6.4 Word Problems and Quantitative Reasoning

Why This Matters for the PUPCET

Word problems are where the PUPCET Mathematics portion (about 21% of the test) separates prepared students from the rest. The algebra itself is usually easy — the difficulty is converting a Filipino-context story about jeepney fares, sari-sari store change, or a bus trip from Manila to Batangas into clean equations. The good news: nearly every word problem on a Philippine entrance exam belongs to one of about five families, and each family has a standard setup. Learn the setups and these items become the fastest points on the test.

The Four-Step Solving Framework

Use the same framework for every word problem, no exceptions:

1. Read and name the unknown. Read the problem twice. On the second pass, write "let x = ..." for exactly what the question asks. If there are two unknowns, define one in terms of the other using the problem's own words.
2. Translate into equations. Convert each sentence into algebra. Every fact in the problem should appear somewhere in your equations — unused facts usually mean a missed relationship.
3. Solve. Use the equation skills from Sections 6.1 and 6.2.
4. Answer the ACTUAL question and check. Exams routinely ask for something one step beyond your x (the older sibling's age, the number of ten-peso coins, the slower speed). Re-read the final sentence before marking an option, then substitute your answer back into the original story to confirm it makes sense.

Type 1: Age Problems

Golden rule: define variables for the PRESENT ages, then shift with + or − for future or past.

Worked Example 1

Mang Tomas is three times as old as his son Paolo. In 12 years, Mang Tomas will be twice as old as Paolo. How old is Paolo now?

Step 1. Let p = Paolo's present age. Then Tomas's present age = 3p. Step 2. In 12 years: Tomas will be 3p + 12, Paolo will be p + 12. Translate the second sentence: 3p + 12 = 2(p + 12). Step 3. Solve: 3p + 12 = 2p + 24, so p = 12. Step 4. Paolo is 12 years old now (Tomas is 36; in 12 years, 48 = 2 × 24 ✓).

Type 2: Number Problems

These are pure translation drills using the phrase table from Section 6.1.

Worked Example 2

The sum of three consecutive integers is 48. Find the largest.

Step 1. Let the integers be n, n + 1, and n + 2. Step 2. n + (n + 1) + (n + 2) = 48, so 3n + 3 = 48. Step 3. 3n = 45, so n = 15. Step 4. The question asks for the LARGEST: n + 2 = 17. Marking 15 would be answering the wrong question — the decoy option is always the unadjusted x.

For consecutive even or odd integers, use n, n + 2, n + 4.

Type 3: Money and Coin Problems (Pesos)

Money problems need two equations: one counting the pieces, one counting the value.

Worked Example 3

Aling Nena's cash box contains 25 bills, some worth ₱20 and some worth ₱50, totaling ₱890. How many ₱50 bills does she have?

Step 1. Let f = number of ₱50 bills. Then ₱20 bills = 25 − f. Step 2. Value equation: 50f + 20(25 − f) = 890. Step 3. 50f + 500 − 20f = 890, so 30f = 390 and f = 13. Step 4. She has 13 fifty-peso bills (and 12 twenty-peso bills: 13 × 50 + 12 × 20 = 650 + 240 = 890 ✓).

The same skeleton handles tickets (adult vs. student), coins (₱5 vs. ₱10), and sales of two products.

Type 4: Mixture Problems

Track the amount of the pure substance before and after mixing: (concentration × volume) must balance.

Worked Example 4

How many liters of a 20% salt solution must be mixed with 30 liters of a 50% salt solution to get a 35% solution?

Step 1. Let x = liters of the 20% solution. Step 2. Salt in = salt out: 0.20x + 0.50(30) = 0.35(x + 30). Step 3. 0.20x + 15 = 0.35x + 10.5, so 4.5 = 0.15x and x = 30. Step 4. 30 liters of the 20% solution. Sanity check: equal parts of 20% and 50% should average to exactly 35% — and (20 + 50)/2 = 35. ✓

The sanity check reveals a powerful shortcut: when equal volumes mix, the resulting concentration is the simple average of the two. If the target is exactly midway between the two concentrations, the answer is always equal amounts.

Type 5: Distance–Rate–Time

Everything rests on one formula and its rearrangements:

Movement patterns: two objects moving TOWARD each other close the gap at the sum of their speeds; moving in the SAME direction, the gap changes at the difference of their speeds.

Worked Example 5

A bus leaves Manila for Batangas at 60 kph. One hour later, a car leaves Manila on the same route at 90 kph. How long after the car departs does it catch the bus?

Step 1. Let t = the car's travel time in hours. The bus has then been traveling t + 1 hours. Step 2. Catching up means equal distances: 60(t + 1) = 90t. Step 3. 60t + 60 = 90t, so 60 = 30t and t = 2. Step 4. The car catches the bus 2 hours after it departs (both have gone 180 km ✓).

Estimation and Answer Elimination

The PUPCET is multiple-choice, which means the correct answer is printed in front of you. Use that.

• Estimate first. If a ₱1,250 jacket is discounted 15%, you know the savings are a bit over ₱120 (10% = ₱125). Any option near ₱1,050 or above ₱1,200 can be crossed out before you compute 1250 − 187.50 = ₱1,062.50.
• Eliminate impossible signs and sizes. Ages and counts of coins must be positive whole numbers; a probability cannot exceed 1; a hypotenuse must exceed each leg. Often two of four options die instantly.
• Check units. If the question asks for hours and two options are clearly in minutes, the unit mismatch itself may identify the answer.

Working Backwards from the Options

When the setup feels tangled, plug the choices into the story. Start with a middle value so a wrong guess still tells you whether to go higher or lower.

Worked Example 6 (Solved Backwards)

"Three times a number, decreased by 8, equals 34. What is the number?" Options: 10, 12, 14, 16.

Test 12: 3(12) − 8 = 28, too small — so try larger. Test 14: 3(14) − 8 = 34. ✓ The answer is 14, found without writing an equation. Working backwards shines on age and money problems where the equations have parentheses, and it doubles as your verification method even when you solve directly.

Common Traps and Final Tips

• Answering x when the question asks for 2x, the larger number, or the other person's age — re-read the last sentence every time.
• In age problems, adding years to only one person's age; time passes for everyone.
• In coin problems, writing one equation for value but forgetting the count equation (or vice versa).
• In distance problems, mixing units like kph with minutes — convert minutes to hours first (30 min = 0.5 h).
• Spending four minutes on one hard item. The PUPCET gives roughly a minute per question; mark, skip, and return. An easy point banked beats a hard point chased.

Section Takeaway

Name the unknown, translate every sentence, solve, then answer the question actually asked. Memorize the five standard setups — present-age variables, consecutive-integer chains, count-plus-value money equations, salt-balance mixtures, and d = rt with sum-or-difference speeds. And remember that on a multiple-choice exam, the options are a tool: estimate, eliminate the impossible, and work backwards when the forward path is slow.