The naming step decides the whole problem
Every word problem on the SAT is solvable in two moves: name a quantity, then write one sentence of the story as one equation. Nearly all lost marks come from the first move, not the second.
Write the definition as a full sentence with a unit attached: “Let $a$ be the number of adult tickets sold.” Never write “let $a$ be adults” — that is ambiguous between a count, a price and a total.
If two quantities in the story add to a known total, name only one of them. The other is $(\text{total}-x)$. This turns a two-variable story into a one-variable equation.
Quy tắc rẻ nhất mà hiệu quả nhất: ẩn phải kèm đơn vị. “Gọi $x$ là số vé người lớn” khác hẳn “gọi $x$ là tiền vé người lớn”. Hai cách đặt đều đúng, nhưng phương trình viết ra hoàn toàn khác nhau; sai ở đây thì mọi bước sau đều sai mà vẫn ra một con số đẹp.
The phrase patterns that become symbols
- “$k$ more than $x$” $\rightarrow x+k$; “$k$ less than $x$” $\rightarrow x-k$ (the order is reversed from the words).
- “$k$ times as many $A$ as $B$” $\rightarrow A=kB$ (the quantity named first is the big one).
- “$x$ decreased by $p\%$” $\rightarrow (1-p/100)\,x$; “increased by $p\%$” $\rightarrow (1+p/100)\,x$.
- “a fixed fee of $f$ plus $r$ per unit” $\rightarrow C=f+rn$.
- “the result is the same as” $\rightarrow$ the equals sign, and it goes exactly there.
The sum of two numbers is $48$. The larger number is $3$ less than twice the smaller. What is the larger number?
Let $x$ be the smaller number. Then the larger is $2x-3$, and $x+(2x-3)=48$ gives $3x=51$, so $x=17$ and the larger number is $2(17)-3=\boxed{31}$. Check: $17+31=48$.
Three less than half a number is the same as the number decreased by $15$. Find the number.
“Three less than half a number” is $\tfrac12 n-3$, not $3-\tfrac12 n$. So $\tfrac12 n-3=n-15$, giving $12=\tfrac12 n$ and $\boxed{n=24}$. Check: $12-3=9$ and $24-15=9$.
“$3$ less than $n$” viết là $n-3$, không phải $3-n$. Cụm “less than” đảo thứ tự so với lời đọc; cụm “less” đứng một mình thì không đảo (“$n$ less $3$” vẫn là $n-3$). Đây là chỗ mất điểm nhiều nhất của cả chủ đề, và bản đảo dấu luôn có mặt trong bốn phương án.
Rate times quantity: money, mixtures, work
Every one of these stories is the same equation wearing different clothes: \[ (\text{value per unit})\times(\text{number of units})\;+\;(\text{value per unit})\times(\text{number of units}) \;=\;\text{total value}. \]
A theater sold $320$ tickets. Adult tickets cost $15 and child tickets cost $9, and the total revenue was $3,960. How many child tickets were sold?
Let $c$ be the number of child tickets; then $320-c$ adult tickets were sold. \[ 15(320-c)+9c=3960 \;\Longrightarrow\; 4800-6c=3960 \;\Longrightarrow\; 6c=840 \;\Longrightarrow\; \boxed{c=140}. \] Check: $180$ adult tickets give $2,700 and $140$ child tickets give $1,260; the sum is $3,960.
A chemist mixes a $20\%$ acid solution with a $50\%$ acid solution to obtain $60$ mL of a $30\%$ solution. How many millilitres of the $50\%$ solution are used?
Let $x$ be the millilitres of $50\%$ solution; the rest, $60-x$, is the $20\%$ solution. Track the acid, not the liquid: \[ 0.50x+0.20(60-x)=0.30(60)=18 \;\Longrightarrow\; 0.30x+12=18 \;\Longrightarrow\; \boxed{x=20}. \] Check: $20$ mL at $50\%$ is $10$ mL of acid, $40$ mL at $20\%$ is $8$ mL; total $18$ mL out of $60$ mL is $30\%$.
Bài hỗn hợp luôn có hai phương trình ẩn trong đề: một cho tổng lượng dung dịch, một cho lượng chất tan. Nếu tổng đã biết (như $60$ mL ở trên) thì dùng ngay $60-x$ và chỉ còn một phương trình. Nếu tổng chưa biết — kiểu “thêm bao nhiêu lít vào $12$ lít sẵn có” — thì tổng là $x+12$ và nó phải xuất hiện ở vế phải.
A $30\%$ salt solution is added to $12$ litres of a $10\%$ salt solution to produce a $25\%$ solution. How many litres of the $30\%$ solution are added?
Let $x$ be the litres added. The final volume is $x+12$, so \[ 0.30x+0.10(12)=0.25(x+12) \;\Longrightarrow\; 0.30x+1.2=0.25x+3 \;\Longrightarrow\; 0.05x=1.8 \;\Longrightarrow\; \boxed{x=36}. \] Check: $36$ L at $30\%$ is $10.8$ L of salt, plus $1.2$ L, gives $12$ L of salt in $48$ L, and $12/48=25\%$.
Fixed fee plus a rate, given two data points
A great many SAT stories hide the model $C=f+rn$ and hand you two values of $C$ instead of $f$ and $r$. Subtracting the two equations kills $f$ in one line.
A plumber charges a flat call-out fee plus an hourly rate. A $2$-hour job costs $175 and a $5$-hour job costs $325. What does an $8$-hour job cost?
With $C=f+rh$: $\;f+2r=175$ and $f+5r=325$. Subtracting, $3r=150$, so $r=50$ and $f=175-2(50)=75$. Then $C=75+50(8)=\boxed{\$475}$.
The difference of the two totals divided by the difference of the two counts is the per-unit rate, every time — no algebra needed. Here $\dfrac{325-175}{5-2}=50$.
Percent change, discount, and tax
Successive percent changes multiply; they never add.
A jacket is marked down $15\%$, and then a $5 coupon is applied. The customer pays $63. What was the original price?
Let $p$ be the original price. The order in the story is the order in the equation: $0.85p-5=63$, so $0.85p=68$ and $\boxed{p=\$80}$. Check: $15\%$ off $80 is $68, minus the coupon is $63.
A shirt priced at $d$ dollars is discounted $30\%$, and then $8\%$ sales tax is applied to the discounted price. Which expression gives the amount paid?
Discount first: $0.70d$. Tax on that: $1.08(0.70d)=\boxed{0.756d}$. Note $0.70+0.08\neq$ anything useful here — $0.78d$ is the answer to a different, wrong story.
Cộng phần trăm thay vì nhân. Giảm $30\%$ rồi cộng thuế $8\%$ không ra $-22\%$. Cũng cẩn thận thứ tự: giảm $15\%$ rồi trừ phiếu $5 khác với trừ $5 rồi giảm $15\%$ ($63 so với $63,75 nếu giá gốc là $80). Đề luôn nói rõ thứ tự — đọc đúng thứ tự đó.
Read the last sentence again
The variable you named and the quantity the question wants are often different. This is a deliberate design feature of the test, not an accident.
The length of a rectangle is $5$ cm more than twice its width, and the perimeter is $82$ cm. What is the area?
Let $w$ be the width; the length is $2w+5$. Then $2\bigl(w+(2w+5)\bigr)=82$ gives $6w+10=82$, so $w=12$ and the length is $29$. The question asks for area: $12\times 29=\boxed{348\ \text{cm}^2}$.
Ba câu hỏi bắt buộc đọc lại sau khi ra số: (i) đề hỏi ẩn hay hỏi một biểu thức của ẩn? (ii) đơn vị đề hỏi có trùng đơn vị mình đang dùng không (phút hay giờ, cm hay m)? (iii) nếu đề hỏi “nhiều nhất bao nhiêu” thì kết quả phải làm tròn xuống số nguyên, kể cả khi phần thập phân là $0{,}9$.
Nam has $95. He buys one backpack for $32 and spends the rest on notebooks costing $4.75 each. What is the greatest number of notebooks he can buy?
Remaining money: $95-32=63$. Then $4.75n\le 63$ gives $n\le 13.26\ldots$, so $\boxed{n=13}$. Fourteen notebooks would cost $66.50, more than he has.