Word Problems

Quick Tip: The Arithmetic Reasoning (AR) subtest is entirely word problems—16 questions in 39 minutes. That's about 2.5 minutes per question, so practice translating words to math quickly.

Translating Words to Math

Key Translation Words

Setting Up Equations

Pattern: "[Some quantity] is [some relationship to] [another quantity]"

Example: "Three more than twice a number is 17"

• "Twice a number" = $2x$
• "Three more than" = $2x + 3$
• "is 17" = $= 17$
• Equation: $2x + 3 = 17$

Rate, Time, and Distance Problems

The Fundamental Formula

$\text{Distance} = \text{Rate} \times \text{Time}$

Or rearranged:

• $\text{Rate} = \frac{\text{Distance}}{\text{Time}}$
• $\text{Time} = \frac{\text{Distance}}{\text{Rate}}$

Worked Example: Basic Rate Problem

Problem: A car travels at 55 mph. How far will it travel in 3 hours?

Solution: $\text{Distance} = \text{Rate} \times \text{Time} = 55 \times 3 = 165 \text{ miles}$

Worked Example: Meeting in the Middle

Problem: Two cars start 360 miles apart and drive toward each other. Car A travels at 50 mph and Car B at 40 mph. When will they meet?

Solution: Step 1: Combined rate = 50 + 40 = 90 mph (they're closing the gap together)

Step 2: Time = Distance / Rate = 360 / 90 = 4 hours

Worked Example: Catch-Up Problem

Problem: A train leaves at 2 PM traveling at 60 mph. Another train leaves from the same station at 3 PM traveling at 80 mph in the same direction. When will the second train catch up?

Solution: Step 1: Head start distance = 60 × 1 = 60 miles

Step 2: Closing speed = 80 − 60 = 20 mph

Step 3: Time to catch up = 60 / 20 = 3 hours (at 6 PM)

Work Problems

The Work Formula

If Person A completes a job in $a$ hours and Person B in $b$ hours:

$\text{Combined time} = \frac{ab}{a + b}$

Or use rates: Add the work rates (job per hour): $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$

Worked Example: Two Workers

Problem: John can paint a room in 6 hours. Mary can paint the same room in 4 hours. How long will it take them working together?

Solution: Method 1 (Formula): $t = \frac{6 \times 4}{6 + 4} = \frac{24}{10} = 2.4 \text{ hours} = 2 \text{ hours } 24 \text{ minutes}$

Method 2 (Rates):

• John's rate: $\frac{1}{6}$ room per hour
• Mary's rate: $\frac{1}{4}$ room per hour
• Combined: $\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ room per hour
• Time: $\frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4$ hours

Percentage Word Problems

Common Percentage Problem Types

Worked Example: Discount Problem

Problem: A jacket originally costs $80. It's on sale for 30% off. What is the sale price?

Solution: Step 1: Discount = 80 × 0.30 = $24

Step 2: Sale price = 80 − 24 = $56

Shortcut: 30% off means you pay 70%: $80 \times 0.70 = \$56$

Worked Example: Percent Change

Problem: A stock price increased from $40 to $52. What was the percent increase?

Solution: $\text{Percent increase} = \frac{52 - 40}{40} \times 100 = \frac{12}{40} \times 100 = 30\%$

Ratio and Proportion Problems

Setting Up Proportions

If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$ (cross multiply)

Worked Example: Ratio Problem

Problem: The ratio of boys to girls in a class is 3:5. If there are 24 boys, how many girls are there?

Solution: $\frac{3}{5} = \frac{24}{x}$

Cross multiply: $3x = 5 \times 24 = 120$

$x = \frac{120}{3} = 40$ girls