many students, how many more, what is the total value, the value of $x+y$. Ba trong bốn phương án nhiễu của câu dạng này là những số bạn đã thực sự tính đúng trên nháp — chỉ là không phải số được hỏi.
Giải ra $x$ đẹp rồi khoanh ngay. Ở ví dụ rạp phim, cả $105$ và $75$ đều xuất hiện trong bốn phương án; đáp án đúng là $30$, con số duy nhất chưa ai viết ra trên nháp.