SAT-94: The Direct Translation Method for Word Problems

SAT-94: The Direct Translation Method for Word Problems

Description: SAT-91 gave you the dictionary; this lesson gives you the system. The Direct Translation Method is a repeatable four-step routine that turns any word problem into equations you can solve — even the multi-sentence ones.

The four steps

1. Define variables: write "let x = …" for each unknown.
2. Translate each sentence into an equation (using the SAT-91 dictionary).
3. Solve the equation(s).
4. Answer the question asked (check what it actually wants).

(Oʻzbekcha: oʻzgaruvchini belgilang, har bir gapni tenglamaga aylantiring, yeching, soʻng aynan soʻralganini toping.)

Worked Example 1 — one unknown

"A number tripled, then decreased by 4, equals 17. Find the number."

• Let x = the number. Translate: 3x − 4 = 17.
• Solve: 3x = 21 → x = 7.

Worked Example 2 — two unknowns, one relationship

"Anna is 5 years older than Ben. The sum of their ages is 27. How old is Ben?"

• Let Ben = b. Then Anna = b + 5. Translate the sum: b + (b + 5) = 27.
• 2b + 5 = 27 → 2b = 22 → b = 11. (Ben is 11; the question asked for Ben, so stop here.)

(Oʻzbekcha: soʻralgani Ben boʻlgani uchun b ni topib toʻxtaymiz.)

Worked Example 3 — a rate problem

"A taxi charges $3 plus $2 per mile. A ride cost $19. How many miles?"

• Let m = miles. Translate: 3 + 2m = 19.
• 2m = 16 → m = 8 miles.

Worked Example 4 — answer the right quantity

"The width of a rectangle is x. The length is twice the width. The perimeter is 30. Find the length."

• Width = x, length = 2x. Perimeter: 2(x) + 2(2x) = 30 → 6x = 30 → x = 5.
• But the question wants the length = 2x = 10 (don't stop at x = 5 — that's the trap).

Rule: the last step is the most missed one. After solving, re-read and make sure you reported the exact quantity asked. (Oʻzbekcha: oxirgi qadam — aynan soʻralgan miqdorni berish; uni unutmang.)

Choosing which unknown to call the variable

When a problem has two related quantities, define your variable as the smaller or simpler one, then express the other in terms of it. In the ages example, calling Ben's age b (and Anna's b + 5) is cleaner than the reverse, because it avoids subtraction. A good choice of variable keeps the equation simple and reduces sign errors. If your first choice leads to ugly fractions, it's fine to restart with the other quantity as the variable. (Oʻzbekcha: ikki bogʻliq miqdor boʻlsa, oddiyrogʻini oʻzgaruvchi deb oling — tenglama soddaroq boʻladi.)

When to use two equations

If a problem gives two genuinely separate facts about two unknowns, you can set up two equations and solve the system (review the linear-systems lessons). But many SAT word problems only need one variable if you express the second quantity in terms of the first — fewer variables means fewer chances to slip. Reach for a full system only when the two quantities can't be linked in a single expression. (Oʻzbekcha: ikkita alohida fakt berilsa, ikkita tenglama tuzing; aks holda bitta oʻzgaruvchi yetarli.)

Practice 1

"A number doubled, then increased by 6, equals 20. Find the number."

Practice 2

"Maria has twice as many books as Sam. Together they have 18. How many does Sam have?"

Key words — Kalit soʻzlar

• Define a variable — oʻzgaruvchini belgilash
• Translate — tarjima qilish
• Equation — tenglama
• Unknown — nomaʼlum
• Relationship — bogʻlanish (munosabat)
• Rate — tezlik (narx sur'ati)
• Perimeter — perimetr
• Sum of ages — yoshlar yigʻindisi
• Solve — yechish

Summary

• Four steps: define → translate → solve → answer the question.
• Name each unknown clearly before translating.
• The most common error is stopping at x instead of the quantity actually asked.