11.3 Word Problems and Applied Math

11.3 Word Problems and Applied Math

Quick Answer: USTET word problems reward a fixed routine: identify the unknown, translate English into an equation or proportion, solve carefully, then verify that the result answers the question asked and makes sense with the units. Algebra and arithmetic you already know become exam points only when translation is accurate.

Applied math sits at the intersection of General Mathematics and the number-sense skills from earlier chapters. Statistics questions may also arrive dressed as stories (“the mean of five scores is 80; four scores are known; find the fifth”). This section trains the multi-step habits that prevent leaving easy points on the table.

A Reliable Four-Step Routine

1. Read twice. First pass for the story; second pass to mark the asked quantity and every number with its unit (pesos, hours, students, percent).
2. Define a variable for the main unknown. If two unknowns appear, look for a relationship (one is twice the other; their sum is 50) so you can reduce to one variable or a simple system.
3. Write the math. Prefer an equation, proportion, or clear multi-line plan over mental juggling. Entrance-exam wrong options are built from common translation errors (using the wrong base for a percent, forgetting to subtract a discount, averaging averages incorrectly).
4. Solve and check. Plug the answer back into the story. Confirm you found what was asked — remaining amount versus original, time still needed versus total time, one person’s share versus the combined total.

Percent Applications in Stories

Percent problems dominate applied stems.

• Part from whole: $\text{part} = \text{percent} \times \text{whole}$.
• Whole from part: $\text{whole} = \text{part} \div \text{percent}$ (percent as a decimal).
• Percent change: $\dfrac{\text{new} - \text{old}}{\text{old}} \times 100%$.

Successive percents. A fee of PHP 600 increases by 10%, then the new amount decreases by 10%. Final amount is not 600. After +10%: $600 \times 1.10 = 660$. After −10%: $660 \times 0.90 = 594$. Successive equal percent up and down do not cancel.

Discount then tax (or tax then discount). Always apply operations in the order the problem states, and watch whether tax applies to the discounted price.

Ratio, Proportion, and Scaling

If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ (cross products). Unit rates help: kilometers per hour, pages per minute, pesos per kilo.

Worked example. A map scale is 1 cm : 5 km. A road measures 7.2 cm on the map. Actual length: $7.2 \times 5 = 36$ km.

Sharing in a ratio. Divide PHP 1,200 in the ratio 2 : 3. Total parts $= 5$. One part $= 240$. Shares: $480$ and $720$. Check: $480 + 720 = 1{,}200$.

Distance, Rate, and Time

$d = rt, \quad r = \frac{d}{t}, \quad t = \frac{d}{r}.$

Keep units consistent (hours with km/h, not minutes unless you convert).

Opposite directions. Speeds add for the rate at which distance between travelers grows. Same direction: subtract speeds for the catching-up rate.

Worked example. A bus leaves Manila at 60 km/h. A car leaves 1 hour later at 90 km/h on the same route. Let $t$ be hours the car travels until catching up. Bus has a 60 km head start and travels $t+1$ hours: $60(t+1) = 90t$ → $60t + 60 = 90t$ → $60 = 30t$ → $t = 2$ hours.

Work-Rate Problems

If A completes a job in $a$ hours, A’s rate is $\frac{1}{a}$ job per hour. Together:

$\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$

for time $t$ working together (assuming constant rates and one complete job).

Worked example. Liza can finish a review packet in 4 hours; Marco in 6 hours. Together: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$, so $t = \frac{12}{5} = 2.4$ hours.

Mixture and Average Chains

Mixture problems balance amounts of a quantity (salt, juice concentrate, priced tickets).

Ticket mix. Adult tickets cost PHP 120 and student tickets PHP 80. If 50 tickets bring in PHP 4,800, let $a$ be adult tickets. Then $a + s = 50$ and $120a + 80s = 4{,}800$. Substitute $s = 50 - a$: $120a + 80(50 - a) = 4{,}800$ → $120a + 4{,}000 - 80a = 4{,}800$ → $40a = 800$ → $a = 20$.

Missing score from a mean. Mean of 5 scores is 84. Four scores: 80, 88, 79, 90. Sum of five scores must be $84 \times 5 = 420$. Known sum $= 337$. Missing score $= 420 - 337 = 83$.

Consecutive Integers and Age Patterns

Consecutive integers: $n, n+1, n+2, \ldots$ Consecutive even/odd: $n, n+2, n+4, \ldots$

Age problems set ages now, then adjust by the same number of years for “in 5 years” or “10 years ago.” Write expressions for each person at the same time reference before equating.

Multi-Step Story Blueprint

Many USTET items look like this:

A school ordered 240 workbooks. After giving 15% to Grade 11 STEM and 25% of the remainder to ABM, how many workbooks are left?

Step 1: STEM gets $0.15 \times 240 = 36$. Remainder $= 204$. Step 2: ABM gets $0.25 \times 204 = 51$ (not 25% of 240). Step 3: Left $= 204 - 51 = 153$.

The planted wrong answer $240 - 0.15(240) - 0.25(240) = 144$ ignores “of the remainder.” Underline remainder language every time.

Estimation and Choice Elimination

With four options, estimate:

• 18% of 450 is a bit less than 20% of 450 (= 90), so about 81.
• If a combined work time must be less than each individual time, eliminate any option larger than both solo times.
• Distances cannot be negative; probabilities cannot exceed 1; counts of people should be whole numbers unless the problem allows otherwise.

Connecting Back to Statistics

Word problems and statistics meet when stems ask for a new mean after a score changes, the probability implied by a survey table, or which average to report after an outlier appears. Use Section 11.1 and 11.2 tools inside the same four-step routine: define the quantity, translate, compute, check.

Timing Tips for the Mathematics Subtest

Applied items can eat minutes. If translation is unclear after one careful reread, mark and move — then return with leftover time. When you return, rewrite the equation from scratch rather than staring at a half-finished expression. Neat scratch work is faster than dense overlapping arithmetic when you need to find an error.

Build pattern recognition: percent of remainder, catching up at relative speed, together-work rates, and missing value from a mean. Recognizing the pattern early is the highest-leverage applied-math skill you can bring to USTET.