5.2 Problem Solving

5.2 Problem Solving on NMAT Quantitative

After fundamental operations, CEM NMAT Quantitative ( 30 items, recommended ~40 minutes ) stresses problem solving: short word problems that require organizing information, choosing a model, and computing accurately. These items measure quantitative reasoning — can you turn a story into mathematics without losing a unit, a base, or a step?

You are not expected to invent college-level modeling. You are expected to handle rates, work, mixture, averages, age, proportion, and simple linear algebra the way a careful high-school problem solver would — under time pressure and without a calculator.

Quick frame: Every problem-solving item is three jobs: (1) translate, (2) compute, (3) verify. Most errors happen in (1) or (3), not in the arithmetic itself.

The translate → equation pipeline

1. Underline the question — what quantity is asked? (time, amount, rate, age next year, concentration…)
2. List givens with units: numbers without units are unfinished notes.
3. Assign a variable only when needed; many items are pure proportions.
4. Write one equation or proportion that must be true.
5. Solve with fraction-friendly arithmetic.
6. Check: units match the question; magnitude is plausible; plug back if easy.

Word → math dictionary (high-yield)

Unit consistency (silent score-saver)

Before multiplying rates, convert to one system:

• Hours ↔ minutes (×60 or ÷60)
• mL ↔ L (×1000 or ÷1000)
• mg ↔ g
• "per day" vs "per hour" when totals span different windows

Mini example: 90 mL/h for 2.5 h → total volume ( 90 \times 2.5 = 225 ) mL. If options are in L, convert: ( 0.225 ) L. Computing ( 90 \times 2.5 ) then forgetting to convert is a classic trap when options exist in both units.

Rates and distance / amount problems

Core templates:

• ( \text{distance} = \text{speed} \times \text{time} )
• ( \text{amount} = \text{rate} \times \text{time} )
• Average speed for equal distances is not the arithmetic mean of speeds; use total distance ÷ total time.

Worked problem 1 — split rates (original)
A shuttle travels 60 km to a community clinic at 40 km/h and returns 60 km at 60 km/h. What is the average speed for the round trip?

• Time going: ( 60/40 = 1.5 ) h
• Time return: ( 60/60 = 1 ) h
• Total distance 120 km; total time 2.5 h
• Average speed: ( 120 / 2.5 = 48 ) km/h

Trap: ( (40+60)/2 = 50 ) looks "reasonable" but is wrong for equal distances at different speeds.

Worked problem 2 — medical infusion framing (original)
An IV is set to deliver 500 mL over 4 hours at a constant rate. After 1.5 hours, how much remains?

• Rate: ( 500/4 = 125 ) mL/h
• Delivered in 1.5 h: ( 125 \times 1.5 = 187.5 ) mL
• Remaining: ( 500 - 187.5 = 312.5 ) mL

Alternatively: fraction of time remaining is ( 2.5/4 = 5/8 ); ( (5/8)\times 500 = 312.5 ) mL. Same answer, fraction route often cleaner.

Work problems ("together" and "opposite")

If A completes a job in ( a ) hours, A's rate is ( 1/a ) job per hour. Working together: ( 1/a + 1/b = 1/t ).

Worked problem 3
Pump A fills a tank in 6 hours; pump B in 3 hours. Together, time to fill one tank?

• ( 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 ) → ( t = 2 ) hours.

If B empties while A fills, subtract rates: ( 1/a - 1/b ) when B is faster at emptying; interpret signs carefully from the story.

Mixture and concentration

Track pure amount = concentration × total volume (or mass).

Worked problem 4 (original)
How much 20% saline must be mixed with 30% saline to make 10 L of 24% saline?

Let ( x ) = liters of 20% solution; then ( 10 - x ) liters of 30%.

Pure salt:
( 0.20x + 0.30(10 - x) = 0.24 \times 10 )
( 0.20x + 3 - 0.30x = 2.4 )
( -0.10x = 2.4 - 3 = -0.6 )
( x = 6 ) L of 20%, and 4 L of 30%.

Check: pure salt = ( 0.2(6) + 0.3(4) = 1.2 + 1.2 = 2.4 ) kg (or consistent mass units) in 10 L → 24%. ✓

Alligation shortcut (when comfortable): differences from target 24%:
30 − 24 = 6 parts of the 20%; 24 − 20 = 4 parts of the 30%; ratio 6:4 = 3:2, same 6 L : 4 L in 10 L total.

Averages and weighted averages

Simple average: sum of values ÷ count.
Weighted: ( \dfrac{\sum w_i x_i}{\sum w_i} ).

Worked problem 5
Scores 70, 80, and 90 on quizzes of equal weight average 80. If a fourth equal-weight quiz score of 60 is added, new average:
( (70+80+90+60)/4 = 300/4 = 75 ).

Worked problem 6 — weighted (original)
A station records 20 readings with mean 98 and 30 readings with mean 102. Combined mean:
( \dfrac{20\cdot 98 + 30\cdot 102}{50} = \dfrac{1960 + 3060}{50} = \dfrac{5020}{50} = 100.4 ).

Trap: Averaging 98 and 102 to get 100 ignores unequal sample sizes.

Age problems

Ages move in lockstep: if today A is ( x ) and B is ( y ), in 5 years they are ( x+5 ) and ( y+5 ). Ratios change even though the difference stays constant.

Worked problem 7
Today Mira is 3 times as old as Jun. In 8 years, she will be twice as old as Jun. Find Jun's age today.

Let Jun = ( j ); Mira = ( 3j ).
( 3j + 8 = 2(j + 8) )
( 3j + 8 = 2j + 16 )
( j = 8 ). Mira is 24 today.
Check: in 8 years, 32 and 16 → Mira twice Jun. ✓

Proportion and scaling (including maps and doses)

( \dfrac{a}{b} = \dfrac{c}{d} ) ↔ ( ad = bc ) (cross multiply).

Worked problem 8 — dose proportion (original)
If 5 mL of syrup contain 125 mg of drug, how many mg are in 8 mL of the same syrup?

( \dfrac{5}{125} = \dfrac{8}{x} ) or better ( \dfrac{125}{5} = \dfrac{x}{8} )
( x = 125 \times 8 / 5 = 25 \times 8 = 200 ) mg.

Worked problem 9 — inverse proportion
If 4 identical machines complete a batch in 9 hours (constant total work), 6 machines need:
( 4 \times 9 = 6 \times t ) → ( t = 6 ) hours (more machines, less time).

Simple algebra that shows up constantly

• Linear: ( ax + b = c )
• Two unknowns with two relations (substitution or elimination)
• "Consecutive integers," "even integers" (( n, n+2 ))
• Translating "five less than twice a number" → ( 2n - 5 )

Worked problem 10
Twice a number decreased by 7 is 31. Find the number.
( 2n - 7 = 31 ) → ( 2n = 38 ) → ( n = 19 ).

Multi-step NMAT-style synthesis (fully worked)

Worked problem 11 (original multi-step)
A barangay clinic receives a shipment of 240 rapid-test kits. On Monday it uses 15% of the shipment. On Tuesday it uses ( \dfrac{1}{4} ) of what remained after Monday. How many kits are left after Tuesday?

• Monday use: ( 0.15 \times 240 = 36 )
• Remaining after Monday: ( 240 - 36 = 204 )
• Tuesday use: ( \dfrac{1}{4} \times 204 = 51 )
• Left: ( 204 - 51 = 153 )

Trap chain: 15% + 25% of original 240 = 40% used → 144 left — wrong because Tuesday's quarter applies to the remainder, not the original.

Worked problem 12 (original multi-step with units)
A nurse walks 2.4 km to a remote site at 4 km/h, spends 45 minutes on patient education, and returns at 6 km/h. Total time away from base?

• Out: ( 2.4/4 = 0.6 ) h = 36 min
• Education: 45 min
• Return: ( 2.4/6 = 0.4 ) h = 24 min
• Total: ( 36 + 45 + 24 = 105 ) min = 1 h 45 min

Convert everything to minutes or hours before adding — mixed units cause false options like 105 hours or 1.05 h mistaken for 1 h 5 min.

Estimation to eliminate options

Under time pressure, estimate before fine arithmetic:

1. Round to 1–2 significant figures.
2. Compute a rough value.
3. Kill options far from that magnitude or with impossible units.
4. Only then compute exactly among remaining choices.

Example: 19.6% of 505 ≈ 20% of 500 = 100. Options 9.9, 99, 198, 505 — only ~99 is plausible (exact: ( 0.196 \times 505 \approx 99 )).

Example: Tank work rates 1/7 + 1/8 ≈ 0.14 + 0.125 = 0.265 jobs/h → time ≈ 1/0.265 ≈ a bit under 4 h. Exact: ( 1/7 + 1/8 = 15/56 ), ( t = 56/15 \approx 3.73 ) h.

When algebra beats trial-and-error

Trial-and-error (plugging options) can work on multiple choice, but it is slower when:

• The stem has two linked conditions (ages, mixtures, consecutive constraints)
• Options are messy fractions or decimals
• You must produce an intermediate value not listed
• More than one option "almost" fits unless you track both conditions

Prefer algebra or proportion when the relationship is linear and you can write one equation in under 20 seconds.
Prefer smart trial when options are spaced widely and the equation would be heavier than testing two candidates with a quick plug-in.

Hybrid (often best): Estimate → drop two options → plug the remaining two into the condition.

Error log tags for problem solving

When you miss a practice item, tag it:

• Translation — wrong equation from English
• Units — mixed hours/minutes or volume units
• Base — percent of wrong quantity; remainder vs original
• Average — arithmetic mean of rates instead of total/total
• Arithmetic — pure computation slip after correct setup
• Skipped verify — answer failed plug-back or magnitude check

Reviewing tags weekly beats re-solving the same numbers without diagnosis.

Study protocol (offline)

1. Ten translations only: read stems, write equations, do not solve — trains the bottleneck.
2. Five fully timed multi-step items (~2 minutes each max), then untimed error analysis.
3. Mixture + work pair each session — these recycle fraction fluency from Section 5.1.
4. One "trap rewrite": take a correct solution and invent the wrong path (e.g., average of speeds) so you recognize that distractor later.

Link forward

Problem solving shares the Quantitative block with data interpretation (tables, charts, multi-step reads). The same habits — define the asked quantity, watch units, estimate first — transfer directly. Speed tactics for the whole 30-item / ~40-minute Quantitative subtest build on both direct operations and the word-problem pipeline in this section.

Section checkpoint

You are ready to leave this section when you can: (1) write a proportion or linear equation from a stem in one pass, (2) finish mixture, work, and weighted-average standards without searching for a formula sheet, (3) catch remainder-vs-original percent traps, and (4) choose algebra vs option-plug deliberately based on time and structure — not habit alone.