Word problems

The math is easy. The English is the test.

How to solve them

1. 1

   Underline what is asked, with its unit

   The last sentence is the target. Write it down as a thing with a unit: “total paid, in dollars.” Then check every answer choice against it. A choice in the wrong unit is wrong even if the arithmetic behind it is right.

   Example

   A trip takes 3 hours. The train leaves at 9:40 a.m. When does it arrive?

   1. Asked: a time of day, not a number of hours.
   2. 9:40 a.m. + 3 hours = 12:40 p.m.
   3. A choice that says “3 hours” answers a different question.

   Answer 12:40 p.m.

   Next step: Units decide ↓
2. 2

   Let the units choose the operation

   Keyword lists (“each means divide”, “per means divide”) fail on this test, because the same word can call for two different operations. Units never lie. Write every number with its unit and pick the operation that leaves the unit the question wants.

   $/hour × hours = $

   Multiply

   Paid $14 per hour for 6 hours. Total pay?

   $/hour × hours = $ → 14 × 6 = $84

   Divide

   Paid $84 for 6 hours. Pay per hour?

   $ ÷ hours = $/hour → 84 ÷ 6 = $14

   Divide the other way

   Paid $84 at $14 per hour. How many hours?

   $ ÷ $/hour = hours → 84 ÷ 14 = 6

   Next step: Word → symbol ↓
3. 3

   Translate the words that really are math

   A few words do translate reliably. Learn these, and read everything else through the units.

   is, was, will be=“The total is 40” → T = 40

   of (with a fraction or %)דhalf of 30” → ½ × 30

   more than, older than+“5 more than x” → x + 5

   less than, younger than− (flipped)“5 less than x” → x − 5

   how many more, difference−bigger − smaller

   twice, triple× 2, × 3“twice a number” → 2n
   Shortcut

   “Less than” flips the order

   “7 less than a number” is n − 7, not 7 − n. The amount after “than” is where you start.

   Example

   “Four more than twice a number is 26.” Find the number.

   1. Twice a number: 2n. Four more than that: 2n + 4. “Is”: =.
   2. 2n + 4 = 26, so 2n = 22.
   3. n = 22 ÷ 2 = 11. Check: 2 × 11 + 4 = 26.

   Answer 11

   Next step: Name the unknown ↓
4. 4

   Name the unknown before you solve

   When a problem has two people or two amounts, write one line saying what your letter stands for. It takes five seconds, and it prevents the most common mistake on easy questions: answering for the wrong person.

   Example

   Lena has 3 times as many stamps as Omar. Together they have 48. How many does Lena have?

   1. Let O = Omar’s stamps. Lena has 3O.
   2. O + 3O = 48, so 4O = 48 and O = 12.
   3. The question asks for Lena: 3 × 12 = 36. Answering 12 answers for Omar.

   Answer 36 stamps

   Next step: Two-step problems ↓
5. 5

   Two-step problems: keep the in-between number

   Many questions need one result to find another. Write the first result down with its unit, then use it. The in-between number is almost always waiting among the choices, because it is exactly what a rushed student picks.

   Example

   Kim’s gym: $25 to join, $18 a month, 7 months.

   1. Monthly part first: 18 × 7 = 126 dollars.
   2. Add the fee: 126 + 25 = 151.
   3. $126 is the in-between number. It is not what was asked.

   Answer $151

   Next step: Check it ↓
6. 6

   Check the answer against the question

   Before you choose, read the question one more time and run three quick checks. They take a few seconds and catch most careless errors.

   The right thing?

   Asked: Lena’s stamps.

   Lena’s count, not Omar’s.

   The right unit?

   Asked: a time of day.

   A clock time, not a number of hours.

   A sensible size?

   Asked: a sale price.

   Below the original price, never above it.

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

5 questions at test level

Pick an answer, then read why. No timer, no account.

10 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.