4.3 Word Problems

Word Problems

Word problems combine everything from the previous two sections — order of operations, fractions, and percentages — into realistic scenarios. The key skill is translating words into an equation. This section walks through the six word-problem types that appear most often on the CSE-PPT Subprofessional exam: age, work/rate, distance-rate-time, ratio and proportion, mixture, and average problems.

General Strategy

1. Identify the unknown and assign it a variable (usually x).
2. Translate each sentence into a mathematical expression.
3. Write one equation that connects the expressions.
4. Solve the equation step by step.
5. Check that your answer makes sense in the context of the problem.

Age Problems

Age problems describe relationships between people's ages now and at a different point in time (in the past or future). The trick is that when time passes, everyone's age increases by the same number of years.

Worked Example: Age

Maria is 4 times as old as her son. In 5 years, she will be 3 times as old as her son. What is the son's current age?

• Setup: Let the son's current age = x, so Maria's current age = 4x. In 5 years: son = x + 5, Maria = 4x + 5.
• Equation: 4x + 5 = 3(x + 5)
• Solve: 4x + 5 = 3x + 15 → 4x − 3x = 15 − 5 → x = 10.

Answer: The son is currently 10 years old (and Maria is 40).

Work / Rate Problems

Work problems ask how long a combined effort takes when individual rates are known. Convert each worker's time into a rate (fraction of the job done per hour), add the rates together, then take the reciprocal of the combined rate to get the combined time.

Worked Example: Combined Work

Employee A can finish a filing task alone in 6 hours. Employee B can finish the same task alone in 3 hours. Working together, how long will it take them?

• Setup: Rate of A = 1/6 job per hour. Rate of B = 1/3 job per hour. Let t = time working together.
• Equation: 1/6 + 1/3 = 1/t
• Solve: 1/6 + 2/6 = 3/6 = 1/2, so 1/2 = 1/t, meaning t = 2.

Answer: Working together, they finish in 2 hours.

Distance-Rate-Time Problems

These problems use the formula Distance = Rate × Time (often written d = rt). You can rearrange it as Rate = Distance ÷ Time or Time = Distance ÷ Rate.

Worked Example: Distance-Rate-Time

A messenger travels from the city hall to a barangay office 45 kilometers away, driving at a steady speed of 30 kilometers per hour. How long does the trip take?

• Setup: d = 45, r = 30, solve for t.
• Equation: t = d ÷ r = 45 ÷ 30
• Solve: 45 ÷ 30 = 1.5

Answer: The trip takes 1.5 hours (1 hour and 30 minutes).

Ratio and Proportion Problems

A ratio compares two quantities (written a : b or a/b). A proportion states that two ratios are equal, and you solve it by cross-multiplying.

Worked Example: Ratio and Proportion

In an office, the ratio of male to female employees is 3:5. If there are 24 male employees, how many female employees are there?

• Setup: 3/5 = 24/x, where x is the number of female employees.
• Equation (cross-multiply): 3 × x = 5 × 24
• Solve: 3x = 120 → x = 40.

Answer: There are 40 female employees.

Mixture Problems

Mixture problems combine two quantities with different values (concentrations, prices, or grades) to form a blend, using the idea that the combined value of the ingredients before mixing must equal the value of the finished blend.

Worked Example: Mixture

A canteen wants to mix coffee worth ₱200 per kilogram with coffee worth ₱320 per kilogram to make 6 kilograms of a blend worth ₱260 per kilogram. How many kilograms of the ₱200 coffee are needed?

• Setup: Let x = kilograms of ₱200 coffee, so (6 − x) = kilograms of ₱320 coffee.
• Equation: 200x + 320(6 − x) = 260(6)
• Solve: 200x + 1920 − 320x = 1560 → −120x = 1560 − 1920 → −120x = −360 → x = 3.

Answer: 3 kilograms of the ₱200 coffee (and 3 kilograms of the ₱320 coffee) are needed.

Average (Mean) Problems

The average of a set of values is the sum of the values divided by how many values there are: Average = (Sum of values) ÷ (Number of values).

Worked Example: Average

A clerk processed the following number of documents over 5 days: 12, 15, 9, 18, and 16. What is the average number of documents processed per day?

• Setup: Average = (12 + 15 + 9 + 18 + 16) ÷ 5
• Solve: 12 + 15 + 9 + 18 + 16 = 70; 70 ÷ 5 = 14.

Answer: The average is 14 documents per day.

Exam Strategy

Always write the equation before solving — resist the urge to guess from the answer choices. If you finish early, plug your answer back into the original word problem to verify it makes logical sense (an age can't be negative, a percentage can't exceed what's reasonable for the context, and so on).